See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/331021490 BOOK. « EPRB Paradox Resolution. A New Quantum Mechanical Formalism Based on the Probability Representation of Quantum States. Bell inequalities revisited» Preprint * February 2019 DOI: 10.13140/RG.2.2.14240.43521 CITATIONS 0 READS 61 3 authors: Some of the authors of this publication are also working on these related projects: Cognitive Fractal Radar and Fractal Radio Physics. View project Quantum field theory in curved distributional spacetime View project Jaykov Foukzon Technion Israel Institute of Technology 244 PUBLICATIONS 257 CITATIONS SEE PROFILE Elena R. Men'kova All-Russian Research Institute for Optical and Physical Measurements 110 PUBLICATIONS 86 CITATIONS SEE PROFILE A. A. Potapov Russian Academy of Sciences 412 PUBLICATIONS 498 CITATIONS SEE PROFILE All content following this page was uploaded by Jaykov Foukzon on 11 February 2019. The user has requested enhancement of the downloaded file. 1 TABLE OF CONTENTS Preface....................................................................... 7 Introduction................................................................. 8 I.1. Bell's type inequalities violations.................................. 8 I.2. A new quantum mechanical formalism based on the probability representation of quantum states........................ 19 I.3. EPR Paradox resolution by using Postulate of EPRNonlocality.................................................................... 31 I.4. EPR-B Paradox resolution and Postulate of EPR-BNonlocality.................................................................... 36 Chapter I. The postulate of EPR-B nonlocality................................................................... 43 I.1. The EPR paradox..................................................... 43 I.2. Einstein's 1927 gedanken experiment ........................... 43 I.3. The continuous variable EPR paradox EPR-Reid's criteria......................................................................... 46 I.3.1. The weak postulate of nonlocality for continuous variables...................................................................... 49 I.3.2. The strong postulate of nonlocality for continuous variables....................................................................... 50 I.4. The EPR-Bohm paradox. Reid's criteria for EPR-Bohm paradox ....................................................................... 58 I.4.1. The weak postulate of nonlocality. Heisenberg spin uncertainty relations........................................................ 60 I.4.2. The strong postulate of nonlocality ............................. 61 2 Chapter II. A new quantum mechanical formalism based on the probability representation of quantum states and observables................................................................. 64 II.1. Generalized Postulates for Continuous Valued Observables ..................................................................... 64 II.2. The nonclassical collapse models with spontaneous localizations based on generalized measurement postulates... 74 II.2.1. Quantum Mechanics with Nonclassical Spontaneous Localizations................................................................. 74 II.2.2. The generalization of nonclassical collapse models.......................................................................... 77 Chapter III. Einstein's 1927 gedanken experiment revisited...................................................................... 81 III.1. Single-photon space-like antibunching......................... 81 III.2. The measure algebra of physical events in Minkowski space-time............................................................................ III.2.1. Beamsplitter transformation.................................... III.2.2. Splitting a two-photon state..................................... 86 91 94 III.3. Einstein's 1927 gedanken experiment explained............. 99 Chapter IV. The EPR paradox resolution........................ 110 IV.1. The relaxed locality principle..................................... 110 IV.2. Generalized EPR argument and postulate of nonlocality.................................................................... 116 IV.2.1. The EPR-Reid criterion.......................................... 117 IV.2.2. The Postulate of Nonlocality and signature of the EPR paradox........................................................................ 126 IV.2.3. The EPR-nonlocality criteria.................................... 131 IV.3. Nonlocal Schrödinger equation implies the Postulate of Nonlocality................................................................... 132 3 IV.4. Position-momentum entangled photon pairs in non-linear wave-guide................................................................... 141 IV.5. Position-momentum entangled photon pairs and the experimental verification of the postulate of nonlocality.......... 147 IV.6. The EPR Paradox Resolution by using quantum mechanical formalism based on the probability representation of quantum states.......................................................... 154 IV.6.1. Preliminaries....................................................... 154 IV.6.2. The EPR Paradox Resolution................................. 157 Chapter V. EPR-B Paradox Resolution............................ 163 V.1. EPR-B experiment................................................... 163 V.2. EPR-B paradox resolution.......................................... 170 V.2.1. First step EPR-B: Spin measurement of A.................. 172 V.2.2. Second step EPR-B: Spin measurement of B.............. 174 V.2.3. Resolution of the EPR-B experiment in de BroglieBohm interpretation by the "relaxed locality principle"............ 175 V.2.3.1. Step 1: spin measurement of A in de Broglie-Bohm interpretation................................................................. 176 V.2.3.2. Step 2: Spin measurement of B  in de BroglieBohm interpretation........................................................ 177 V.2.3.2.1. The prediction of the result of the spin measurement of B  under assumption of canonical postulate of locality..................................................................... 177 V.2.3.2.2. The prediction of the result of the spin measurement of B  under assumption of postulate of nonlocality.................................................................... 178 V.2.4. Physical explanation of non-local influences using the relaxed principle of locality............................................... 179 V.3. EPR-B paradox resolution by using quantum mechanical formalism based on the probability representation of quantum states.......................................................................... 180 V.3.1. Preliminaries........................................................ 180 V.3.2. The EPR-B paradox resolution................................. 193 4 Chapter VI. Schrödinger's cat measured spin. Schrödinger's cat paradox resolution............................. 208 VI.1. Stern-Gerlach experiment revisited............................. 208 VI.2. Schrödinger's cat which measures spin. Schrödinger's cat paradox resolution..................................................... 220 Chapter VII. The Bell inequalities revisited.......................... 223 VII.1. Bell theorem without the hypothesis of locality and without the introduction of hidden variables......................... 223 VII.1.1. Clauser-Horne-Shimony-Holt (CHSH) inequality......... 223 VII.1.2. Clauser Horne inequality....................................... 228 VII.1.3. Violation of Bell's inequality under strict Einstein locality conditions........................................................... 232 VII.1.4. CHSH theorem without the hypothesis of locality........ 240 VII.1.5. CHSH theorem without the introduction of hidden variables...................................................................... 244 VII.2. Physical nature of the violation of the Bell inequalities................................................................... 246 VII.2.1. Physical interpretation of the Bell test experiment under EPR-B nonlocality postulate.................................... 246 VII.2.2. EPR-B nonlocality is the physical nature of the violation of the Bell inequalities.......................................... 266 VII.2.3. Canonical Local Realistic Hidden Variable Theory......................................................................... 269 VII.2.4. Local Hidden Variable Theory revisited. Generalized Local Hidden Variable Theory Validity of CHSH-inequality for correlations taking into account EPR-B nonlocality.................................................................... 272 VII.3. Bell inequalities revisited.......................................... 276 VII.3.1. Clauser-Horne-Shimony-Holt (CHSH) inequality revisited. Validity of revised CHSH inequality....................... 276 VII.3.2. Clauser-Horne inequality revisited. Validity of revised Clause-Horne inequality.................................................. 282 5 VII.3.3. Revised CHSH inequality without the hypothesis of locality......................................................................... 287 VII.3.4. Revised CHSH inequality without the introduction of hidden variables............................................................. 294 VII.4. Leggett inequality revisited....................................... 298 VI.4.1. Classical Leggett inequality.................................... 298 VII.4.2. Derivation of the canonical Leggett inequality............ 310 VII.4.3. Leggett inequality revisited. Validity of revised Leggett inequality..................................................................... 317 Appendices................................................................. 329 Appendix A. Heisenberg's noise-disturbance uncertainty principle................................................................................ 329 Appendix B. Conditional probability density functions...... 335 Appendix C. Fourier transform and Heisenberg uncertainty principle................................................... 336 References................................................................... 337 6 7 PREFACE This book is devoted to the presentation of the new quantum mechanical formalism based on the probability representation of quantum states. In the 20s and 30s it became evident that some properties in quantum mechanics can be assigned only to the quantum mechanical system, but not necessarily to its constituents. This led Einstein, Podolsky and Rosen (EPR) to their remarkable 1935 paper where they concluded that quantum mechanics is not a complete theory of nature (EPR paradox). In order to avoid the contradiction which arises from instantaneous action at a distance mentioned above we introduce an extension of the canonical relativity by using measure algebra of physical events in Minkowski space-time. The canonical QM formalism is extended by additional new postulate of EPRB nonlocality for continuous and discrete observables, chapter I. The postulate of EPRB nonlocality is supported by new quantum mechanical formalism based on the probability representation of quantum states. Chapter II is devoted to the new quantum mechanical formalism based on the probability representation of quantum states. Chapter III is devoted to the Einstein's 1927 gedanken experiment resolution. Chapter IV is devoted to the EPR paradox resolution. Chapter V is devoted to the EPR-B paradox resolution. Chapter VI is devoted to the Schrödinger's cat (measured spin) paradox resolution. Chapter VII is devoted to the Bell inequalities revisited. Remind that the canonical arguments which were presented by many authors, namely, that violations of Bell type inequalities signal us that the classical Kolmogorovian model of probability is inapplicable to quantum phenomena. We claimed that the canonical assumption, under which Bell type inequalities were derived, is not supported by real physical nature of the EPRB experiments. The fundamental physical nature violations of the canonical Bell type inequalities explained by Postulate of EPR-Nonlocality and Heisenberg noise-disturbance uncertainty relations. 8 INTRODUCTION I.1. Bell's type inequalities violations The canonical argument which were presented by many authors, namely, that violations of Bell type inequalities [1], [2] signal us that the classical model of probability [3] (Kolmogorov, 1933) is inapplicable to quantum phenomena. It well known that any attempt to explain to these violations by some additional value e.g., to philosophize about (non)locality and (un)reality, is not helpful [4], [5], [6]. Remind that one of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., of the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of local realism. Let   ,,P be a Kolmogorov probability space. For two random variables A, B :    we set EAB    ABdP 1.1 Theorem I.1.1. (CHSH-inequality). Let Ai,Bj, i; j  1,2 , be random variables with values in 1,1 . Then the corresponding combination of correlation satisfies the CHSH-inequality: S  EA1B1   EA1B2   EA2B1   EA2B2   2. 1.2 Theorem I.1.2. (Bell's no-go theorem [2]). Bell's no-go theorem says that Bell type inequalities, e.g., the CHSH-inequality (1.2), which are derived in Kolmogorovian model of probability are violated for correlations calculated in the quantum probability model. 9 Remark I.1.1. By using Bell's no-go theorem many authors were concluded that the Kolmogorovian model of probability has to be rejected in general as inapplicable to these correlations in any cases, see for example [4]-[8]. In papers [7]-[9] authors claimed that the fair sampling assumption is not supported by real EPRB experiments. In papers [8]-[9] complete, probability spaces consistent with EPR-Bohm-Bell experimental data by taking into account random choice of settings, were obtained. Fig. I.1.1. Timing-experiment with optical switches(C 1 and C 2 ). Adopted from [10] The switch C 1 followed by the two polarizers in orientations a and a' is equivalent to a single polarizer switched between the orientations a and a . A switching occurs approximatively each 10 ns. A similar setup, independently driven, is implemented on the second side. In experiment, the distance L between the switches was large enough (13 m) that the time of travel of a signal between the switches at the velocity of light (43 ns) was significantly larger than the delay between two switchings (about 10 ns) and the delay between the emission between the two photons (5 ns average). Let us consider now an experiment taking into account random choice of settings [9]. (a) There is a source of entangled photons. 10 (b) There are 4 PBSs and corresponding pairs of detectors for each PBS, totally 8 detectors. PBSs are labeled as i  1,2 (at the left-hand side, LHS) and j  1,2 (at the right-hand side, RHS). (c) Directly after source there are 2 distribution devices, one at LHS and one at RHS. At each instance of time, t  0,,2. . . each device opens the port to only one (of two) optical fibers going to the corresponding two PBSs. For simplicity, we suppose that each pair of ports i, j,1,1,1,2,2,1,2,2, can be opened with equal probabilities Pi, j  1/4. We introduce the observables measured in this experiment. They are modifications of the polarization observables ai , i  1,2, and bj , j  1,2, We define the "LHS-observables": (1) Ai  1, i  1,2 if the corresponding (up or down) detector is coupled to i-th PBS (at LHS) fires and the i-th channel is open, (2) A i  0 if the i-th channel (at LHS) is blocked, (3) in the same way we define the "RHS-observables": Bj  0,1, corresponding to PBSs j  1,2. Remark I.1.2. [9]. Thus unification of 4 incompatible experiments of the CHSH-test into a single experiment modifies the range of values of polarization observables for each of 4 experiments; the new value, zero, is added to reflect the random choice of experimental settings. We emphasize that this value has no relation to the efficiency of detectors. In this model we assume that detectors have 100% efficiency. The observables take the value zero when the optical bers going to the corresponding PBSs are blocked. Let ,,P be an arbitrary probability space and let 0  ,0  ,P0   0 and  let be arbitrary random variable  :   . Then the conditional expectation of the random variable  conditioned to the event 0 is defined as follows: E|0    dP0, 1.3 11 where the conditional probability P0 is defined by Bayes formula P0X  PX|0   PX 0  P0  1.4 Let us now consider the sample space of points  :   1 , 0,1  ,0,1 , 0,0,2  ,0,2 ,1  ,0,0,2 ,0,2  , 1.5 where ,  1,1. We define the following probability measure on  : P1 , 0,1  , 0  1 4 p111 ,1  ,P1 , 0,0,2    1 4 p121 ,2   P0,2 ,1  , 0  1 4 p212 ,1  ,P0,2 , 0,2    1 4 p222 ,2   1.6 where p ij is any collection of probabilities, i.e., pij  0,, pij,  1,,  1,1. 1.7 We define random variables Ai,Bj : A11 , 0,1  , 0  A11 , 0,0,2    1 ,A20,2 ,1  , 0  A20,2 , 0,2    2 , B11 , 0,1  , 0  B11 , 0,0,2    1  ,B20,2 ,1  , 0  B20,2 , 0,2    2  1.8 and we put these variables equal to zero in other points. We define the random variables which are responsible for selections of pairs of ports to PBSs. For the device at LHS: L1 ,0,0,2    L1 ,0,1  ,0  1,L0,2 ,0,2    L0,2 ,1  ,0  2. 1.9 For the device at RHS: 12 R1 ,0,1  ,0  R0,2 ,1  ,0  1,R0,2 ,0,2    R1 ,0,0,2    2. 1.10 We choose now 0  ij  |L  i,R  j. We set ECAiBj  EAiBj|L  i,R  j    AiBjdPij, 1.11 and SC  EA1B1 |L  1,R  1  EA1B2 |L  1,R  2  EA2B1 |L  2,R  1  EA2B2 |L  2,R  2. 1.12 Theorem I.1.3. [9]. (CHSH-inequality for conditional correlations.) Let Ai,Bj, i, j  1,2 , be random variables defined by Eq. (1.8). Then the corresponding combination of conditional correlation SC satisfies the inequality: SC  4. 1.13 However in papers [9], [10], [11], Bell's type inequality were derived in its traditional form, without resorting to the hypothesis of locality and hidden-variable theory the only assumption being that the probability distributions are nonnegative. The starting point of these papers that is a recognition of the existence of a positive-definite probability distribution function. Let A,A,B,B be random variables with values in the set 1,1, i.e., A  1,A  1,B  1,B  1. 1.14 Assume that there exists joint probability distribution function PA,A,B,B of A,A,B,B defining probabilities for each possible set of 13 outcomes such that:  A,A ,B,B  PA,A ,B,B   1,PA,A ,B,B   0, 1.15 and PA,A,B,B   PA,A,B,B  PA,B,B. 1.16 The main point of papers [9], [10], [11] also be a recognition of the existence of a positive-definite probability distribution function without any reference to the full classical Kolmogorovian model of probability [3]. We abbreviate now for short [11]: P1  PA,A ,B,B      PA  1,A   1,B  1,B  1, P2  PA,A ,B,B      PA  1,A   1,B  1,B  1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P16  PA,A ,B,B      PA  1,A   1,B  1,B  1. 1.17 For the quantities AB, A B, AB  and A B  using Eq. (1.17) one obtains the representatives AB  PAB   PAB   PAB   PAB , etc., 1.18 where PAB   PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B    , etc. 1.19 Theorem I.1.4. [11]. Assume that there exists joint probability distribution function PA,A ,B,B of A,A,B,B such that Eq. (1.15) Eq .(1.16) is satisfied. Then Bell inequality of the form 14 |AB  AB  AB   AB |  2. 1.20 holds. Remark I.1.3. We claim that the even general assumption given by Eqs.(1.15) - (1.16) is not supported by fundamental physical nature of the EPRB experiments. This fundamental physical nature formally explained by (i) Postulate of EPR-Nonlocality (see subsection I.4) and by (ii) Heisenberg noise-disturbance uncertainty relation, (see Appendix A). Remind that in a typical Bell experiment, (see Fig. I.1.2), two particles A and B which may have previously interacted for instance they may have been produced by a common source are now spatially separated and are each measured by one of two distant observers, Alice and Bob. Fig. I.1.2. Scheme of a "two-channel" Bell experiment Alice may choose one out of several possible measurements to perform on her system and we let x denote her measurement choice. For instance, x may refer to the position of a knob on her measurement apparatus. Similarly, we let y denote Bob's measurement choice. Once the measurements are performed, they yield random outcomes a and b on the two systems. The 15 actual values assigned to the measurement choices x,y and outcomes a,b are purely conventional; they are mere macroscopic labels distinguishing the different possibilities. These outcomes a and b are thus in general governed by a Kolmogorovian probability distribution pab|xy, which can of course depend on the particular experiment being performed. The assumption of locality implies that we should be able to identify a set of past factors, described by some variables λ, having a joint causal influence on both outcomes, and which fully account for the dependence between a and b. Once all such factors have been taken into account, the residual indeterminacies about the outcomes must now be decoupled, that is, the Kolmogorovian probabilities for a and b should factorize: pab|xy,  pa|x,pb|y,. 1.21 The different values of across the runs should thus be characterized by a probability distribution qλ. Combined with the above factorability condition, we can thus write pab|xy    dqpa|x,pb|y,, 1.22 where we also implicitly assumed that the measurements x and y can be freely chosen in a way that is independent of , i.e., that qλ|x,y  qλ . Let us consider for simplicity an experiment where there are only two measurement choices per observer x,y  0,1 and where the possible outcomes take also two values labelled a,b  1,1. Let axby  be the expectation value of the product ab for given measurement choices x,y : λ λ 16 axby   a,b abpab|xy. 1.23 Consider the following expression S  a0b0   a0b1   a1b0   a1b1 , 1.24 which is a function of the probabilities pab|xy. If these probabilities satisfy the locality decomposition (1.21), we necessarily have that S  a0b0   a0b1   a1b0   a1b1   2, 1.25 which is known as the Clauser-Horne-Shimony-Holt (CHSH) inequality [12]. Remark I.1.4. Note that particles A and B in real EPRB experiments cannot collapse simultaneously. Taking into account postulate of EPR-Nonlocality (see subsection I.4) and Heisenberg noisedisturbance uncertainty relation we obtain that random outcomes a and b mentioned above in real EPRB experiments they is not simply random variables a and b but time dependent random functions at1 and bt2. Remark I.1.5. These time dependent outcomes at1 and bt2 are thus in general governed by a time dependent Kolmogorovian joint probability distribution pa, t1 ;b, t2 |x t1yt2, which can of course depend on the particular experiment being performed. By repeating the experiment a sufficient number of times and collecting the observed data, one can get a fair estimate of such time dependent Kolmogorovian joint probabilities distribution. The assumption of locality implies that we should be able to identify a set of past factors, described by some variables λ, having a joint causal influence on both outcomes at1 and bt2, and which fully account for the dependence between a t1 and bt2 . Once all such factors have been taken into account, the residual indeterminacies about the outcomes must now be decoupled, that is, the time 17 dependent Kolmogorovian joint probabilities for a and b should factorize: pa, t1 ;b, t2 |xy,  pa, t1 |x,pb, t2 |y,. 1.26 The variable will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because may involve physical quantities that are not fully controllable. The different values of across the runs should thus be characterized by a probability distribution qλ, t1 , t2. Combined with the above factorability condition, we can thus write instead (1.22) pa, t1 ;b, t2 |xy    dq, t1 , t2 pa, t1 |x,pb, t2 |y,, 1.27 where we also implicitly assumed that the measurements x t1 and yt2 can be freely chosen in a way that is independent of , i.e., that qλ, t1 , t2 |x t1 ,yt2  qλ, t1 , t2 . Let us consider for simplicity an experiment where there are only two measurement choices per observer x t1 ,yt2  0,1 and where the possible outcomes take also two values labelled at1 ,bt2  1,1. Let at1x t1 bt2y t2 be the expectation value of the product at1bt2 for given measurement choices x t1 ,yt2 : at1x t1 bt2y t2  a,b abpab, t1 , t2 |x t1yt2. 1.28 We assume now that t1 , t2   2 ,    ,0    1, pab, t1 , t2 |x t1 yt2  pab, t1  t2 |x t1 yt2  |t1  t2 |    |t1  t2 |  0, q, t1 , t2   q, t1  t2   |t1  t2 |    |t1  t2 |  0 . 1.29 λ λ λ λ 18 Thus at1x t1 bt2y t2  a,b abpab, t1  t2 |x t1yt2. 1.30 We denote at1x t1 bt2y t2  axby  1.31 iff |t1  t2 |  0, at1x t1 bt2y t2  axby  1.32 iff |t1  t2 |   and t1  t2 , at1x t1 bt2y t2  axby  1.33 iff |t1  t2 |   and t1  t2 , axby   axby   axby   axby . 1.34 Consider now the following expression S  a0b0   a0b1   a1b0   a1b1 , 1.35 which is a function of the probabilities pab, t1 , t2 |x t1yt2. If these probabilities satisfy the locality decomposition (1.26) and Eq. (1.28), we necessarily have that S  S  S  S  a0b0   a0b1   a1b0   a1b1   6, 1.36 where 19 S  a0b0   a0b1   a1b0   a1b1 , S  a0b0   a0b1   a1b0   a1b1 , S  a0b0   a0b1   a1b0   a1b1 . 1.37 Remark I.1.6. Note that in contrast with CHSH-inequality for conditional correlations (1.13) the inequality (1.36) is obtained without any references to random choice of settings x t1 and yt2 . I.2. A new quantum mechanical formalism based on the probability representation of quantum states Definition I.2.1. In probability theory, the sample space (observation space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible outcomes are listed as elements in the set. It is common to refer to a sample space by the label Ω. Remark I.2.1. A well-defined sample space (observation space) is one of three basic elements in a probabilistic model (a probability space   ,,P); the other two are a well-defined set of possible events (a sigma-algebra ) and a probability assigned to each event (a probability measure function P). A simple example of a sample phase space and corresponding probability space closer to our Stern-Gerlach experiment is a coin toss. Consider 1000 coin tosses. If the coin is tossed without bias, you will find close to 500 heads and 500 tails, corresponding to prob heads  0.5 and prob tails  0.5. Here the sample space consists of the 1000 detailed trajectories of the toss, which your eye cannot follow, but which if analyzed by a very fast computer could predict which toss would give a head and which a tail (Fig. I.2.1). Again, the probabilities are just reflections of our ignorance of the details, but the details are there. So we have the questions are there hidden details underlying the probabilities in quantum 20 mechanics? Is there a hidden sample space and corresponding probability space? Fig. I.2.1. A sample space. Trajectories in a coin toss. Adapted from [7] In the de Broglie-Bohm interpretation: a particle has an initial position and follows a path whose velocity at each instant is given by a classical equation. On the basis of this assumption we conduct a simulation experiment by drawing random initial positions of the electrons in the initial wave packet ("quantum equilibrium hypothesis"). Fig. I.2.2 shows, after its initial starting position, 100 possible quantum trajectories of an electron passing through one of the two slits: We have not represented the paths of the electron when it is stopped by the first screen. Fig. I.2.3 shows a close-up of these trajectories just after they leave their slits. Remark I.2.2. The different trajectories explain both the impact of electrons on the detection screen and the interference fringes. This is the simplest and most natural interpretation to explain the impact positions: "The position of an impact is simply the position of the particle at the time of impact." This was the view defended by Einstein at the Solvay Congress of 1927. The position is the only measured variable of the experiment. 21 Fig. I.2.2. A sample space in Bohmian QM. 100 electron trajectories for the Jӧnsson experiment. Adapted from [8] Fig. I.2.3. Close-up on the 100 trajectories of the electrons just after the slits. Adapted from [8] 22 Fig. I.2.4. Ten silver atom trajectories within initial spin orientation θ0 = π/3 and initial position z0; arrows represent the spin orientation θ(z, t) along the trajectories. Adapted from [8] Fig. I.2.4 presents, for a silver atom with the initial spinor orientation (θ0 = π/3, ɸ0=0), a plot in the (Oyz) plane of a set of 10 trajectories whose initial position z0 has been randomly chosen from a Gaussian distribution with standard deviation σ0. The spin orientations θz, t are represented by arrows. Now let us consider a mixture of pure states where the initial orientation (θ0 ,φ0 ) from the spinor has been randomly chosen. These are the conditions of the initial Stern and Gerlach experiment. Fig. I.2.5 represents a simulation of 10 quantum trajectories of silver atoms from which the initial positions z0 are also randomly chosen. 23 Fig. I.2.5. Ten silver atom trajectories where the initial spin orientation  0 0,  has been randomly chosen; arrows represent the spin orientation θ(z, t) along the trajectories. Adapted from [8] Definition I.2.2. A probability space consists of three parts: 1. A sample space (observation space) Ω, which is the set of all possible single outcomes   . 2. A set of events , where each event is a set containing  or more outcomes. 3. The assignment of probabilities to the events; that is, a function P from events to probabilities. An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a -algebra . Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, P :  0,1. Remark I.2.3. Note that: (i) In conventional quantum mechanics we dealing with a probabilities without any probability space   ,,P. σ 24 (ii) However a wave function  in quantum mechanics is a description of the quantum state | of a quantum system . The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements of an observable Q  Q (represented by operator Q) made on the quantum system  in state | can be derived from a wave function  . (iii) From (ii) it follows that there exists an probability space   ,,P and random variable Q Q| : Ω  E, i.e. X Q| is a measurable function from the set of possible outcomes Ω to some set E. Example. We now, consider as an example, the simple case of a non-relativistic single particle, without spin, in one spatial dimension. Note that: (i) The state of such a particle is completely described by its position-space wave function, x where x is the position of a particle. This is a complex-valued function of real variable x. For one spinless particle in 1D, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number |x|2  xx  x 2.1 is interpreted as the probability density that the particle is at x . (ii) If the particle position is measured, its location cannot be determined from the wave function, but is described by a probability distribution. The probability that its position x will be in the interval a  x  b is the integral of the density over this interval: Pa  x  b   a b |x|2dx. 2.2 This leads to the normalization condition 25    |x|2dx  1 2.3 because if the particle is measured, there is 100% probability that it will be somewhere. (iii) Assume that particle is in state |. From a statement (ii) it follows that the coordinate x of the particle wave function, x under measurement by a measuring device is a random variable x  Xx|, Xx|Ω|  E | which is well defined on a probability space  |  | ,| ,P. 2.4 (iv) However in conventional quantum mechanics as mentioned above such probability space |  | ,| ,P is missing. Remark I.2.4. For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers. Note that: (i) Technically, because of the normalization condition, wave functions form a projective space Hp rather than an ordinary infinitedimensional vector space H. Also H is a Hilbert space, because the inner product of two wave functions 1 and 2 can be defined as the complex number 1 |2      1x2xdx. 2.5 (ii) Hp  S   H. (iii) The all values of the wave function x are components of a vector |. There are uncountably infinitely many of them and integration is used in place of summation. In Bra-ket notation, this 26 vector is written |     dxx|x , 2.6 where x  |x  x   x. Let us consider QM system which consists of one particle with a wave function x, x  a,b, such that supp x  a,b and   |x|2dx  1. We go to construct now corresponding probability space |  | ,| ,P. In one dimension, the position x of such a particle can range over the values a  x  b. Consider now the measurement of coordinate of such QM particle. Obviously a sample space for such a coordinate measurement is Ω|  a,b  a,b. Note that in practice observable x is measured to an accuracy x determined by the measuring device. Thus xx1x2x  x1 ,x  x2   a,b  x  x1 ,x  x2   |  and therefore -algebra a,b  Ba,b is the Borel algebra on the set a,b. Let B be the Borel algebra B  ab a,b , we choose the probability measure PB :B 0,1 of the form PBA   A xdx, 2.7 where A  B ,  xdx  1 and dx is the Lebesgue measure. Definition I.2.3. We choose the probability measure P|:B  0,1 corresponding to a wave function x,x2 2  1, in the following form: σ 27 P|A   A |x|2dx, 2.8 where A  B and dx is the Lebesgue measure. Definition I.2.4. A random variable X | : Ω|  E | is a measurable function from the set of possible outcomes Ω to some set E | . The technical axiomatic definition requires Ω | to be a probability space and E | to be a measurable space. Note that although X | is usually a real-valued function X | : Ω|  a,b, it does not return a probability. The probabilities of different outcomes or sets of outcomes (events) in our case are already given by the probability measure P| with which Ω | is equipped above. Definition I.2.5. (Real-valued random variables.) In a case mentioned above the observation space is a set a,b. Recall, ,B,PB  is the probability space. For real observation space, the function X | : Ωa,b  a,b is a real-valued random variable, i.e. aba  b|a  X |  b  a,b . Definition I.2.6. Let |  H. We define now a signed measure P|A  : B    by formula P|A A   A xp |A xdx, 2.9 where p|A x  |x|A | 2 . Remark I.2.5. We assume now that ,,P  ,B,PB  and P|  PB , i.e. P | is absolutely continuous with respect to P . By RadonNicodym theorem there exists a random variable X | :    such that for any A  B : P|A   A X |dPB, 2.10 28 Using Eq. (2.10) we define random variable X | : Ω|   by formula X |  dP| dPB . 2.11 Definition I.2.7. The cumulative distribution function of a real-valued random variable X |  is the function given by FX | x  P  B |X |   x , 2.12 where the right-hand side represents the probability that the random variable X |  takes on a value less than or equal to x. The probability that X |  lies in the semi-closed interval a1 ,b1  a,b, where a1  b1 , is therefore P|a1  X |  b1   FX | b1   FX |a1 . The CDF of any continuous random variable X |  X | can be expressed as the integral of its probability density function pX| x as follows: FX | x    x pX | tdt    x |t|2dt. 2.13 From Eq. (2.9) Eq. (2.10) we obtain EX |   a,b X |dP     xpX | xdx. 2.14 Using canonical QM-abbreviation |     |x x|dx, 2.15 29 where x|  x, |  S  H, from Eq. (2.14) Eq. (2.15) we obtain  |x |    X |dP     xpX | xdx. 2.16 where  x |  x|. Remark I.2.6. We assume now that: (i) for any x  H : (a)   |X | |dP  , (b)   X | 2 dP  , (ii) for any x  H : X |  1,2dP  1dP 2dP. Definition I.2.8. We will write the Eq. (1.16) in the following form  |x |    Xx |dP     xpXx | xdx, 2.17 where  x |  x|. This form remind that continuous random variable X |x  X |x corresponds to the coordinate of a particle with a state vector |. Remark I.2.7. We assume that particle A is initially in the state |A   H. We assume now that: if on performing a measurement of x on particle A with an accuracy x, and the result is obtained in the range xA  x,xA  x at instant t, then unconditional measure PB immediately after the measurement at instant t collapses to conditional measure P A X||A xA,x, where X  B : P  A X |A x A,x  PB X  X  A x A,x PB X  A x A,x , 2.18 30 and where X  A xA,x  |xA  x  X |A   xA  x. Remark I.2.8. (i) From Eq. (2.18) it follows that unconditional probability density function pAx  |x|A | 2 immediately after the measurement at instant t collapses to the following conditional probability density function pA x|X  A x A,x  pAx PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x 2.19 (ii) From Eq. (2.18) it follows that immediately after the measurement on particle B at instant t a wave function Ax  x|A  collapses to the following wave function A coll x  Ax PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x. 2.20 Remark I.2.9. Note that in contrast with the usual 'Copenhagen' interpretation in quantum mechanical formalism based on the probability representation of quantum states, coordinate of a particle with a state vector | that continuous random variable X |x  X |x and such a random variable in contrast with a state vector | does not collapses under the measurement. However unconditional measure PB immediately after the measurement at instant t collapses to conditional measure PBX||A xA,x (is given by Eq.(2.18)) on whole probability space   ,B,PB . Remark I.2.10. Thus immediately after measurement on any particle A probability space   ,B,PB  collapses to probability space  A  ,B ,PB X |A xA,x 31 ,B,PB  collaps  ,B,P  A X |A xA,x . 2.21 I.3. EPR Paradox resolution by using Postulate of EPRNonlocality

In the 20s and 30s it became evident that some properties in quantum mechanics can be assigned only to the quantum mechanical system, but not necessarily to its constituents. This led Einstein, Podolsky and Rosen (EPR) to their remarkable 1935 paper where they concluded that quantum mechanics is not a complete theory of nature (EPR paradox). The conclusion was derived from some common sense requirements that EPR postulated: 1. Completeness: Each element of realism should have its correspondence in a theory. 2. Realism: If a property can be assigned to a physical system with certainty then there exists an element of realism that corresponds to this property. 3. Locality: Measurements of different elements of realism in spatially separated systems can not influence each other. In original paper [13], Einstein, Podolsky and Rosen describe two particles A and B with perfectly correlated position xB  xA  x0 3.1 and perfectly anti-correlated momentum pB  pA, 3.2 see Fig. I.3.1. 32 Fig. I.3.1. Particles A and B with correlated position and anticorrelated momentum EPR originally argued as follows. Consider two spatially separated subsystems at A and B. EPR considered two observables x (the position) and  p (momentum) for subsystem A, where x and  p do not commute, so that C  0 x,p  2C. 3.3 Suppose now that one may predict with certainty the result of measurement x based on the result of a measurement performed at B. Also, for a different choice of measurement at B, suppose one may predict the result of measurement  p at A . Such correlated systems are predicted by quantum theory. Assuming local realism EPR deduce the existence of an element of reality, x , for the physical quantity x and also an element of reality,  p , for  p . Local realism implies the existence of two hidden variables x and  p that simultaneously predetermine, with no uncertainty, the values for the result of an x or  p measurement on subsystem A , should it be performed. This hidden variable state for the subsystem A alone is not describable within quantum mechanics, since simultaneous eigenstates of x and  p do not exist. Hence, EPR argued, if quantum mechanics is to be compatible with local realism, we must regard quantum mechanics to be incomplete. In the idealized entangled state proposed by EPR, |EPR     |x,xdx     |p,pdp 3.4 33 the positions and momenta of the two particles are perfectly correlated. Note that: this state is non-normalizable and cannot be realized in the laboratory. When coordinates x A and pA are measured in independent realizations of the same state, the correlations allow for an exact prediction of x B and pB . EPR assumed that such exact predictions necessitate an element of reality which predetermines the outcome of the measurement. Quantum mechanics however prohibits the exact knowledge of two noncommuting variables like x B and pB , since their measurement uncertainties are subject to the Heisenberg relation xBpB  /2. 3.5 Remark I.3.1. A most critical component of the EPR argument was the principle of EPR-locality. Indeed, one may regard the EPR paradox as a statement of the mutual incompatibility of EPR-locality, entanglement, and completeness. We accept now the following postulate: The postulate of nonlocality for continuous variables. The Heisenberg uncertainty relations xApA  1 3.6 cannot be violated in any cases: (i) according to quantum mechanics, the Heisenberg uncertainty relations (1.3.4) cannot be violated if the coordinate x A and momentum pA of the particle A are measured directly by measurements performed on the particle A, (ii) the Heisenberg uncertainty relations (3.8) cannot be violated even if the coordinate xA and momentum pA of the particle A are measured indirectly, i.e. by using measurement on particle B , as 34 required in EPR gedanken experiment, (iii) in any cases true coordinate x A and momentum pA of the particle A cannot be predicted simultaneously with a sufficiently small uncertainty x A and pA such that the Reid's inequality [14]: xApA  1 3.7 based on local realism cannot be satisfied, i.e., always xApA  1. 3.8 Remark I.3.2. Obviously under postulate of nonlocality EPR paradox disappears. However postulate of nonlocality is supported by quantum mechanical formalism based on the probability representation of quantum states. Using probability representation Eq. (2.11) of quantum states |A  and |B  from Eq. (3.4) we obtain X |B   X |A   x0 ,a.s. 3.9 We assume that particle A is initially in the state |A   H. We assume now that a measurement of x performing on particle B with an accuracy x, and the result is obtained in the range xB  x,xB  x at instant t. Then from Eq.(3.9) we obtain |x B  x  X |B   x B  x  |x B  x 0   x  X |B   x 0  x B  x 0   x  |x A  x  X |A   x A  x. 3.10 We let now for short 35 X B x B ,x  |x B  x  X |B   x B  x, X  A x A,x  |x A  x  X |A   x A  x. 3.11 Then from Eq. (3.10) Eq. (3.11) we obtain X B xB ,x  X  A xA,x. 3.12 Since event X B xB ,x   was occurred by performing a measurement on particle B, then unconditional measure PB immediately after measurement at instant t collapses to conditional measure P B X||B xB ,x, where X  B and therefore from Eq.(2.18) we obtain PX B X  A x A,x X B x B ,x  PB X B x B ,x  X  A x A,x PB X B x B ,x  PB X B x B ,x  X  A x A,x PB X B x B ,x  PB X B x B ,x PB X B x B ,x  1. 3.13 Therefore since event X B xB ,x   was occurred by performing a measurement on particle B immediately after measurement event X  A xA,x occurs with a probability = 1. From Eq. (3.13) it follows that unconditional probability density function pAx  |x|A | 2 immediately after measurement at instant t collapses to the following conditional probability density function pA x|X  A x A,x  pAx PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x 3.14 36 a wave function Ax  x|A  collapses to the following wave function A coll x  Ax PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x. 3.15 From Theorem C.1, (see Appendix C) it follows that a wave function Ap  p|A  collapses in accordance with Heisenberg uncertainty relations (3.6). I.4. EPR-B Paradox resolution and Postulate of EPR-B-Nonlocality Entangled states are very interesting states because they exhibit correlations that have no classical analog. They are of particular importance in quantum computation and quantum information. As an example let us take the entangled bi-partite pure state: 1  1 2 |01AB  |10AB   HA  HB . 4.1 Obviously this state can not be decomposed as a simple product state |qA |pB,q,p  0,1. Remark I.4.1. Note that 1 mentioned above is one of the so-called four Bell states: 1  1 2 |01AB  |10AB ,2  1 2 |01AB  |10AB , 3  1 2 |00AB  |11AB ,4  1 2 |00AB  |11AB . 4.2 37 They form a convenient basis of bi-partite quantum states of twodimensional Hilbert spaces. The state is maximally entangled, i.e. when we trace over the state B then the reduced density operator A of the system will be a multiple of the identity operator. This means that if we measure in system A in any basis the result will be completely random (0 or 1 with equal probability 1/2). Remark I.4.2. However, there is a perfect correlation: Whenever we measure with certainty 1 in system A then we will measure 0 in system B with certainty and vice versa. Remark I.4.3. However, despite the randomness, the choice of basis for measurement in system A clearly has a nonlocal effect on the state of the system B: it gives it a definite orientation in the basis |0B, |1B , which it did not have before the measurement. Remark I.4.4. Obviously the process described above is nonlocal: the state changes instantly even though the systems A and B could be space-like separated. We are accustomed to saying that this sort of instantaneous action at a distance is forbidden by canonical relativity. Remark I.4.5. In order to avoid the contradiction which arises from instantaneous action at a distance mentioned above we introduce an extension of the canonical relativity by using measure algebra of physical events in Minkowski space-time, see Chapter III.2. Definition I.4.1. A measure algebra   B,P with a probability measure P, is a Boolean algebra B with a countably additive probability measure. Definition I.4.2. (i) A measure algebra of physical events ph  B,P with a probability measure P, is a Boolean algebra of physical events B with an countably additive probability measure. (ii) A Boolean algebra of physical events can be formally defined as a set B of elements a,b, . . . with the following properties: 1. B has two binary operations,  (logical AND, or "wedge") and  (logical OR, or "vee"), which satisfy: 38 the idempotent laws: (1) a  a  a  a  a, the commutative laws: (2) a  b  b  a, (3) a  b  b  a, and the associative laws: (4) a  b  c  a  b  c, (5) a  b  c  a  b  c. 2. The operations satisfy the absorption law: (6) a  a  b  a  a  b  a. 3. The operations are mutually distributive (7) a  b  c  a  b  a  c, (8) a  b  c  a  b  a  c. 4. B contains universal bounds 0 and 1 which satisfy (9) 0  a  0, (10) 0  a  a, (11) 1  a  a, (12) 1  a  1. 5. B has a unary operation a (or a  ) of complementation (logical negation), which obeys the laws: (13) a  a  0, (14) a  a  1 . All properties of negation including the laws below follow from the above two laws alone. 6. Double negation law: a  a . 7. De Morgan's laws: (i) a  b  a  b, (ii) a  b  a  b. 8. Operations composed from the basic operations include the following important examples: The first operation, a  b (logical material implication): (i) a  b  a  b. The second operation, a  b, is called exclusive. It excludes the possibility of both a and b (ii) a b  a  b  a  b. The third operation, the complement of exclusive or, is equivalence or Boolean equality: (iii) a  b  a  b . 9. B has a unary predicate Occa, which meant that event a has occurred, and which obeys the laws: (i) Occa  b  Occa  Occb, (ii) Occa  b  Occa  Occb, (iii) Occa  Occa. 39 Definition I.4.3. (i) Let B be a Boolean algebra of physical events. A Boolean algebra BM4 of physical events in Minkowski spacetime M4  1,3 that is cartesian product BM4  B M4 . (ii) Let BM4 be a Boolean algebra of physical events in Minkowski spacetime. A measure algebra of physical events M4 ph  BM4 ,P in Minkowski spacetime that is a Boolean algebra BM4 with a probability measure P such that A A  M4 ph B B  M4 ph A  B  PA  PB, APA  PAOc . 4.3 (iii) Let BM4 be Boolean algebra of the all physical events in Minkowski spacetime and let M4 ph be an measure algebra M4 ph  BM4 ,P with a probability measure P. We denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc. (iv) We will write for a short AOcx,BOcx, . . . etc., instead OccAx,OccBx, . . . etc. Remark I.4.6. Note that Boolean algebra BM4 of physical events in Minkowski spacetime obviously contains a pairs Ax1 ,Bx2  of a Boolean equivalent events Ax1  and Bx2  such that x1x2Ax1   Bx2 , 4.4 i.e., Boolean equality Ax1   Bx2  always holds even the events Ax1  and Bx2  are space-like separated. Definition I.4.4. A probability measure P on a measure space , gives a probability measure algebra   ,,P  on the Boolean algebra of measurable sets modulo null sets, i.e., sets P measure zero. Remark I.4.7. We assume now that a measure algebra 40 M4 ph  BM4 ,P admit a representation  : M4 ph  ,,P  of the measure algebra M4 ph  BM4 ,P in a probability measure algebra   ,,P , such that (ii) PX  P1X for any X  . Definition I.4.5. Given two events Ax1  and Bx2  from the measure algebra M4 ph  BM4 ,P the conditional probability of Ax1  given Bx2  is defined as the quotient of the probability of the joint of events Ax1  and Bx2  , and the probability of B : PAx1 |Bx2   PAx1   Bx2  PBx2  4.5 where PBx2   0. Remark I.4.8. Assume that: (i) BOcx2 , then since event Bx2  is occurred, unconditional probability measure P on algebra BM4 collapses to the conditional probability measure PBOc Ax1   PAx1 |BOcx2   PAx1 |Bx2   PAx1   Bx2  PBx2  4.6 and measure algebra M4 ph  BM4 ,P collapses to the measure algebra M4 ph  BM4 ,PBOc  . (ii) BOcx2  and Ax1   Bx2 , then PBOc Ax1   PAx1 |B Ocx2   PAx1   Bx2  PBx2   PBx2  PBx2   1, 4.7 i.e., PBOc Ax1   1 and therefore, since event Bx2  is occurred in point x2 , the event Ax1  is occurred with probability 1 in point x2 even points x1 and x2 are space-like separated. Remind that Bohm [15] considered two spatially-separated spin-1/2 particles at A and B produced in an entangled singlet state (often referred to as the EPR-Bohm state or the Bell-state): 41 |  1 2 1 2 A  1 2 B   1 2 A 1 2 B 4.8 Here | 1 2 A are eigenstates of the spin operator  J z A , and we use  J z A ,  J x A ,  J y A to define the spin-components measured at location A . The spin-eigenstates and measurements at B are defined similarly. By considering different quantization axes, one obtains different but equivalent expansions of | in Eq. (4.4), just as EPR suggested. Fig. I.4.1. The Bohm gedanken EPR experiment. Two spin-1/2 particles prepared in a singlet state from the source into spatially separated regions A and B, and give anti-correlated outcomes for Jθ A and JB, where θ is x, y or z Bohm's paradox is based on the existence, for Eq. (4.1), of a maximum anti-correlation between not only  J z A and  J z B , but  J y A and  J y B , and also  J x A and  J x B . An assumption of local realism would lead to the conclusion that the three spin components of particle A were simultaneously predetermined, with absolute definiteness. Since no such quantum description exists, this is the situation of an EPR paradox. We accept now the following postulate: The postulate of nonlocality for observables with discrete values. The Heisenberg spin uncertainty relations 42 JxAJyA  JzA  /2,JxAJzA  JyA  /2,JzAJyA  JxA  /2 4.9 cannot be violated in any cases: (i) if the three spin components of the particle A are measured directly by measurements performed on the particle A, (ii) and even if some spin components of the particle A are measured indirectly as required in Bohm gedanken EPR experiment. Remark I.4.11. Obviously under postulate of nonlocality EPR paradox disappears. However postulate of nonlocality supported by quantum mechanical formalism based on the probability representation of quantum states, see Chapter V.3. 43 Chapter I THE POSTULATE OF EPR-B NONLOCALITY I.1. The EPR paradox In 1935, Einstein, Podolsky and Rosen (EPR) originated the famous EPR paradox [6]. This argument concerns two spatially separated particles which have both perfectly correlated positions and momenta, as is predicted possible by quantum mechanics. The EPR paper spurred investigations into the nonlocality of quantum mechanics, leading to a direct challenge of the philosophies taken for granted by most physicists. The EPR conclusion was based on the assumption of local realism, and thus the EPR argument pinpoints a contradiction between local realism and the completeness of quantum mechanics. I.2. Einstein's 1927 gedanken experiment Einstein never accepted orthodox quantum mechanics because he did not believe that its nonlocal collapse of the wave function could be real. When he first made this argument in 1927 [7], he considered just a single particle. The particle's wave function was diffracted through a tiny hole so that it `dispersed' over a large hemispherical area before encountering a screen of that shape covered in photographic film. Since the film only ever registers the particle at one point on the screen, orthodox quantum mechanics must postulate a `peculiar mechanism of action at a distance, which prevents the wave... from producing an action in two places on the screen'. That is, according to the theory, the detection at one point must instantaneously collapse the wave function to nothing at all other points. 44 Remark 1.2.1. It was only in 2010, nearly a century after Einstein's original proposal, that a scheme to rigorously test Einstein's `spooky action at a distance' [7], [8] using a single particle (a photon), as in his original conception, was conceived [9]. In this scheme, Einstein's 1927 gedanken experiment is simplified so that the single photon is split into just two wave packets, one sent to a laboratory supervised by Alice and the other to a distant laboratory supervised by Bob. However, there is a key difference, which enables demonstration of the nonlocal collapse experimentally: rather than simply detecting the presence or absence of the photon, homodyne detection is used. This gives Alice the power to make different measurements, and enables Bob to test (using tomography) whether Alice's measurement choice affects the way his conditioned state collapses, without having to trust anything outside his own laboratory. Fig. 1.2.1. Simplified version of Einstein's original gedanken experiment. Adapted from [10] A single photon is incident on a beam splitter of reflectivity R and then subjected to homodyne measurements at two spatially separated locations. 45 Alice is trying to convince Bob that she can steer his portion of the single photon to different types of local quantum states by performing various measurements on her side. She does this by using different values of her LO phase, and extracting only the sign s  , of the quadrature she measures. Meanwhile, Bob scans his LO and performs full quantum-state tomography to reconstruct his local quantum state. He reconstructs unconditional and conditional local quantum states to test if his portion of the single photon has collapsed to different states according to Alice's LO setting θ, and result s see Fig . 1.2.1. The key role of measurement choice by Alice in demonstrating `spooky action at a distance' was introduced in the famous EinsteinPodolsky-Rosen (EPR) paper [6] of 1935. In its most general form, this phenomenon has been called EPR-steering, to acknowledge the contribution and terminology of Schrödinger [11], who talked of Alice `steering' the state of Bob's quantum system. From a quantum information perspective, EPR-steering is equivalent to the task of entanglement verification when Bob (and his detectors) can be trusted but Alice (or her detectors) cannot. This is strictly harder than verifying entanglement with both parties trusted [12], but strictly easier than violating a Bell inequality [13], where neither party is trusted [12]. Remark 1.2.2. A recent experimental test of entanglement for a single photon via an entanglement witness has no efficiency loophole [14], however, it demonstrates a weaker form on nonlocality than EPR-steering. In [10], it was demonstrated experimentally that there exists Einstein's elusive `spooky action at a distance' for a single particle without opening the efficiency loophole without claim to have closed the separation loophole. That is the one-sided device-independent verification of spatial-mode entanglement for a single photon. θ 46 I.3. The continuous variable EPR paradox. EPR-Reid's criteria We remind that EPR treated the case of a non-factorizable pure state | which describes the results for measurements performed on two spatially separated systems at A and B (Fig. 1.3.1). Nonfactorizable means entangled, that is, we cannot express | as a simple product |  |A  |B , where |A and |B are quantum states for the results of measurements at A and B, respectively. Fig.1.3.1. The original EPR gedanken experiment. Two particles move from the source into spatially separated regions A and B, and yet continue to have maximally correlated positions: xA+x0=xB and anti-correlated momenta: pA=-pB. Adapted from [15] In the first part of their paper, EPR point out in a general way the problematic aspects of such entangled states. The key issue is that one can expand | in terms of more than one basis, that corresponds to different experimental settings, which we parametrize by  . Let us consider the state |   dx|x ,A  |ux ,B , 1.3.1 where the x eigenvalue could be continuous or discrete. The parameter setting  at the detector B is used to define a particular orthogonal measurement basis |ux ,B . On measurement at B, that 47 projects out a wave-function |x ,A at A , the process called reduction of the wave packet. Remark 1.3.1. The locality assumption postulates no action-at-adistance, so that measurements at a location B cannot immediately disturb the system at a spatially separated A location. Remark 1.3.2. The problematic issue is that different choices of measurements  at B will cause reduction of the wave packet at A in more than one possible way. EPR states that, as a consequence of two different measurements at B, the second system may be left in states with two different wave functions. Yet, no real change can take place in the second system in consequence of anything that may be done to the first system. The problem was established by EPR by a specific example, shown in Fig.1.3.1. EPR considered two spatially separated subsystems, at A and B, each with two observables x and p where x and p are noncommuting quantum operators, with commutator x,p   xp  px  2C  0. 1.3.2 The results of the measurements x and p are denoted x and p respectively, and this convention we follow throughout the paper. We note that EPR assumed a continuous variable spectrum and considered wave function  defined in a position representation by x,xB    eip/xxBx0 dp , 1.3.3 where x 0 is a constant implying space-like separation. Here the pairs x and p refer to the results for position and momentum measurements at A , while x B and pB denote the position and 48 momentum measurements at B. We leave off the superscript for A system, to emphasize the inherent asymmetry that exists in the EPR argument, where one system A is steered by the B other. Remark 1.3.3. According to canonical quantum mechanics, one can predict with certainty that a measurement x will give result xB  x0 , if a measurement x B , with result x B , was already performed at B. One may also predict with certainty the result of p measurement, for a different choice of measurement at B. If the momentum at B is measured to be p , then the result for p is p . These predictions are made without disturbing the second system at A , based on the assumption, implicit in the original EPR paper, of locality. Remark 1.3.4. The locality assumption can be strengthened if the measurement events at A and B are causally separated (such that no signal can travel from one event to the other, unless faster than the speed of light). Remark 1.3.5. The remainder of the EPR argument may be summarized as follows. Assuming local realism, one deduces that both the measurement outcomes, for x and p at A , are predetermined. The perfect correlation of x with xB  x0 implies the existence of an element of reality for the x measurement. Similarly, the correlation of p with pB implies an element of reality for p . Although not mentioned by EPR, it will prove useful to mathematically represent the elements of reality for x and p by the respective variables xA and p A , whose possible values are the predicted results of the measurement. Remark 1.3.6. To continue the argument, local realism implies the existence of two elements of reality, xA and p A , that simultaneously predetermine, with absolute definiteness, the results for measurement x or p at A . These elements of reality for the localized 49 subsystem A are not themselves consistent with quantum mechanics. Simultaneous determinacy for both the position and momentum is not possible for any quantum state. Hence, assuming the validity of local realism, one concludes quantum mechanics to be incomplete or even inconsistent! Remark 1.3.7. We claim that any assumption of local realism is completely wrong. Such a claim meant as minimum the weak postulate of nonlocality. I.3.1. The weak postulate of nonlocality for continuous variables The Heisenberg uncertainty relations xApA  1 1.3.4 cannot be violated in any cases: (i) of course, according to quantum mechanics, the Heisenberg uncertainty relations (1.3.4) cannot be violated if the coordinate x A and momentum pA of the particle A are measured directly by measurements performed on the particle A, (ii) the Heisenberg uncertainty relations (1.3.4) cannot be violated even if the coordinate xA and momentum pA of the particle A are measured indirectly, i.e. by using measurement on particle B , as required in EPR gedanken experiment, (iii) in any cases true coordinate x A and momentum pA of the particle A cannot be predicted simultaneously with a sufficiently small uncertainty x A and pA such that the Reid's inequality [16]: xApA  1 1.3.5 50 based on local realism would be satisfied, i.e., always xApA  1. 1.3.6 We claim strictly stronger assumptions of nonlocality than mentioned above. I.3.2. The strong postulate of nonlocality for continuous variables Let | t x  A and | t x  B be a state vector in x -representation at instant t of the particle A and particle B correspondingly. Let  t p  A and  t p  B be a state vector in p -representation at instant t of the particle A and particle B correspondingly. Let t Ax  x |t x  A ,t Bx  x |t x  B be a wave functions in x representation of the particle A and particle B correspondingly. Let t Ap  p|t p  A ,t Bp  p|t p  B be a wave functions in p representation of the particle A and particle B correspondingly. Let t A/BxA,xB  be corresponding two-particle wave function in x representation and let t A/BpA,pB  be corresponding two-particle wave function in p -representation. We claim that: (i) whenever a measurement of the coordinate x of a particle B is performed at instant t with result x B  xB  ,xB  ,  1, then: (a) according to quantum mechanics a state vector | t x  B collapses at instant t to the state vector  t,,,xB x B ~  LxB B ,|t x  B 1.3.7 51 given by law (1.2.20), where  Lx B B , is a norm-reducing, positive, self-adjoint, linear operator in the 2 -particle non projective Hilbert space H, representing the localization of particle B around the point x B , (see subsection II.2.); (b) according to postulate of nonlocality a state vector | t x  A immediately collapses at instant t to the state vector  t,,,xA x A ~  LxBx0 A ,| t x  A 1.3.8 given by law (1.2.20) and this is true independent of the distance in Minkowski space-time M4  1,3 that separates the particles. Thus | t x  B col lapse   t,,,xB x B  |t x  A col lapse   t,,,xBx0 x A 1.3.9 (ii) under conditions given by Eq. (1.3.7) Eq. (1.3.9) two-particle wave function t A/BxA,xB  collapses at instant t by law t A/BxA,xB  col lapse   LxBx0 A  LxB B ,t A/BxA,xB  1.3.10 (iii) whenever a measurement of the momentum pB of a particle B is performed at instant t with result p B  pB  ,pB  ,  1, then: (a) according to quantum mechanics a state vector  t p  B collapses at instant t to the state vector  t,,,p B p B ~  Lp B B , t p  B , 1.3.11 where  Lp B B , is a norm-reducing, positive, self-adjoint, linear operator in the 2 -particle non projective Hilbert space H, representing the localization of momentum of the particle B 52 around the value pB . The localization operators  Lp B B , have been chosen to have the following form:  Lp B B ,  1  3/4 exp  1 2 p  pB 2 1.3.12 where   0,1 and lim0   . (b) according to postulate of nonlocality a state vector  t p  A immediately collapses at instant t to the state vector  t,,,xA p A ~  Lp B A , t p  A 1.3.13 and this is true independent of the distance in Minkowski space-time M4  1,3 that separates the particles. Thus  t p  B col lapse   t,,,p B p B   t p  A col lapse   t,,,pB p A 1.3.14 (iv) under conditions given by Eq. (1.3.11) Eq. (1.3.13) two-particle wave function t A/BpA,pB  collapses at instant t by the law t A/BpA,pB  col lapse   LpB A  LpB B , t A/BpA,pB . 1.3.15 Remark 1.3.8. Let p t A and p t B be the momentum at instant t of the particle A and particle B correspondingly. Note that whenever a measurement of the coordinate x of a particle B is performed at instant t with the accuracy xB  1 then: (i) immediately after this measurement the momentum p t B at instant t changed according to quantum mechanics by the Heisenberg uncertainty relations (1.3.4); (ii) immediately after this measurement the momentum p t A at instant 53 t changed according to postulate of nonlocality by the Heisenberg uncertainty relations (1.3.4). Remark 1.3.9. Let x t A and x t B be the coordinate at instant t of the particle A and particle B correspondingly. Note that whenever a measurement of the momentum p of a particle B is performed at instant t with the accuracy pB  1 then: (i) immediately after this measurement the coordinate x t B at instant t changed according to quantum mechanics by the Heisenberg uncertainty relations (1.3.4); (ii) immediately after this measurement the momentum at instant changed according to postulate of nonlocality by the Heisenberg uncertainty relations (1.3.4). Remark 1.3.10. Schrödinger [11] pointed out that the EPR twoparticle wave function in Eq. (1.3.3) was verschränkten which he later translated as entangled i.e., not of the separable form AB . Schrödinger considered as a possible resolution of the paradox that this entanglement degrades as the particles separate spatially, so that EPR correlations would not be physically realizable. Definition 1.3.1.Quantum inseparability (entanglement) for a general mixed quantum state is defined as the failure of    dP A   B , 1.3.16 where  dP  1 and  is the density operator. Here  is a discrete or continuous label for component states, and  A and  B correspond to density operators that are restricted to the Hilbert spaces A and B respectively. x t A t 54 Remark 1.3.11. The definition of inseparability extends beyond that of the EPR situation, in that one considers a whole spectrum of measurement choices, parametrized by  for those performed on A system, and by  for those performed on B. We use canonical notation x  A and x  B to describe all measurements at A and B. Denoting the eigenstates of x  A by |x  A , we define PQxA |,  xA |  A |x A and PQx B |,  x B |  B |x B , which are the localized probabilities for observing results x  A and x  B respectively. The separability condition (1.3.9) then implies that joint probabilities Px A,x B are given as [16]: Px A,x B    dPPQx A |PQxB | . 1.3.17 Remark 1.3.12. We note the canonical restriction 2xA |2pA |  1 1.3.18 where 2xA | and 2pA | are the variances of PQx  A|, for the choices  corresponding to position x and p momentum, respectively. Thus, 2xA |2pA |  1 1.3.19 is an EPR criterion, meaning that this would imply an EPR "paradox". Remark 1.3.13. Note that the original EPR state of Eq. (1.3.3) is not separable. Suppose that, based on a result x B for the measurement at B , an estimate x estxB  is made of the result x at A. We may define the average error  infx of this inference as the root mean square (RMS) of the deviation of the estimate from the actual value, so that [16-18]: 55 inf 2 x   dxdxBPtx,xB x  x estxB 2 . 1.3.20 An inference variance  inf 2 p is defined similarly, i.e. inf 2 p   dpdpBPtp,pB p  pestpB 2 . 1.3.21 Remark 1.3.14. Let t A/BxA,xB  be corresponding two-particle wave function in x -representation and let t A/BpA,pB  be corresponding two-particle wave function in p -representation. Note that: (i) Ptx,x B  is the joint probability of obtaining an outcome x at A and x B at B at t instant is of the form Ptx,xB   t A/BxA,xB  2 , 1.3.22 (ii) Ptp,p B  is the joint probability of obtaining an outcome p at A and pB at B at t instant is of the form Ptp,pB   t A/BpA,pB  2 . 1.3.23 The best estimate, which minimizes  infx , is given by choosing x est for each x B to be the mean x|xB  of the conditional distribution Ptx|xB  . This is seen upon noting that for each x B result, we can define the RMS error in each estimate as inf 2 t,x|xB    dxP tx|xB x  x estxB 2 . 1.3.24 The average error in each inference is minimized for x est  x|xB , when each inf 2 t,x|xB  becomes the variance 2t,x|xB of Ptx|x B  . We thus define the minimum inference error  infx for position, 56 averaged over all possible values of x B , as VA|B x  min inf 2 x   dxBPtxB 2t,x|xB  , 1.3.25 where Px B  is the probability density for a result x B upon measurement of xB . This minimized inference variance is the average of the individual variances for each outcome at B . Similarly, we can define a minimum inference variance VA|B p for momentum, i.e. VA|B p  min inf 2 p   dpBP tpB 2t,p|pB  . 1.3.26 Remark 1.3.15. Let t A/BxA,xB  be corresponding two-particle wave function in x -representation and let t A/BpA,pB  be corresponding two-particle wave function in p -representation. Note that: (i) according to local realism the conditional distributions densities Plocx|xB  and Plocp|p B  vary given by formulae Plocx|xB  ~  LxB B ,t A/Bx,xB  1.3.27 and Plocp|pB  ~  Lp B B , t A/BpA,pB . 1.3.28 (ii) distribution densities Ploct,x B  and Ploct,p B  are given by formulae Ploct,xB   dxP loct,x|xB  1.3.29 and P loct,pB   dpP loct,p|pB . 1.3.30 57 Remark 1.3.16. Let t A/BxA,xB  be corresponding two-particle wave function in x -representation and let t A/BpA,pB  be corresponding two-particle wave function in p -representation. Note that: (i) according to postulates of nonlocality the conditional distributions densities Pn .loct,x|x B  and Pn .loct,p|p B  are given by formulae Pn .loct,x|xB    LxBx0 A  LxB B ,t A/Bx,xB  1.3.31 and Pn .loct,p|pB  ~  Lp B A  Lp B B ,t A/Bp,pB , 1.3.32 see Eq. (1.3.10) and Eq. (1.3.15) respectively. (ii) distributions Pn .loct,x B  and Pn .loct,p B  are given by formulae Pn .loct,xB   dxPn .loct,x|xB  1.3.33 and Pn .loct,p|B   dpPn .loct,p|pB  1.3.34 Thus we can define corresponding RMS errors as  loc.inf 2 t,x|x B    dxP loct,x|xB x  x estx B 2  loc.inf 2 t,p|pB    dxP loct,p|pB p  x estpB 2 1.3.35 and n .loc.inf 2 t,x|x B    dxP loct,x|x B x  x estx B 2 , n .loc.inf 2 t,p|pB    dxP loct,p|pB p  x estpB 2 1.3.36 respectively. We thus define the minimum inference error  infx for position, averaged over all possible values of x B and pB as min  loc.inf 2 x   dxBP loct,x B  loc.2 t,x|xB  , min  loc.inf 2 p   dpBP loct,pB  loc.2 t,p|pB  1.3.37 58 and min n .loc.inf 2 x   dx BPn .loct,x B n .loc.2 t,x|x B  , min n .loc.inf 2 p   dpBPn .loct,pB n .loc.2 t,p|pB . 1.3.38 respectively. From Eq. (1.3.37) and Eq. (1.3.38) we obtain the EPRnonlocality criteria min loc.inf. 2 x  min n .loc.inf. 2 x   dx BP loct,x B  loc.2 t,x|x B   Pn .loct,x B n .loc.2 t,x|x B   0, min loc.inf. 2 p  minn .loc.inf. 2 p   dpBP loct,pB  loc.2 t,p|pB   Pn .loct,pB n .loc.2 t,p|pB   0 1.3.39 and minloc.inf. 2 x minloc.inf. 2 p  min n .loc.inf. 2 x n .loc.inf 2 p  0. 1.3.40 I.4. The EPR-Bohm paradox. Reid's criteria for EPRBohm paradox Bohm [19]-[20] considered two spatially-separated spin-1/2 particles at A and B produced in an entangled singlet state (often referred to as the EPR-Bohm state or the Bell-state): |  1 2 1 2 A  1 2 B   1 2 A 1 2 B 1.4.1 Here | 1 2 A are eigenstates of the  J z A spin operator, and we use  J z A ,  J x A ,  J y A to define the spin-components measured at location A . The spin-eigenstates and measurements at B are defined similarly. By considering different quantization axes, one obtains different but equivalent expansions of | in Eq. (1.3.1), just as EPR suggested. 59 Fig. 1.4.1. The Bohm gedanken EPR experiment. Two spin-1/2 particles prepared in a singlet state from the source into spatially separated regions A and B, and give anti-correlated outcomes for Jθ A and JB, where θ is x, y or z. Adapted from [16] Bohm's paradox is based on the existence, for Eq. (1.9.1), of a maximum anti-correlation between not only  J z A and  J z B , but  J y A and  J y B , and also  J x A and  J x B . An assumption of local realism would lead to the conclusion that the three spin components of particle A were simultaneously predetermined, with absolute definiteness. Since no such quantum description exists, this is the situation of an EPR paradox. Remark 1.4.1.Bohm's paradox is based on the existence, for Eq.(1.4.1), of a maximum anti-correlation between not only  J z A and  J z B , but  J y A and  J y B , and also  J x A and  J x B . Remark 1.4.2. Note that an assumption of local realism would lead to the conclusion that the three spin components of particle A were simultaneously predetermined, with absolute definiteness. Since no such quantum description exists, this is the situation of an EPR paradox. Remark 1.4.3. Criteria sufficient to demonstrate Bohm's EPR paradox can be derived using Reid's canonical inferred uncertainty approach [16]. Using the Heisenberg spin uncertainty relation JxAJyA  JzA  /2, 1.4.2 60 one obtains the following canonical spin-EPR criterion that is useful for the Bell state given by Eq. (1.4.1) infJxAinfJyA  1 2  Jz B PJzB  JzA  Jz B . 1.4.3 Here Jz A  Jz B is the mean of the conditional PJz A |Jz B  distribution. Calculations for Eq. (1.4.1) including the effect of detection efficiency  reveals this EPR criterion to be satisfied for   0.62. The concept of spin-EPR has been experimentally tested in the continuum limit with purely optical systems for states with Jz A   0. In this case the EPR criterion linked closely to definition of spin squeezing infJxAinfJyA  JzA  . 1.4.4 Remark 1.4.4. We claim that any assumption of local realism is completely wrong. The three spin components of particle A were simultaneously predetermined, does not with absolute definiteness but only with uncertainties which required by Heisenberg spin uncertainty relations (1.4.5). Such claim meant as minimum the weak postulate of nonlocality. I.4.1. The weak postulate of nonlocality Heisenberg spin uncertainty relations The Heisenberg spin uncertainty relations JxAJyA  JzA  /2,JxAJzA  JyA  /2,JzAJyA  JxA  /2 1.4.5 does not violate in any cases: (i) if the three spin components of the particle A are measured directly by measurements performed on the particle A, (ii) and even if some spin components of the A particle are 61 measured indirectly as required in Bohm gedanken EPR experiment. Think of the following situation: a particle with zero spin decays into two particles (A and B), each with 1/2-spin. Due to the fact that spin angular momentum must be conserved during the decay, if initially the total spin angular momentum was zero, then after the decaying process it must still be zero. Therefore, particles A and B have opposite spin. Take as an example the dissociation of an exited hydrogen molecule into two hydrogen atoms. If the decaying mechanism does not change total angular momentum, then the spins on the hydrogen atoms will be anti-correlated. Remark 1.4.5. Whenever a measurement of the spin of A is found to be positive with respect of the z -axis (we shall note this state as |z, then, under local realism, we could infer that the spin of the B particle must be negative |z, and this is true independent of the distance that separates the particles. The spin of these particles are then entangled. Remark 1.4.6. We claim again that any assumption of local realism is completely wrong. I.4.2. The strong postulate of nonlocality Let | t A and | t B are states at instant t of the particle A and particle B correspondingly. Let |z,A/B be eigenstates of the spin operator SA/B z : SA/B z  1 0 0 1 1.4.6 We claim that: (i) whenever a measurement of the spin of a particle A is performed at instant t1  t and particle A is found in the state |z, i.e., a state | t1 A collapses at instant t1 to the state |z,A with respect of the 62 Heisenberg spin uncertainty relations (1.4.5), then a state | t1 B immediately collapses at instant t1 to the state |z,B with respect of the Heisenberg spin uncertainty relations (1.4.5), and this is true independent of the distance in Minkowski space-time that separates the particles: |t1 A col lapse  |z,A  |t1 B col lapse  |z,B 1.4.7 (ii) whenever a measurement of the spin of a particle A is performed at instant t1  t and particle A is found in the state |z, i.e., a state | t1 A collapses at instant t1 to the state |z,A with respect of the Heisenberg spin uncertainty relations (1.4.5), then a state | t1 B immediately | t B collapses at instant t1 to the state |z,B with respect of the Heisenberg spin uncertainty relations (1.4.5), and this is true independent of the distance in Minkowski space-time that separates the particles: |t1 A col lapse  |z,A  |t1 B col lapse  |z,B 1.4.8 Note that, we can not predict which spin will be positive (or negative) with respect of the z -axis, so the state that describes the spins of the particles could be for instance the spin singlet state |  1 2 |  | 1.4.9 We have 50% probability for the spin of particle A to be positive (and the spin of B negative) and 50% probability of it being the other way around. Remark 1.4.7. So far we have assumed that we are performing a measurement along the z -axis, but measurements are not restricted to this particular election, we could measure for instance the spin of 63 particle A along the a -axis and the spin of B along the b -axis. Let us see what happens if we decide to measure the spin along the x axis: a  b  x . As it known for 1/2-spins, the spin operator SA/B x can be represented by the 2  2 Hermitian matrix SA/B x  0 1 1 0 1.4.10 By performing a change of basis we can rewrite the state | in terms of the eigenstates of the spin operator SA/B x : |u  1 2 |  |, |v   1 2 |  |, 1.4.11 and using Eq. (1.4.10), we can rewrite the state | as |  1 2 |vu  |uv . 1.4.12 The strong postulate of nonlocality in this case takes the form similarly mentioned above. Just like before, by choosing to measure the spin of A along the x -axis we can determine its value and infer the value of the spin of particle B B  B in the state | x,B  |ux,B  |x,B without the need to measure it (and vice versa). Furthermore, it turns out that this is the case independent of the election of the axis we choose to measure! (Provided that a  b  v ). This is exactly the same situation such that a simple choice of the axis along which to measure the spin A allow us to establish the value of the spin of B along this same axis without the need to measure it. And this is also the case (as we already saw) for physical properties described by non-commuting operators (Sx and Sz do not commute). 64 Chapter II A NEW QUANTUM MECHANICAL FORMALISM BASED ON THE PROBABILITY REPRESENTATION OF QUANTUM STATES AND OBSERVABLES II.1. Generalized Postulates for Continuous Valued Observables Suppose we have an n -dimensional physical quantum system. I. Then we claim the following: Q.I.1. Any given n -dimensional quantum system is identified by a set Q : Q  H,,,2,1 ,H,G, |t  where: (i) H that is some infinite-dimensional complex Hilbert space, (ii)   ,,P that is complete probability space , (iii)   n , that is measurable space , (iv) 2,1 that is complete space of complex valued random variables X: Ω → Cn such that   XdP  ,   X2dP   2.1.1 (v) G : CHH 2,1 that is one to one correspondence such that 65  |Q|    G Q, |  dP  E G Q|  , G  1, |   1 2.1.2 for any |  H and for any Hermitian adjoint operator Q : H  H such that ?? ∈HC∗(H), where CH is C algebra of the Hermitian adjoint operators in H and H an commutative subalgebra of CH. (vi) | t  is a continuous vector function |t  :   H which represented the evolution of the quantum system Q. Q.I.2. For any |1 , |2   H and for any Hermitian operator Q : H  H such that 1 Q 2  2 Q 1  0 2.1.3 the equality holds G Q|1   |2    G Q|1    G Q|2  . 2.1.4 Definition 2.1.1. A random variable X :   E is a measurable function from the set of possible outcomes  to some E set. Definition 2.1.2. Given a probability space   ,,P and a measurable space   n , , any n -valued stochastic process that is a collection of n -valued random variables on  , indexed by a totally ordered set T ("time"). That is, a stochastic process X t is a collection Xt|t  T, where each X t is an n -valued random variable on . The space n is then called the state space of the process. Q.I.3. Suppose that the evolution of the quantum system is represented by continuous vector function |t  :   H. Then any process of continuous measurements on measuring observable Q 66 for the system in state | t  one can to describe by an continuous n valued stochastic process X t  X t ; Q t  X Qt  given on the probability space ,,P and the measurable space n , . Remark 2.1.1. We assume now for short but without loss of generality that n  1. Remark 2.1.2. Let X be random variable X  2,1 such that X  G|, then we denote such random variable by X |. The probability density of X | random variable we denote by p|q,q  . Definition 2.1.3. The classical pure states correspond to vectors v  H of v1 norm. Thus the set of all classical pure states corresponds to the unit sphere S  H in a H Hilbert space. Definition 2.1.4. The projective Hilbert space PH of a complex Hilbert space H is the set of equivalence classes v of vectors v in H, with v  0, for the equivalence relation given by v Pw  v w for some non-zero complex number   . The equivalence classes for the relation P are also called rays or projective rays. Remark 2.1.3. The physical significance of the projective Hilbert space PH is that in canonical quantum theory, the | and | states represent the same physical state of the quantum system, for any   0 . It is conventional to choose a state | from the ray | so that it has unit norm |  1. 67 Remark 2.1.4. In contrast with canonical quantum theory we have used instead contrary to P equivalence relation Q, a H Hilbert space, see Definition 2.1.7. Definition 2.1.5. The non-classical pure states correspond to the vectors v  H of a v1 norm. Thus the set of all non-classical pure states corresponds to the set H\S  H in the H Hilbert space. Suppose we have an observable Q of a quantum system that is found through an exhaustive series of measurements, to have a set  of values q   such that   i1 m 1 i ,2 i ,m  2, 1 i ,2 i   1 j ,2 j  , i  j. Note that in practice any observable Q is measured to an accuracy q determined by the measuring device. We represent now by |q the idealized state of the system in the limit q  0, for which the observable definitely has the value q. II. Then we claim the following: Q.II.1. The states |q : q   form a complete set of  -function normalized basis states for the state space H of the system. That the states |q : q   form a complete set of basis states means that any state |  H of the system can be expressed as: |    c qdq, 2.1.5 where supp cq   and while  -function normalized means that q|q   q  q from which follows cq  q| so that |    |qq|dq. 2.1.6 The completeness condition can then be written as 68   |qq|dq   1H . 2.1.7 Q.II.2. For the system in state | the probability Pq,q  dq; | of obtaining the result q   lying in the range q,q  dq   on measuring observable Q is given by Pq,q  dq; |  p | qdq 2.1.8 for any |  H . Remark 2.1.5. Note that in general case p|q  |cq| 2 . Q.II.3. The observable Q is represented by a Hermitian operator Q : H H whose eigenvalues are the possible results q : q  , of a measurement of Q , and the associated eigenstates are the states |q : q  , i.e. Q |q  q|q,q  . Remark 2.1.6. Note that the spectral decomposition of the operator Q is then Q   q|qq|dq. 2.1.9 Definition 2.1.6. A connected set in  is a set X   that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. Definition 2.1.7. The well localized pure states | with a support   1 ,2  correspond to vectors of the norm 1 and such that: supp cq   is a connected set in  . Thus the set of all well 69 localized pure states corresponds to the unit sphere S   S  H in the Hilbert space H  H. Suppose we have an observable Q of a system that is found through an exhaustive series of measurements, to have a continuous range of values q : 1  q  2 . III. Then we claim the following: Q.III.1. For the system in well localized pure state | such that: (i) |  S and (ii) supp cq  q|cq  0 is a connected set in , then the probability Pq,q  dq; | of obtaining the result q lying in the range q,q  dq on measuring observable Q is given by Pq,q  dq; |  |q||2dq  |cq| 2 dq. 2.1.10 Q.III.2. p|qdq  |q|| 2dq  |cq| 2 dq. Q.III.3. Let |1  and |2  be well localized pure states with 1  1 1 ,2 1  and 2  1 2 ,2 2  correspondingly. Let X1  X |1  and X2  X |2  correspondingly. Assume that 1 2   (here the closure of i, i  1,2 is denoted by i, i  1,2) then random variables X1 and X2 are independent. Q.III.4. If the system is in well localized pure state | the state | described by a wave function q,  q|| and the value of observable Q is measured once each on many identically prepared system, the average value of all the measurements will be 70 Q     q|q,|2dq   |q,|2dq . 2.1.11 The completeness condition can then be written as |qq|dq   1H . Completeness means that for any state |  S it must be the case that |q|| 2dq  0, i.e. there must be a non-zero probability to get some result on measuring observable Q. Q.III.5. (von Neumann measurement postulate) Assume that (i) |  S and (ii) supp cq   is a connected set in  . Then if on performing a measurement of Q with an accuracy q, the result is obtained in the q  1 2 q,q  1 2 q range, then the system will end up in the state Pq,q|  |Pq,q|   |qq  |q/2 |q q ||dq  |qq  |q/2 |q |||2dq . 2.1.12 IV. We claim the following: Q.IV.1. For the system in state |a  a|  H, where: (i) |  S, |a|  1, (ii) supp cq is a connected set in  and (iii) |  1 2 cq|qdq 71 G Q| a  |a|2G Q| . 2.1.13 Q.IV.2. Assume that the system in state |a  a|  H, where (i) |  S, |a|  1, (ii) supp cq is a connected set in  and (iii) |  1 2 cq|qdq. Then if the system is in state |a described by a wave function aq;  q||a and the value of observable Q is measured once each on many identically prepared system, the average value of all the measurements will be Q     q|aq;|2dq. 2.1.14 Q.IV.3. The probability Pq,q  dq; |adq of obtaining the result q lying in the range q,q  dq on measuring Q is Pq,q  dq; |adq  |a|2 cq|a|2  2 dq. 2.1.15 Remark 2.1.7. Note that Q.IV.3 immediately follows from Q.IV.1 and Q.III.2. Q.IV.4. (Generalized von Neumann measurement postulate) If on performing a measurement of observable Q with an accuracy q, the result is obtained in the q  1 2 q,q  1 2 q range, then the system immediately after measurement will end up in the state 72 Pq,q|a  |Pq,q|   |qq  |q/2 |q q ||adq   |qq  |q/2 |q |||2dq  a  |qq  |q/2 |q q ||dq  |qq  |q/2 |q |||2dq  H. 2.1.16 V. We claim the following: Q.V.1. Let |a1,a21 ,2   |1 a11   |2 a22   H1,2  H1 H2  H, where (i) |i a ii   ai|ii   Hi , |i   |ii   Si  , |ai |  1, i  1,2; (ii) supp ciiq, i  1,2 are the connected sets in ; (iii) suppc11 q  suppc22 q   and (iv) |ii   1 2 ciiq|qdq, i  1,2. Then if the system is in a state |a1,a21 ,2  described by a wave function a1,a2q;1 ,2   q||a1,a21 ,2 ,q  1 2 and the value of observable Q1,2 is measured once each on many identically prepared system, the average value of all the measurements will be Q1,2    12 q|a 1,a 2q;1 ,2 |2dq. 2.1.17 Q.V.2. The probability of getting a result q with an accuracy q such that q  1 2 q,q  1 2 q  suppc1q or q  1 2 q,q  1 2 q  suppc2q given by  |qq  |q/2 |q ||1 a 11 | 2  |q ||2 a 22 | 2 dq. 2.1.18 73 Remark 2.1.8. Note that Q.IV.3 immediately follows from Q.III.3. Q.V.3. Assume that the system is initially in the state |a1,a21 ,2 . If on performing a measurement of Q1,2 with an accuracy q, the result is obtained in the q  1 2 q,q  1 2 q range, then the state of the system immediately after measurement given by Pqi,q|a 1,a 2 1 ,2   |Pqi,q|   |q iq  |q/2 |q q  ||1 a 11   |q q ||2 a 2 2 dq  |q iq  |q/2 |q ||11 | 2  |q ||22 | 2 dq    |q iq  |q/2 a1 |q q  ||11   a2 |q  q ||22 dq  |q iq  |q/2 |q ||11 | 2  |q ||22 | 2 dq   H i , qi   i, i  1,2. 2.1.19 Definition 2.1.8. Let H1,2 be H1,2  H1  H2 . Definition 2.1.9. Let |a  be a state |a   a|, where |  S, |a|  1 and |  1 2 cq|qdq. Let |a  be an state such that |a   S. States |a  and |a  is a Q -equivalent: | a  Q |a  iff Pq,q  dq; |a   |a|2 cq|a|2  2 dq  Pqq  dq; |a dq 2.1.20 Q.V.4. For any state |a   a|, where |  S, |a|  1 and |   1 2 cq|qdq there exists a state |a   S such that: | a  Q |a . Definition 2.1.10. Let |a  be a state |a   a|, where |  S, |a|  1 and |  1 2 cq|qdq. Let |a  be a state such that |a   S. States |a  and |a  is a Q-equivalent ( | a   Q |a  ) iff: 74 a | Q|a   a | Q|a . Q.VI. For any state |a   a|, where |  S, |a|  1 and |   1 2 cq|qdq there exists a state |a   S such that: |a   Q |a  II.2. The nonclassical collapse models with spontaneous localizations based on generalized measurement postulates The nonclassical collapse models attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and nonlocal stochastic modifications of the Schrödinger equation. The proposed new nonlocal dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wave packet reduction and for the classical behavior of macroscopic systems. II.2.1. Quantum Mechanics with Nonclassical Spontaneous Localizations Quantum Mechanics with Nonclassical Spontaneous Localizations is based on the following assumptions: (1) Each particle of a system of n distinguishable particles experiences, with a mean rate λi, a sudden spontaneous localization process. (2) In the time interval between two successive spontaneous 75 processes the system evolves according to the usual Schrödinger equation. (3) Let |cl be the classical pure state correspond to a vector |cl  S  H in a nonprojective Hilbert space H, see Subsection II.1, Def. 2.1.1-2.1.2. Then the sudden spontaneous process is a localization given by: |cl ,-localization  ,,x i  cl ,,x i  cl ,x  3 ,  0,1,  1, 2.2.1 where ,,x i  cl   Lx i ,|cl. 2.2.2 Here  Lx i , is a norm-reducing, positive, self-adjoint, linear operator with a symbol Lxi , in the n -particle nonprojective H Hilbert space, representing the localization of particle i around the point x. Definition 2.2.1. Such localization as mentioned above is called ,localization or , -collapse of the state |cl. (4) The probability density pix,, for the occurrence of the localization at point x is assumed to be pix,,  ,,x i  cl 2  3 ,,x i  cl 2 d3x . 2.2.3 (5) Let |n.cl be the nonclassical pure state correspond to a vector |   |  H\S, where |  S, ||  1, see subsection II.1, Def. 2.1.10. Then the sudden spontaneous process is a localization given by: 76 |n.cl ,localization   ,,x i  n.cl ,,x i  n.cl ,x  3 , 2.2.4 where ,,x i  n.cl   Lx i ,|n.cl. 2.2.5 Definition 2.2.2. Such localization as mentioned above is called ,localization or ,-collapse of the state |n.cl. (6) The probability density pix,,,,  for the occurrence of the localization at point x  3 in accordance to postulate Q.IV.3 (see Subsection II.1, Eq.(2.1.15)) is assumed to be pix,,,,   ||6 ,,| |2x i n.cl 2  3 ,,x i  cl 2 d3x . 2.2.6 (7) The localization operators  Lx i , have been chosen to have the form:  Lx i ,  1  3/4 exp  1 2 q i  x2 iff q i  x    1, 0 iff q i  x  . 2.2.7 Here   0,1  d3xLxi , 2  1 and lim0   . Remark 2.2.1. In one dimension case it follows that  Lx i ,  1  1/4 exp  1 2 q i  x2 iff |qi  x |    1, 0 iff |qi  x |  . 2.2.8 Remark 2.2.2. Note that from Eq. (2.2.3) and Eq. (2.2.7) it follows that a probability density pix,,,,  for the occurrence of the 77 localization inside sphere Sx,  qi   3 |q i  x   is given by pix,,  ,,x i  cl 2 , ,,   3 ,,x i  cl 2 d3x, ,,x i  cl 2  1  3/2  q ix d3qi iqi exp  1  q i  x2 ,  iqi   qi || i cl, 2.2.9 and therefore pix,  0 lim pix,,  0 lim 1, 1  3/2  qix d3qi iqi exp  1  q i  x2   ix. 2.2.10 Remark 2.2.3. In one dimension case it follows that a probability density pix,,,  for the occurrence of the localization inside interval x  ,x   is given by pix,,  ,,x i  cl 2  1  1/2  |q ix | d3qi iqi exp  1 qi  x2 ,  iqi   qi || i cl, 2.2.11 and therefore pix,  0 lim pix,,   0 lim1, 1  1/2  |q ix | dqi iqi exp  1 qi  x2   ix. 2.2.12 II.2.2. The generalization of nonclassical collapse models (1) Let |t cl, t  0,T be the classical pure states correspond to a vector-function |t cl : 0,T  S   S such that 78 |t cl S   H , t  0,T, where is a nonprojective H Hilbert space, see Subsection II.1, Def.2.1.1-2.1.2. Then the sudden spontaneous process is the localization along classical trajectory xt : 0,T  3  3 given by: | t cl ,,x tlocalization   t,,,x t i  cl  t,,,x t i  cl ,   0,1,  1,xt  3 , t  0,T, 2.2.13 where t,,,x t i  cl   Lx t i ,|t cl. 2.2.14 Here  Lx t i , is a norm-reducing, positive, self-adjoint, linear operator with a symbol Lx t i , in the n -particle nonprojective H Hilbert space, representing the localization of particle i at each instant t  0,T around the point x t. Definition 2.2.3. Such localization as mentioned above is called ,,xt -localization or ,,xt -collapse of the state | t cl. (2) The probability density pit,xt,, for the occurrence of the localization at point x t at instant t is assumed to be pit,xt,,   t,,,x t i  cl 2 t,, ,t,,   3 ,,x t i  cl 2 d3x. 2.2.15 (3) Let | t n.cl be the nonclassical pure state corresponding to a vector-function t   |t   H\S, where | t   S, ||  1, t  0,T , see Subsection II.1, Def.2.1.10. 79 Then the sudden spontaneous process is the localization along classical trajectory xt : 0,T  3  3 given by: | t n.cl ,,x tlocalization    t,,,x t i  n.cl  t,,,x t i  n.cl , xt 3 , t  0,T 2.2.16 where t,,,x t i  n.cl   Lx t i ,|t n.cl. 2.2.17 Definition 2.2.4. Such localization as mentioned above is called ,,xt -localization or ,,xt -collapse of the state |n.cl. (4) The probability density pit,xt,,,,  for the occurrence of the localization at point x t  3 at instant t  0,T in accordance to postulate Q.IV.3 (see Subsection II.1, Eq. (2.1.14)) is assumed to be pit,xt,,,,   ||6  t,,,| |2x t i n.cl 2 t,, ,t,,   3 ,,x t i  cl 2 d3x. 2.2.18 (5) The localization operators  Lx t i , have been chosen to have the form:  Lx t i ,  1  3/4 exp  1 2 q i  xt 2 iff qi  xt    1, 0 iff q i  xt  . 1.2.19 Here   0,1 and lim0   . Remark 2.2.4. In one dimension case it follows that  Lx t i ,  1  1/4 exp  1 2 q i  x t 2 iff |qi  x t |    1, 0 iff |qi  x t |  . 2.2.20 80 Remark 2.2.5. Note that from Eq. (2.2.18) and Eq. (2.2.19) it follows that a probability density pit,xt,,,,  for the occurrence of the localization at instant t inside sphere Sxt,  qi   3 |q i  xt   is given by pit,xt,,   t,,,x t i  cl 2 t,,  t,,,x t i  cl 2  1  3/2  q ix t d3qi t iq i exp  1  q i  xt 2 ,  t iqi   qi |  t i  cl , 2.2.21 and therefore pit,x,  0 lim pit,x,,   0 lim1t,, 1  1/2  |q ix | dqi iqi exp  1 qi  x t 2   ix t . 2.2.22 81 Chapter III III. EINSTEIN'S 1927 GEDANKEN EXPERIMENT REVISITED III.1. Single-photon space-like antibunching During the famous 5-th Solvay conference in 1927, Einstein [7] considered a single particle which, after diffraction in a pin-hole encounters a detection plate (e.g. in the case of photons, a photographic plate), see Fig 3.1.1. We simplify this thought experiment, though keeping the essence, by replacing the detection plate by two detectors. Einstein noted that there is no a question that only one of them can detect the particle, otherwise energy would not be conserved. However, he was deeply concerned about the situation in which the two detectors are spacelike separated, as this prevents according to relativity any possible coordination among the detectors: It seems to me, Einstein continued, that this difficulty cannot be overcome unless the description of the process in terms of the Schrödinger wave is supplemented by some detailed specification of the localization of the particle during its propagation. I think M. de Broglie is right in searching in this direction. But what happened to Einstein's original Gedanken experiment? This simple with today's technology experiment had been done originally by T. Guerreiro, B. Sanguinetti, H. Zbinden N. Gisin, and A. Suarez, see [21]. This experiment consists in verifying that when a single photon is thrown at a beam splitter, it is detected in only one arm, i.e. the probability PAB of getting a coincidence between the two detectors A and B is much smaller than the product of the probabilities of detection on each side PA  PB, as would be expected in the case of uncorrelated events. 82 Fig. 3.1.1. Einstein's 1927 gedanken experiment. A and B are points on the photographic plate, for which the events of detection can be space-like separated from each other. Adapted from [7] The experimental setup is shown in Fig. 3.1.2 and consists of a source of heralded single photons which is coupled into a single mode fiber and injected into a fiber beam splitter (BS). Each of the two outputs of the beam splitter goes to a single photon detector (IDQ ID200), A detector being close to the source and B detector being separated by a distance of approximately 10 meters. Fig. 3.1.2. Experimental setup: photon pairs regenerated by Spontaneous Parametric Down Conversion at the wavelengths of 1550 nm and 810 nm. These pairs are splitted by a dichroic mirror (DM), and 810 nm photon is sent to detector D, used to herald the presence of the 1550 nm photon that follows to the beam splitter (BS). Arbitrary electronic delays were applied before TDC to ensure the coincidence peak would remain on scale. Adapted from [21] 83 If we ensure that the fiber lengths before each detector are equal by inserting a 10 m (50 ns) fiber delay loop before detector A, the detections will happen simultaneously in some reference frame, thus being space-like separated (a signal would take 33 ns to travel between the two detectors at the speed of light; simultaneity of detection is guaranteed to within 1ns by the matched length of fiber both before and inside the detectors). It is also possible to make the detections time-like separated by removing the 10m delay line from detector A and adding it to detector B. Fig. 3.1.3. Spacetime diagrams for spacelike (i) and timelike (ii) configurations. A and B represent the locations of the detectors. Adapted from [21] First, one measures the probabilities of detecting a photon at detector A or at detector B given that a heralding photon has been detected at H. We denote RHA the total number of coincident counts at detector H and detector A during the time of measurement, and RHA the total number of counts at detector H alone during the same measurement; RHB and RHB denote similar quantities for the measurement with H and B. Next we measure the probability of detectors A and B clicking at the same time, again given a heralding signal. RHAB denotes the number 84 of triple coincident counts at the detectors H, A and B, and RHAB denotes the total number of counts at detector H alone during the same measurement. All these quantities are measured directly for both a space-like configuration and a time-like configuration. Fig. 3.1.4. Coincidences between the heralding detector and each of the detectors A (red) and B (blue) with spacelike separation, measured in a window of 1 ns during a time period of 10 minutes. RHA=9.49×104/10 min, RHB=6.39×104/10 min. The noise is on average: RN=50/10 min. Adapted from [21] Fig. 3.1.5. Coincidences between the heralding detector and each of the detectors A (red) and B (blue) with timelike separation, measured in a window of 1 ns during a time period of 10 minutes. RHA=9.90×104/10 min, RHB=6.22×104/10 min. Adapted from [21] 85 The raw TDC data is shown in Figures 3.1.4-3.1.5 and the results are summarized in Table 3.1.1. Table 3.1.1. Summary of results. Values obtained for the different counting rates and corresponding probabilities defined in the text, measured with spacelike and timelike separation. Adapted from [21] The number of counts given by detector noise and two-photon events can be estimated by looking at the counts away from the peak. As an example, for the space-like configuration (Figure 3.1.4) in a window of 1 ns the noise rate is on average RHN = 50/10 for a 10 minutes integration time [21]. This corresponds to a noise probability PN = 9*10 6 (1.310 6 ). From the values in Table 1 one derives the following probability values for spacelike separation: PA SL  PB SL  1.86  0.01*104 , PAB SL  0.002  0.001*104 . 3.1.1 For timelike separation one derives the values: 86 PA TL*PB TL  1.65  0.01*104 , PAB TL  0.002  0.001*104 . 3.1.2 For the probability PN SL that A and B detect photons coming from different pairs (noise) one derives the value: PN SL1,1  PN SL*PA SL  PN SL*PB SL  0.0025  0.0026*104 3.1.3 III.2. The measure algebra of physical events in Minkowski space-time Definition 3.2.1. [22]. A measure algebra   B,P with a probability measure P, is a Boolean algebra B with a countably additive probability measure. Definition 3.2.2. (i) A measure algebra of physical events ph  B,P with a probability measure P, is a Boolean algebra of physical events B with a countably additive probability measure. (ii) A Boolean algebra of physical events can be formally defined as a set B of elements a,b, . . . with the following properties: 1. B has two binary operations,  (logical AND, or "wedge") and  (logical OR, or "vee"), which satisfy: the idempotent laws:(1) a  a  a  a  a, the commutative laws:(2) a  b  b  a, (3) a  b  b  a, and the associative laws: (4) a  b  c  a  b  c, (5) a  b  c  a  b  c. 2. The operations satisfy the absorption law: (6) a  a  b  a  a  b  a. 3. The operations are mutually distributive 87 (7) a  b  c  a  b  a  c, (8) a  b  c  a  b  a  c. 4. B contains universal bounds 0 and 1 which satisfy (9) 0  a  0 , (10) 0  a  a , (11) 1  a  a , (12) 1  a  1. 5. B has an unary operation a (or a  ) of complementation (logical negation), which obeys the laws: (13) a  a  0, (14) a  a  1. All properties of negation including the laws below follow from the above two laws alone. 6. Double negation law: a  a . 7. De Morgan's laws: (i) a  b  a  b, (ii) a  b  a  b. 8. Operations composed from the basic operations include the following important examples: The first operation, a  b (logical material implication): (i) a  b  a  b. The second operation, a  b, is called exclusive. It excludes the possibility of both a and b (ii) a b  a  b  a  b. The third operation, the complement of exclusive or, is equivalence or Boolean equality: (iii) a  b  a  b . 9. B has a unary predicate Occa , which meant that event a has occurred, and which obeys the laws: (i) Occa  b  Occa  Occb, (ii) Occa  b  Occa  Occb, (iii) Occa  Occa. 88 Remark 3.2.1. A probability measure P on a measure space , gives the probability measure algebra   ,,P on the Boolean algebra of measurable sets modulo null sets. Definition 3.2.3. (i) Let B be a Boolean algebra of physical events. A Boolean algebra BM4 of physical events in Minkowski space-time M4  1,3 that is Cartesian product BM4  B M4 . (ii) Let BM4 be a Boolean algebra of physical events in Minkowski space-time. A measure algebra of physical events M4 ph  BM4 ,P in Minkowski space-time that is a Boolean algebra BM4 with a probability measure P. (iii) Let BM4 be Boolean algebra of the all physical events in Minkowski space-time and let M4 ph be a measure algebra M4 ph  BM4 ,P with a probability measure P. We denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc. (iv) We will write for a short AOcx,BOcx, . . . etc., instead OccAx,OccBx, . . . etc. Definition 3.2.4. Let AutPBM4  be a set of the all measurepreserving automorphism of BM4 . This is a group, being a subgroup of the group AutBM4  of all Boolean automorphism of BM4 . Let P   be Poincaré group. Remark 3.2.2. We assume now that: any element   ,a  P  induced an element   AutPBM4  by formula   Ax  Ax  a  BM4 . Definition 3.2.5. Given two events A and B from the algebra 89 M4 ph  BM4 ,P the conditional probability of A given B is defined as the quotient of the probability of the joint of events A and B, and the probability of B : PA|B  PA  B PB  PAB PB  PA|B , 3.2.1 where PB  0. Definition 3.2.6. (i) Events A and B from the algebra M4 ph  BM4 ,P are defined to be statistically independent or uncorrelated iff PAB  PA  PB, 3.2.2 where PB  0, then this is equivalent to the statement that PA|B  PA. Similarly if PA is not zero, then PB|A  PB is also equivalent. (ii) Events A and B from the algebra   BM4 ,P are defined to be statistically almost independent or almost uncorrelated iff PAB  PA  PB , PAB  PA  PB  A,B, 0  A,B  PA  PB . 3.2.3 Remark 3.2.3. Note that PAB  PA  PB  PAB. 3.2.4 Although mathematically equivalent, this may be preferred philosophically; under major probability interpretations such as the subjective theory, conditional probability is considered a primitive entity. Further, this "multiplication axiom" introduces the symmetry with the summation axiom for mutually exclusive events, i.e. PAB  PA  PB  P AB. 3.2.5 90 Definition 3.2.7. (i) Events A1 ,A2 , . . . ,An  M4 ph  BM4 ,P are said to be exactly mutually exclusive if the occurrence of any one of them implies the non-occurrence of the remaining n  1 events. Therefore, two mutually exclusive events cannot both occur. Formally said, the conjunction of each two of them is 0 (the null event): A  B  0 . In consequence, exactly mutually exclusive events A and B have the property: (ii) Events A1 ,A2 , . . . ,An  M4 ph  BM4 ,P are said to be almost mutually exclusive if A1 ,A2 , . . . ,An have the property: PA1  A2 . . .An   0, PA1  A2 . . .An   PA1   PA2     PAn . 3.2.7 In consequence, almost mutually exclusive events A and B have the property: PA  B  0, PA  B  PA  PB. 3.2.8 Remark 3.2.4. Let Aph ,Bph be events such that detectors A, B detect photon at an instants t1 and t2 correspondingly. Note that (3.1.1) and (3.1.2) show that whether the separation between the detectors is timelike or spacelike, the number of coincidences is three orders of magnitude smaller than what would be expected had the events been statistically almost uncorrelated, i.e., PAB  PA  PB, see Def.3.2.6 (ii). Remark 3.2.5. Let Aph ,Bph be events such that detectors A, B detect photon at an instants t1 and t2 correspondingly. Note that: (i) from Eq. (3.1.1) probability value for spacelike separation follows: PA  B  0. 3.2.6 91 P AphBph SL  0.002  0.001*104  0, 3.2.9 (ii) from Eq. (3.1.2) probability value for timelike separation follows: P AphBph TL  0.002  0.001*104  0. 3.2.10 Therefore in both cases the property (3.2.6) are violated, i.e. PAphBph  0 but however in both cases the property (3.2.8) is satisfied 0.002  0.001*104  P AphBph SL  P Aph SL  P Bph SL  1.86  0.01*104 , 0.002  0.001*104  P AphBph TL  P Aph TL  P Bph TL  1.65  0.01*104 3.2.11 and therefore in both cases the events Aph ,Bph are almost mutually exclusive events. III.2.1. Beamsplitter transformation A beamsplitter is the most simple way to mix two modes, see Fig.3.2.1. From classical electrodynamics, one gets the following amplitudes for the outgoing modes: a1 a2 in  A1 A2 out  t r r t  a1 a2 in . 3.2.12 Fig. 3.2.1. Mixing of two models by a beamsplitter 92 The recipe for quantization is now: `replace the classical amplitudes by annihilation operators'. If the outgoing modes are still to be useful for the quantum theory, they have to satisfy the commutation relations: Aiout,Ajout  ij 3.2.13 These conditions give constraints on the reflection and transmission amplitudes, for example |t |2  |r|2  1. We are now looking for an unitary operator S [the S -matrix] that implements this beamsplitter transformation in the following sense: Ai  SaiS, i  1,2. 3.2.14 Let us start from the general transformation (summation over double indices) ai  Ai  Bijai, a  A  Ba 3.2.15 where we have introduced the matrix and vector notation. Using this S -matrix one can also compute the transformation of the states: |out  S|in. For the unitary transformation, we make the ansatz S  expiJkiak ai 3.2.16 with Jkl a Hermitian matrix (ensuring unitarity). The action of this unitary on the photon mode operators is now required to reduce to ai  Ai  SaiS Bijaj. 3.2.17 93 We compute this `operator conjugation' by using a differential equation: dA i d  iJkiA i. 3.2.18 This is a system of linear differential equations with constant coefficients, so that one obtains a solution A  expiθJ. 3.2.19 We thus conclude that the so-called generator J of the beam splitter matrix is fixed by equation B  expiθJ. 3.2.20 If the transformation B is part of a continuous group and depends on as a parameter, we can expand it around unity. Doing the same for the matrix exponential, we get B  1  iθJ . . . . 3.2.21 Equation (3.2.21) explains the name generator for the J matrix: it actually generates a subgroup of matrices B  Bθ parametrized by the angle. The unitary transformation we are looking for is thus determined via the same J generator. For the two-mode beam splitter, an admissible transformation is given by Bθ  t r r t   cos isin isin cos 3.2.22 The factor i is just put for convenience so that the reflection amplitudes are the same for both sides, r  r, as expected by θ θ 94 symmetry. Expanding for small , the generator is J  0 1 1 0  1 3.2.23 and so that the unitary operator for this beamsplitter is S  expia1 a2  a2 a1 . 3.2.24 herefore, the effective Hamiltonian of the beam splitter is given by Heff  a1  a2  a2  a1 . 3.2.25 III.2.2. Splitting a two-photon state Let us consider two single photon states |in  |1,1 incident on the beam splitter such that mentioned above. Then |  |out  S|in  Sa1  SSa2  SS|0, 0  a1  cosθ  ia2  sinθa2  cosθ  ia1  sinθ|0,0  |2,0  |0,2 sinθ 2  |1,1cosθ. 3.2.26 Let H be a complex Hilbert space such that |cl  H,   0,1  0,1 ,,xi cl  H , ,,x i  cl  Lxi ,|cl. 3.2.27 By postulate Q.I.1 (see section II.1) quantum system with Hamiltonian given by Eq. (3.2.25) is identified with a set θ 95   H,Heff ,,,2,1,G, |t  , where (i) H that is a complex Hilbert space defined above, (ii)   ,,P that is complete probability space, (iii)   n , that is measurable space , (iv) 2,1 that is complete space of random variables X :   n such that   XdP  ,   X2dP  , 3.2.28 (v) G : H  2,1 that is one to one correspondence such that |Q|    G Q|  dP  E G Q|  3.2.29 for any |  H and for any Hermitian operator Q : H  H, (vi) | t  is a continuous vector function |t  :   H which represented the canonical evolution of the quantum system . Remark 3.2.6. Note that M4 ph    M4  ,,P  M4 , where  is a probability measure algebra   ,,P on the Boolean algebra of measurable sets modulo null sets, see Remark 3.2.1. Let BM4 be Boolean algebra of the all physical events in Minkowski space-time M4 and let M4 be a measure algebra M4  BM4 ,  P with a probability measure  P, see Definition 3.2.2 (7). We assume now that there exists subalgebra M4 #  M4 and isomorphism  : M4 #  M4 ph such that for any event Ax  M4 # ,x  t,x1 ,x2 ,x3   M4 (see Definition 3.2.2): 96 Ax  Ax,  PAx  PAx  PAx. 3.2.30 Proposition 3.2.1. Suppose that A and B are events in measure algebra M4  BM4 ,  P . Then following properties is satisfied: 1.PA|B  PA  PB|A  PB  PA  B  PAPB 2.PA|B  PA  PB|A  PB  PA  B  PAPB 3.PA|B  PA  PB|A  PB  PA  B  PAPB 3.2.31 Proposition 3.2.2. Suppose that A and B are events in measure algebra M4  ,,P. Then following properties is satisfied: 1.PA|B  PA  PB|A  PB  PA  B  PAPB 2.PA|B  PA  PB|A  PB  PA  B  PAPB 3.PA|B  PA  PB|A  PB  PA  B  PAPB 3.2.32 Definition 3.2.8. In case (1), A and B are said to be positively correlated. Intuitively, the occurrence of either event means that the other event is more likely. In case (2), A and B are said to be negatively correlated. Intuitively, the occurrence of either event means that the other event is less likely. In case (3), A and B are said to be uncorrelated or independent. Intuitively, the occurrence of either event does not change the probability of the other event. Remark 3.2.7. Suppose that A and B are events in measure algebra 97 M4  ,,P. Note from the result above that if A  B or B  A then A and B are positively correlated. If A and B are disjoint then A and B are negatively correlated. Proposition 3.2.3. Suppose that A and B are events in measure algebra M4  BM4 ,  P . Then: (i) A and B have the same correlation (positive, negative, or zero) as A and B. (ii) A and B have the opposite correlation as A and B (that is, positive-negative, negative-positive, or zero-zero). Proposition 3.2.4. Suppose that A and B are events in measure algebra M4  ,,P. Then: (i) A and B have the same correlation (positive, negative, or zero) as Ac and Bc . (ii) A and B have the opposite correlation as A and Bc (that is, positive-negative, negative-positive, or zero-zero). Definition 3.2.9. Let Ax1   At1 ,r1  and Bx2   Bt2 ,r2  be the events Ax1   M4 # which occur at instant t1 and Bx2   M4 # at instant t2 correspondingly. Let x1,2 be a vector: x1,2  ct1  t2 ,r1  r2  ct1,2 ,r1,2 , t1,2  t1  t2 ,r1,2  r1  r2 . Vectors x1,2  ct1,2 ,r1,2  are classified according to the sign of c2 t1,2 2  r1,2 2 . A vector is (i) timelike if c 2 t1,2 2  r1,2 2 , (ii) spacelike if c 2 t1,2 2  r1,2 2 , and null or lightlike if 98 (iii) c 2 t1,2 2  r1,2 2 . Pairs of events At1 ,r1 ,Bt2 ,r2   M4 # M4 # are classified according to the sign of c 2 t1,2 2  r1,2 2 : (i) a pair At1 ,r1 ,Bt2 ,r2  is timelike separated if c 2 t1,2 2  r1,2 2 , and we denoted such pairs by At1 ,r1 ,Bt2 ,r2 t.l .s. ; (ii) a pair At1 ,r1 ,Bt2 ,r2  is spacelike separated if c 2 t1,2 2  r1,2 2 ; and we denoted such pairs by At1 ,r1 ,Bt2 ,r2 s.l .s. ; (iii) a pair At1 ,r1 ,Bt2 ,r2  is null or lightlike separated if c 2 t1,2 2  r1,2 2 , and we denoted such pairs by At1 ,r1 ,Bt2 ,r2 l .l .s. . Definition 3.2.10. (i) Let M4 # ,t1 ,r1 ,t2 ,r2  t.l .s. be a set of the all timelike separated pairs At1 ,r1 ,Bt2 ,r2 t.l .s. which are corresponding to a given vector t1 ,r1 ,t2 ,r2   M4  M4 , i.e., M4 # ,t1 ,r1 , t2 ,r2  s.l .s.  At1 ,r1 ,Bt2 ,r2   M4 # M4 # c2 t1,2 2  r1,2 2 . 3.2.33a (ii) Let M4 # ,t1 ,r1 ,t2 ,r2  s.l .s. be a set of the all spacelike separated pairs At1 ,r1 ,Bt2 ,r2 s.l .s. which is corresponding to a given vector t1 ,r1 ,t2 ,r2   M4  M4 , i.e., M4 # ,t1 ,r1 , t2 ,r2  s.l .s.  At1 ,r1 ,Bt2 ,r2   M4 # M4 # c2 t1,2 2  r1,2 2 . 3.2.33b Remark 3.2.8. Let M4 # ,t,r1 ,t,r2  s.l .s. be a set of all pairs At,r1 ,Bt,r2  , which corresponds to a given vector t,r1 ,t,r2   M4  M4 , r1  r2 , i.e., 99 M4 # ,t,r1 , t,r2  s.l .s.  At,r1 ,Bt,r2   M4 # M4 # 0  r1,2 2 , r1,2  r1  r2 . 3.2.34 Such pairs obviously are spacelike separated. Note that tr1r2r1  r2  M4 # ,t,r1 , t,r2  s.l .s.   . 3.2.35 Definition 3.2.11. Let At1  Ax1   At1 ,xA  and Bt2  Bx2   Bt2 ,xB  be a symbols such that A t1 and Bt2 represent there is detection events Ax1   M4 # at instant t1 and Bx2   M4 # at instant t2 correspondingly, where symbols x A and x B represent the locations of the detectors A and B correspondingly (see Fig. 3.1.3). We assume that At1 ,Bt2   M4 # ,t1 ,xA ,t2 ,xB  s.l .s. . 3.2.36 Remark 3.2.9. We assume now without loss of generality that t1  t2  t, note that such assumption valid by properties: Ax1   M4 # and Bx2   M4 # , required above, see Remark 3.2.2. III.3. Einstein's 1927 gedanken experiment explained In classical case considered by A. Einstein in his 1927 gedanken experiment, by postulates of canonical QM, both events A t  M4 # and B t  M4 # cannot occur simultaneously, i.e. that is mutually exclusive events with a probability  1, and therefore At  Bt  0. Such exactly mutually exclusive events have the property:  PAt  Bt  0, 3.3.1 100 see Definition 3.2.6. We remind that the probability density pphx,, for the occurrence of a photon localization at point x is assumed to be pph x,,  ,,x ph  cl 2 ,   0,1,  0,1, 3.3.2 where ,,x ph cl  Lx, ph  cl . 3.3.3 and where the localization operators Lx, have been chosen to have the form:  Lx  q,,  1  1/4 exp  1 2 q  x2 iff |q  x |    1, 0 iff |  q  x |  . 3.3.4 see subsection II.2.1. Remark 3.3.1. Note that: (i) from (3.2.27) it follows that ,,x ph  cl  H, (ii) from (3.3.3) and (3.3.4) where   1 it follows that pph x,,  ,,x ph  cl 2   dq ,,xph  |qq| ,,xph    dq Lxq,,ph  |qq| Lxq,,ph    dqLx2q,,ph  |qq| ph   x | ph cl 2  O   x | ph  cl 2 ,   1,  0,1, 3.3.5 101 It follows from postulate Q.I.3 that there exists unique random variable X ;  ph cl given on a probability space ,,P and a measurable space n , by formula X ; ph cl  X ph   G  ph cl 3.3.6 The probability density of random variable X ph  we denote by p ph q,q  . Remark 3.3.2. From postulate Q.II.2 (see subsection II.1) it follows that for the system in state  ph cl the probability P q,q  dq; ph cl of obtaining the result q lying in the range q,q  dq on measuring observable  q is given by P q,q  dq; ph cl  p ph cl qdq  c phcl q 2  q ph cl 2 3.3.7 Now we go to explain Einstein's 1927 gedanken experiment. Let Apht,xA  and Bpht,xA  be events such that detectors A, B detect photon at an instant t correspondingly. By properties (3.2.31) we obtain P A pht,x A   PAph t,x A    PApht,x A , P B pht,x B   PBph t,x B    PBpht,x B . 3.3.8 Note that A t  A pht,x A    |x A    X ph   x A   , Bt  B pht,x B    |x B    X ph   x B   ,   0,,  1, 3.3.9 102 where a small parameter   |xA  xB | is dependent on the measuring device. Thus by general definition of random variable one obtains directly A pht,xA   B pht,xB    3.3.10 and therefore P A pht,xA   B pht,xB   0 3.3.11 The property (3.3.11) follows directly from (3.3.8). Fig. 3.3.1. The plot of the random variable X ph . A t  A pht,x A ,Bt  B pht,x B ,A t  Bt   103 Remark 3.3.3. Let M4 # ,t,xA ,t,xB  s.l .s. be a set of the all pairs At,xA ,Bt,xB  which is corresponding to a given vector t,xA,0,0,t,xB,0,0  M4  M4 , xA  xB, i.e., M4 # ,t,x A , t,x B  s.l .s.  At,x A ,Bt,x B   M4 # M4 # 0  x A  x B 2 . 3.3.12 Such pairs obviously are spacelike separated. Note that txAxBxA  xB  M4 # ,t,xA , t,xB  s.l .s.   . 3.3.13 Now we will go to explain non zero result  PAt  Bt  0 given above by (3.1.1) and (3.1.2):  PA tB t TL  0.002  0.001*104 ,PA t TL*PB t TL  1.65  0.01*104 ,  PA tB t SL  0.002  0.001*104 .PA tB t SL  0.002  0.001*104 . 3.3.14 We consider this problem in general case. Remark 3.3.4. Note that: (i) a probability density px,A,,  for the occurrence of the localization inside interval x  ,x   in arm with detector A (see Fig.3.1.2.) is given by formula px,A ,  |A,,x cl 2 A , , 3.3.15 where 104 |A,,x cl 2  1 AA 1/2  |qx | dq|q|2 exp  1 A q  x2 , q  q||, A ,     |A,,x cl 2 dx, 3.3.16 and where parameter A depends on arm with detector A. (ii) a probability density px,B,,  for the occurrence of the localization inside interval x  ,x   in arm with detector B (see Fig.3.1.2) is given by formula px,B ,  |B,,x cl 2 B , , 3.3.17 where and where parameter B depends on arm with detector B. Remark 3.3.5. Note that parameter  in formula (3.3.18) of course depends on measurement device and there no exist two equivalent devices such that A  B. We assume now that A  B  1, 0  |A  B |,    |x|2  dx  , 3.3.19 |B,,x cl 2  1 BB 1/2  |qx | dq|q|2 exp  1 B q  x2 , q  q||, B ,     |B,,x cl 2 dx, 3.3.18 105 From Eq. (3.3.16) and Eq. (3.3.19) by using Laplace approximation, we obtain: From Eq. (3.3.18) and Eq. (3.3.19) by using Laplace approximation, we obtain: |B,,x cl 2  1 BB 1/2  |qx | dq|q|2 exp  1 B q  x2  |x|2  BO |x|2   |x|2  Bc1 B |x|2  , B ,     |B,,x cl 2 dx  1  c2 BB ,c2 B  O    |x|2  dx . 3.3.21 From Eq. (3.3.15) and Eq. (3.3.17) we obtain px,A ,  |A,,x cl 2 A ,  |x|2  Ac1 A |x|2  1  c2 AA , . 3.3.22a From Eq. (2.2.54) and Eq. (2.2.57) we obtain px,B ,  |B,,x cl 2 B ,  |x|2  Bc1 B |x|2  1  c2 BB , . 3.3.22b Definition 3.3.1. We define now the probability measures P|A,,x A t and P|B,,x A t by formulae |A,,x cl 2  1 AA 1/2  |qx | dq|q|2 exp  1 A q  x2   |x|2  AO |x|2   |x|2  Ac1 A |x|2  , A ,     |A,,x cl 2 dx  1  c2 AA ,c2 A  O    |x|2  dx . 3.3.20 106 P|A,,x A t    A t px,A ,dx, P|B,,x A t    A t px,B ,dx, 3.3.23 where At  a,b and dx is the Lebesgue measure and a,b  Ba,b is the Borel algebra on a set a,b. Definition 3.3.2. We assume now that P A,,x  P and P B,,x  P, i.e. P | is absolutely continuous with respect to P . By RadonNicodym theorem we obtain for any At  a,b : P A,,x A t    A t X A,,x dP, X A,,x   dP A,,x dP , P A,,x A t    A t X A,,x dP, X A,,x   dP A,,x dP . 3.3.24 We write below for a short X 1  X A,,x ,X 2  X B,,x . 3.3.25 Remark 3.3.6. We assume now without loss of generality that X 2  X 1  0 a.s. 3.3.26 see Fig . 3.3.1. Let us consider now the quantity 107 1,2    |X1  X2|dP    X 2  X 1dP. 3.3.27 We assume now that    x|x|2dx  ,   x |x|2  dx  , 3.3.28 From Eq. (3.3.27) by using Eq. (3.3.21) and Eq. (3.3.22) we obtain 1,2    xpx,B ,dx    xpx,A ,dx  1 1  c2 BB   x |x|2  Bc1 B |x|2  dx   1 1  c2 AA   x |x|2  Ac1 A |x|2  dx  1  c2 BB    x |x|2  Bc1 B |x|2  dx  1  c2 AA    x |x|2  Ac1 A |x|2  dx  Bc1 B   x |x|2  dx  c2 BB   x|x|2dx  B 2 c1 Bc2 B   x |x|2  dx  Ac1 A   x |x|2  dx  c2 AA   x|x|2dx  A 2 c1 Ac2 A   x |x|2  dx  Bc1 B  Ac1 A  B 2 c1 Bc2 B  A 2 c1 Ac2 A    x |x|2  dx  c2 AA  c2 BB    x|x|2dx  1c2 AA  c2 BB   2Bc1 B  Ac1 A , 3.3.29 where 1    x|x|2dx,2    x |x|2  dx. 3.3.30 Lemma 3.3.1. Let ,Σ,P be a measure space, and let f be a realvalued measurable function defined on  . Then for any real number t  0 : P  ||f|  t  1 t  |f|t f  dP. 3.3.31 108 From inequality (3.3.31) and Eq. (3.3.29) we obtain P    : |X 1  X 2|  t  1t  X 1 X 2  t |X 1  X 2|dP  1 t2   X 1  X 2  1,2 t  1c2 AA  c2 BB   2Bc1 B  Ac1 A  t . 3.3.32 We define now and chose in (3.3.31) number t  xB  xA  1. Fig. 3.3.2. The plot of the random variables X 1 and X 2. A t  A pht,xA ,Bt  B pht,xB ,A t  Bt  C t A t  A pht,x A    |x A    X 1  x A  , Bt  B pht,x B    |x B    X 2  x B  , 3.3.33 109 Note that PAt  Bt  PC t, 3.3.34 see Fig . 3.3.2. From Eq. (3.3.32) Eq. (3.3.34) it follows that PA t  Bt   1c2 AA  c2 BB   2Bc1 B  Ac1 A  xB  xA 2  1. 3.3.35 110 Chapter IV THE ERP PARADOX RESOLUTION IV.1. The relaxed locality principle The Special Theory of Relativity limits the speed at which any physical influences and any real information can travel to the speed of light, c. The Einstein's principle of locality (EPL): any effects do not propagate faster than the speed of light, i.e. speed of light is a limiting factor. The principle of locality claimed that: (i) Any physical event At1 ,r1  which has occurred at point At1 ,r1   M4 (see Chapt. III, Definition 3.2.9) cannot cause (by physical interaction) a physical event Bt2 ,r2  (result) which has occurred at point Bt2 ,r2   M4 in a time less than T  D/c, where D, is the distance between the points. (ii) Any physical event At,r1  which has occurred at point At,r1   M4 cannot cause a simultaneous physical event Bt,r2  (result) which has occurred at another point Bt,r2   M4 . (iii) Any real physical information about physical event At1 ,r1  at point At1 ,r1  cannot be obtained by observer at point Bt2 ,r2  in a time less than T  D/c, where D, is the distance between the points. Definition 4.1.1. Let M4 # ,t1 ,r1 ,t2 ,r2  t.l .s.  be a set of the all timelike separated pairs of events At1 ,r1 ,Bt2 ,r2 t.l .s.  M4 # ,t1 ,r1 ,t2 ,r2  t.l .s. , (see Chapt. III, Definition 3.2.10(i)) such that t2  t1 and AOct1 ,r1   BOct2 ,r2 . Note that M4 # ,t1 ,r1 ,t2 ,r2  t.l .s.   M4 # ,t1 ,r1 ,t2 ,r2  t.l .s. . Remark 4.1.1. Note that the claim (i) obviously meant that 111 t1  t2 At1 ,r1 Bt2 ,r2  AOct1 ,r1   BOct2 ,r2   At1 ,r1 ,Bt2 ,r2   M4 # ,t1 ,r1 , t2 ,r2  t.l .s. . 4.1.1 Remark 4.1.2. In spacetime diagram, see Fig. 4.1.1, the interval sAB 2 is "time-like", i.e. there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect). Fig. 4.1.1. Spacetime diagram Remark 4.1.3. Note that: (i) the interval sAC 2 in the diagram, see Fig. 4.1.1, is "space-like"; i.e. there is a frame  t of reference in which events At,r1  and Ct,r2  occur simultaneously at instant t , separated only in space. There are also frames in which A precedes C and frames in which C precedes A. 112 (ii) If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result. For example, if A was the cause, and C the effect, then there would be frames of reference in which the effect preceded the cause. Although this in itself will not give rise to a paradox, one can show that faster than light signals can be sent back into one's own past. A causal paradox can then be constructed by sending the signal if and only if no signal was received previously. (iii) Obviously there exist space-like separated pairs of physical events At,r1 ,Bt,r2 s.l .s. such that the events At,r1  and Ct,r2  always occur only simultaneously at any instant t i.e., AOct,r1   COct,r2 . 4.1.2 Example 4.1.1. Let us consider two synchronized clock A and B which at rest on given inertial frame F I . Assume that clock A at rest in point r1 and clock B at rest in point r2 correspondingly. Fig. 4.1.2. Clock A and clock B which at rest on given inertial frame F I . 113 Let At,r1  be the event which consists that time on clock A is t at time t according to clock A and let Bt,r1  be the event which consists that time on clock B is t at time t according to clock B. It is clear that AOct,r1   BOct,r2 . Definition 4.1.2. Let M4 # ,t1 ,r1 ,t2 ,r2  s.l .s.  be a set of the all spacelike separated pairs of events At1 ,r1 ,Bt2 ,r2 s.l .s.  M4 # ,t1 ,r1 ,t2 ,r2  s.l .s. , (see Chapt. III, Definition 3.2.10 (ii)) such that AOct1 ,r1   BOct2 ,r2 . 4.1.3 Remark 4.1.4. Note that the condition (4.1.3) does not violate the Einstein's principle of locality and gives only an additional properties of the algebra M4 # . Remark 4.1.5. Note that from (4.1.3) it follows that M4 # ,t1 ,r1 ,t2 ,r2  s.l .s.   M4 # ,t1 ,r1 ,t2 ,r2  s.l .s. . On the basis of this Gedanken experiment, which is also realized by photons, the EPR-paradox can be derived if the following two principles are taken as postulates. 1. The principle of reality R : If the value A i of an observable A can be determined without altering the quantum system S, then any property PA i which corresponds to this value of A pertains to the system S. 2. The principle of locality L : 2.1. If two quantum systems S1 and S2 cannot interact with each other, then a measurement with respect to one system cannot alter the other system and therefore we can assume the existence of state vectors |S1  and |S2 . 2.2. Let x 1 and  x 2 be two observables measured with respect to systems S1 and S2 mentioned above. Then by result of measurement of the quantity x2  S2 |  x 2 |S2  at instant t, it is impossible to get any information on result of measurement of the 114 quantity x1  S1 |  x 1 |S1  at the same instant t. We assume now the relaxed principle of locality. Intuitively this principle says that for even spacelike separated entangled quantum systems S1 and S2 any measurement at instant t with respect to system S1 always immediately alter the other system S2 at the same instant t. But no additional information about the system S1 can be found out upon measurement on the system S2 except the canonical information which can be predicted by using correlation relations which follows from concrete type of entanglement. 3. The relaxed principle of locality Lrel : 3.1. Any spacelike separated quantum systems S1 and S2 cannot interact with each other and therefore we can assume the existence of state vectors |S1  and |S2  correspondingly. 3.2. Let S12t,r1  and S21t,r2  be two spacelike separated entangled quantum systems located in points t,r1  and t,r2  correspondingly. (i) Assume that a state vector |S12t,r1  suddenly collapses at instant t to state vector S12 s-col t,r1  : |S12t,r1  scol lapse  S12 s-col t,r1  , 4.1.4 then a state vector |S21t,r2  immediately collapses to state vector S21 col t,r2  : |S21t,r2  col lapse  S21 col t,r2  4.1.5 (ii) Assume that a state vector |S12t,r1  after measurement immediately collapses at instant t to state vector S12 m -col t,r1  : |S12t,r1  m col lapse  S12 m -col t,r1  , 4.1.6 then a state vector |S21t,r2  immediately collapses to state vector 115 S21 col t,r2  : |S21t,r2  col lapse  S21 col t,r2  4.1.7 (iii) Let S12 s-col t,r1  and S21 col t,r2  be a physical events defined by formulae (4.1.4) and (4.1.5) correspondingly, then Occ S12 s-col t,r1   Occ S21 col t,r2  , 4.1.8 (see Chapt. III, Definition 3.2.8(2)). (iv) Let S12 m -col t,r1  and S21 col t,r2  be a physical events defined by formulae (4.1.6) and (4.1.7) correspondingly, then Occ S12 m -col t,r1   Occ S21 col t,r2  , 4.1.9 3.3. No any additional information about the system S1 upon measurement at instant t can be found out upon measurement on the system S2 upon measurement at instant t except the canonical information which can be predicted by using correlation relations which follows from concrete type of entanglement. Remark 4.1.6. Note that conditions (1.1.8) - (1.1.9) are very similarly to condition (4.1.3) and give only an additional properties of the algebra M4 # . Remark 4.1.7. Note that from (4.1.8) it follows that S12 s-col t,r1 ,S21 col t,r2   M4 # ,t1 ,r1 , t2 ,r2  s.l .s.  4.1.10 from (4.1.9) it follows that S12 m-col t,r1 ,S21 col t,r2   M4 # ,t1 ,r1 ,t2 ,r2  s.l .s.  4.1.11 116 Remark 4.1.8. Note that: (i) collapse of a state vector |S21t,r2  given by (4.1.5) occurs without any interaction between quantum systems S12  and S21  but only by property given by formulae (4.1.8); (ii) collapse of a state vector |S21t,r2  given by (4.1.7) occurs without any interaction between quantum systems S12  and S21  but only by property given by formulae (4.1.9). Remark 4.1.9. We find that the EPR-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle. However it follows also, that the nonlocalities which are introduced above cannot be explained within the conventional quantum theory. IV.2. Generalized EPR argument and Postulate of Nonlocality Entanglement is one of the most interesting properties of quantum mechanics, and is an important ingredient of quantum information protocols such as quantum dense coding and quantum computation. In the Schrödinger picture, a necessary and sufficient criterion for the emergence of entanglement is that the state describing the entire system is inseparable, i.e. the wave function of the total system cannot be factored into a product of separate contributions from each sub-system. Using the Heisenberg approach, a sufficient criterion for the presence of entanglement is that correlations between conjugate observables of two subsystems allow the statistical inference of either observable in one sub-system, upon a measurement in the other, to be smaller than the standard quantum limit, i.e. the presence of non-classical correlations. The latter approach was originally proposed in the paper of Einstein, Podolsky and Rosen [1]. These two different pictures result in two distinct methods of characterizing entanglement. One is to identify an observable signature of the mathematical criterion for wave function 117 entanglement, i.e. inseparability of the state. The second looks directly for the onset of non-classical correlations. For pure states these two approaches return the same result suggesting consistency of the two methods. However, when decoherence is present, causing the state to be mixed, difference scan occur. IV.2.1. The EPR-Reid criterion We remind now EPR-Reid criterion [2]-[5]. EPR originally argued as follows. Consider two spatially separated subsystems at A and B. EPR considered two observables x (the position) and  p (momentum) for subsystem A, where x and  p do not commute, so that (C is nonzero) x,p  2C. 4.2.1 Suppose now that one may predict with certainty the result of measurement x based on the result of a measurement performed at B. Also, for a different choice of measurement at B, suppose one may predict the result of measurement  p at A . Such correlated systems are predicted by quantum theory. Assuming local realism EPR deduce the existence of an element of reality, x , for the physical quantity x and also an element of reality,  p , for  p . Local realism implies the existence of two hidden variables x and  p that simultaneously predetermine, with no uncertainty, the values for the result of an x or  p measurement on subsystem A , should it be performed. This hidden variable state for the subsystem A alone is not describable within quantum mechanics, since simultaneous eigenstates of x and  p do not exist. Hence, EPR argued, if quantum mechanics is to be compatible with local realism, we must regard quantum mechanics to be incomplete. We remind that in original publication [1] Einstein, Podolsky and Rosen describe two particles A and B with correlated position 118 xB  xA  x0 4.2.2 and anti-correlated momentum pB  pA, 4.2.3 (see Fig. 4.2.1). Fig. 4.2.1. Particles A and B with correlated position and anticorrelated momentum In the idealized entangled state proposed by EPR, |EPR     |x,xdx     |p,pdp the positions and momenta of the two particles are perfectly correlated. Note that: this state is non-normalizable and cannot be realized in the laboratory. When coordinates x A and pA are measured in independent realizations of the same state, the correlations allow for an exact prediction of x B and pB . EPR assumed that such exact predictions necessitate an element of reality which predetermines the outcome of the measurement. Quantum mechanics however prohibits the exact knowledge of two noncommuting variables like x B and pB , since their measurement uncertainties are subject to the Heisenberg relation 119 xBpB  /2. 4.2.4 Classical notion of EPR correlations was generalized to a more realistic scenario, yielding a Reid criterion [6] for the uncertainties x inf B and pinf B of the inferred predictions for x B and pB . The EPR criterion is met if these uncertainties violate the Heisenberg inequality for the inferred uncertainties x inf B pinf B  /2. Reid extended classical EPR argument to situations where the result of measurement x at A cannot be predicted with absolute certainty [2]-[5]. The assumption of local realism allows us to deduce the existence of an element of reality of some type for x at A , since we can make a prediction of the result at A , without disturbing the subsystem at A , under the locality assumption. Let   x A ,  x B be a wave function of composite system A  B. Let x i B be the result of a measurement, x B say, performed at B, where i is used to label the possible results, discrete or otherwise, of the measurement x B . As a result of the measurement of the coordinate, we have a new wave function of composite system A  B which is given by Eq. (4.2.3) (see Remark 4.2.1) x iB x,  x B  x iB  x A,  x B  Rx B  x i B x A,x B  Rx B  x i B x,x B 4.2.5 and therefore adjoint probability density px iB x,  x B  p x,x B x i B at instant at once after measurement is given by px iB x,  x B  p x,x B x i B  Rx B  x i B x,x B 2 4.2.6 Then the conditional probability density px iBx  px|x i B  conditional on a result x i B for QM measurement at B is given by 120 px iBx  px|x i B      px iB x,  x B d  x B     d  x B x iB x,  x B 2     d  x B Rx B  x i B x,x B 2 . 4.2.7 The predicted results for the measurement at A , based on the measurement at B, are however no longer a set of definite numbers with zero uncertainty, but become fuzzy, being described by a set of distributions Px|x i B  giving the probability of a result for the measurement at A, conditional on a result x i B for measurement at B. We define  i 2x to be the variance of the conditional distribution Px|x i B . Similarly we may infer the result of measurement  p at A , based on a (different) measurement,  p B say, at B. Denoting the results of the measurement  p B at B by p j B , we then define the probability distribution, P p|pj B which is the predicted result of the measurement for  p at A conditional on the result p j B for the measurement  p B at B. The variance of the conditional distribution P p|pj B is denoted by  j 2p. Remark 4.2.1. We remind now that the QM-measurement is represented by the canonical scheme | a   |a    a  |,daa  a   1,pa   a 2  |a  a  |, 4.2.8 where pa  is a corresponding probability density. To obtain the probability that the parameter a  turns out to belong to the set  one has to integrate over this set: Pa      dapa  . 4.2.9 121 If the state | is represented by the wave function a the operator a  describing the measurement giving the result a  will be taken in the following form a a  Ra  aa, 4.2.10 where Ra is a function with a support concentrated in some vicinity of zero and representing the 'fuzziness' of the measurement. It is a characteristic function of the measurement and may, for example, be (and typically is) a Gaussian function. The width of this function corresponds to the resolution of the measurement. Normalization daa   a   1 of the operators a  is provided by the corresponding normalization of the function Ra as follows: da|R2a|  1. 4.2.11 If the measurement is described by the Gaussian function Ra  exp  a  a 2 42 4.2.12 it is a minimally disturbing measurement of the coordinate a  with resolution  [7]. Remark 4.2.2. Consider the momentum representation p of the initial wave function q p  1 2  dqqexp  i  pq . 4.2.13 As a result of the measurement of the coordinate, 122 q q  Rq  qq  q q, 4.2.14 we have a new wave function and its momentum representation has the form q p  dp  Rq p  pp , 4.2.15 where  Rq p is a momentum representation of the function Rq q. Note that  Rq p   Rpexp  i  pq ,  Rp  1 2  dqRqexp  i  pq . 4.2.16 Remark 4.2.3. Consider now a coordinate measurement having a Gaussian characteristic function of width of the order of  Rq  exp  q 2 42 . 4.2.17 Then the momentum representation of this function (characterizing the structure of the momentum uncertainty / acquired in the measurement) is also Gaussian with width of the order of / :  Rp  exp p 2   2 . 4.2.18 For a given experiment one could in principle measure the individual variances  i 2x of the conditional distributions Px|x i B (and also  j 2p for the Pp|pj B). Obviously if each of the variances  i 2x and  j 2p satisfy i 2x  0 and j 2p  0 this would imply the demonstration of the original 123 EPR paradox. This situation however is not practical for continuous variable measurements. Instead of considering the problem of simultaneous eigenstates as originally proposed by EPR, one can suggest a different and experimentally realizable criterion based on the Heisenberg Uncertainty Principle: xp  C. For the sake of notational convenience we now consider in the remainder of this subsection that appropriate scaling enables x and p to be dimensionless and C  1. EPR correlations however would be demonstrated in a convincing manner if the experimentalist could measure each of the conditional distributions Px|x i B and establish that each of the distributions is very narrow, in fact constrained such that [2]-[5]: Px|x i B  0 iff |x  i| , Pp|pj B  0 iff |p  j| . 4.2.19 Here i is the mean value of the conditional distribution Px|x i B and  j is the mean value of the conditional distribution Pp|pj B. In this case the assumption of local realism would imply, since the measurement x B at B will always imply the result of x at A to be within the range i  x , that the result of the measurement at A is predetermined to be within a bounded range of width 2 . In a straightforward extension of EPR's argument, we replace the words predict with certainty with predict with certainty that the result is constrained to be within the range i  , and then define an element of reality with this intrinsic bounded by fuzziness  . We now consider the situation where an experimenter has demonstrated that for every outcome x i B (and p j B ) for the measurement x B (and p B ) performed at B, the variance  ix (and  jp ) of the appropriate conditional distribution satisfies ix  1,jp  1 4.2.20 124 for any i, j  . The measurement at B always allows an inference of the result at A to a precision better than given by the uncertainty bound 1 . Remark 4.2.4. In this case we do not predict a result at A with certainty, as in EPR's original paradox. The measurement x B at B however does predict by Eq. (4.2.3) [or by Eq. (4.2.9) in general case] with a certain probability constraints on the result for x at A . Remark 4.2.5. Following the EPR argument, which assumes no action-at-a-distance, so that the measurement at B does not cause any instantaneous influence to the system at A , one can attribute a probabilistic predetermined element of reality to the system at A . Remark 4.2.6. There is a similar predicted result for the measurement p at A based on a result of measurement at B, and a corresponding predetermined description based again on the noaction-at-a-distance assumption. Remark 4.2.7. The important point in establishing the EPR paradox for this more general yet practical situation is that under the EPR premises the predetermined statistics (or generalised elements of reality) for the physical quantities x and p are attributed simultaneously to the subsystem at A . Assuming no action-at-a-distance, the choice of the experimenter (Bob) at B to infer information about either x or p cannot actually induce the result of the measurement at A . As there is no disturbance created by Bob's measurement, the (appropriately extended) EPR definition of realism is that the prediction for x is something (a probabilistic element of reality) that can be attributed to the subsystem at A , whether or not Bob makes his measurement. Remark 4.2.8. This is also true of the prediction for p , and therefore the two elements of reality representing the physical quantities x and p exist to describe the predictions for x and p simultaneously. The paradox can then be established by proving the impossibility of such a simultaneous level of prediction for both x and p for any quantum description of the subsystem A alone. By this we mean 125 explicitly that there can be no procedure allowed, within the predictions of quantum mechanics, to make simultaneous inferences by measurements performed at B or any other location, of both the result x and p at A , to the precision indicated by ix  1, jp  1. Remark 4.2.9. Recall that the inference of the result at A by measurement at B is actually a measurement of x performed with the accuracy determined by the  ix . However simultaneous measurements of x and p to the accuracy (4.2.9) are not possible (predicted by quantum mechanics). The reduced density matrix describing the state at A after such measurements would violate the H.U.P. (Heisenberg Uncertainty Principle). A simpler quantitative, experimentally testable criterion for EPR was proposed by Reid in 1989 see, for example, [2]-[5]. The 1989 inferred H.U.P. criterion is based on the average variance of the conditional distributions for inferring the result of measurement x (and also for p ). The EPR paradox is demonstrated when the product of the average errors in the inferred results for x and p violates the corresponding H.U.P. The spirit of the original EPR paradox is present, in that one can perform a measurement on B to enable an estimate of the result x at A (and similarly for p ). Abbreviation 4.2.1. For the sake of notational convenience we now abbreviate in the remainder of the book: loc.ix and loc.ip instead  ix and  jp for the variance  ix and  jp which were calculated under assumption no action-at-a-distance, see Remarks 4.2.5 4.2.6. We define now [2]:  loc.inf. 2 x   i Px i B  loc.i 2 x,  loc.inf. 2 p   j P pj B  loc.j 2 p. 4.2.21 Here loc.inf. 2 x is the average variance for the prediction (inference) under assumption no action-at-a-distance of the result x for x at A , conditional on a measurement x B at B. Here i   labels all outcomes of the measurement x at A , and i and  ix are the mean and standard deviation, respectively, of the conditional distribution 126 Px|x i B, where x i B is the result of the measurement x B at B. We define a loc.inf. 2 p similarly to represent the weighted variance for the prediction (inference) under assumption no action-at-a-distance of the result p at A , based on the result of the measurement at B. Here Px i B is the probability for a result x i B upon measurement of x B , and Ppj is defined similarly. The Reid's criterion to demonstrate the EPR "paradox", the Reid's local signature of the EPR paradox, is loc.inf. 2 x loc.inf. 2 p  1. 4.2.22 This criterion is a clear criterion for the demonstration of the EPR "paradox", by way of the argument presented above. Such a prediction (4.2.21) for x and p with the average inference variances given, cannot be achieved by any quantum description of the subsystem alone. This EPR criterion has been achieved experimentally. IV.2.2. The Postulate of Nonlocality and signature of the EPR paradox Remark 4.2.10. A most critical component of the EPR argument was the principle of locality. Indeed, one may regard the EPR paradox as a statement of the mutual incompatibility of locality, entanglement, and completeness. Experimental tests of Bell's inequalities have indicated that quantum mechanics is complete by ruling out the possibility of hidden variables. Therefore, it is generally agreed that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state consistent with the result of measurement, regardless of how far apart the particles are. In the situation proposed by EPR, the position or momentum of the unmeasured 127 particle becomes a reality if and only if, the corresponding quantity of the other particle is measured. Remark 4.2.11. The assumption of nonlocality allows us to deduce the existence of a fuzzy element of reality of some type for x at A , since we can make a prediction of the result at A , but with some disturbing of the subsystem at A , under the measurement,  x B say, performed at B. This prediction is subject to the result x i B of a measurement, x B say, performed at B, where i is used to label the possible results, discrete or otherwise, of the measurement  x B . We accept now the following postulate: Postulate of Nonlocality (i) Let A and B two entangled particles. Let   x A ,  x B be a wave function of composite system A  B. Let x i B be the result of a measurement, x B say, performed at B, where i is used to label the possible results, discrete or otherwise, of the measurement x B . As a result of the measurement of the coordinate, we have a new wave function of composite system A  B which [in contrast with Eq.(4.2.5)] is given by x iB  x A ,  x B  R2  x A  x i Ax i B R1  x B  x i B x A ,x B , x i Ax i B   x 0  x i B . 4.2.23 (ii) Let A and B are two entangled particles. Let   p A ,  p B be a wave function of composite system A  B. Let p j B be the result of a measurement,  p B say, performed at B, where j is used to label the possible results, discrete or otherwise, of the measurement  p B . As a result of the measurement of the coordinate, we have a new wave function of composite system A  B which is given by 128 p jB  p A ,  p B   R2  p A  pj A pj B   R1  p B  pj B pA ,pB , pj A pj B  pj B . 4.2.24 Remark 4.2.12. The spirit of the original EPR paradox now is present, in that the canonical EPR correlations (4.2.2) and (4.2.3) well preserved. Remark 4.2.13. Note that EPR correlations x i Ax i B   x0  x i B and pj A pj B  pj B however would be demonstrated in a convincing manner if the experimentalist could measure each of the conditional distributions Px|x i B and establish that each of the distributions is very narrow, in fact constrained so that [2]-[5] px|x i B  0 iff |x  i| , pp|pj B  0 iff |p  j| , Px|x i B  0 iff |x  i| , Pp|pj B  0 iff |p  j| . 4.2.25 Here i is the mean of the conditional distribution Px|x i B and  j is the mean of the conditional distribution Pp|pj B. Remark 4.2.14. We assume now that a coordinate and momentum measurements have a Gaussian characteristic function of width of the order of 2 R1x  R2x  Rx  exp  x 2 42  R1p   R2p   Rp  exp  p 2 42 4.2.26 In this case the Postulate of Nonlocality would imply, since the measurement x B at B will always imply the result of x at A to be within the range i  x , that the result of the measurement at A is predetermined to be within a bounded range of width 2 . In a straightforward extension of EPR's argument, we replace the words 'predict with certainty' with 'predict with certainty that the result is 129 constrained to be within the range i   '. We now consider the situation where an experimenter has demonstrated that for every outcome x i B (and p j B ) for the measurement x B (and p B ) performed at B, the variance  ix (and  jp) of the appropriate conditional distribution satisfies ix  1,jp  1 4.2.27 for all i, j. The measurement at B always allows an inference of the result at A to a precision better than given by the uncertainty bound 1. In this case we do not predict a result at A with certainty, as in EPR original paradox. The measurement x B at B however does predict with a certain probability constraints on the result for x at A. Remark 4.2.15. Note that adjoint probability density p  x A,  x B x i B at instant at once after measurement [in contrast with Eq. (4.1.6)] is given by p  x A ,  x B x i B  x iB  x A ,  x B 2  Rx A  x i Ax i B Rx B  x i B x A ,x B 2 , x i Ax i B   x 0  x B  i. 4.2.27a Then the conditional probability density px iBx  px|x i B  depending on a result x i B for QM measurement at B is given by px iBx  px|x i B      px iB x,  x B d  x B     d  x B x iB x,  x B 2     d  x B Rx A  x i Ax i B Rx B  x i B x,x B 2 . 4.2.28 There is a similar predicted result for the measurement  p at A based on a result of measurement at B, and a corresponding 130 predetermined description based on the QM constraints p jB  p A ,  p B   RpA  pj A pj B   RpB  pj B pA ,pB , pj A pj B  pB . 4.2.29 The spirit of the original EPR "paradox" is present, in that one can perform a measurement on B to enable an estimate of the result x at A (and similarly for  p ). Abbreviation 4.2.2. For the sake of notational convenience we now abbreviate in the remainder of the book: nonloc.ix and nonloc.ip instead  ix and  jp for the variance  ix and  jp which were calculated under nonlocality assumption (postulate) by conditional probability density given by Eq. (4.1.28). We define now nonloc.inf. 2 x   i Px i B nonloc.i 2 x, nonloc.inf. 2 p   j P pj B nonloc.j 2 p. 4.2.30 Here nonloc.inf. 2 x is the average variance for the prediction (inference) of the result x for  x at A, conditional on a measurement  x B at B. Here i labels all outcomes of the measurement x at A, and μi and  ix are the mean and standard deviation, respectively, of the conditional distribution Px|x i B, where x i B is the result of the measurement  x B at B. We define a nonloc.inf. 2 p similarly to represent the weighted variance for the prediction (inference) of the result  p at A, based on the result of the measurement at B. Here Px i B is the probability for a result x i B upon measurement of  x B , and Ppj B is defined similarly. The criterion to demonstrate the EPR paradox, the signature of the EPR paradox, is the criterion to demonstrate the EPR paradox, the nonlocal signature of the EPR paradox, is given by 131 nonloc.inf. x nonloc.inf. p  1. 4.2.31 This criterion is a clear criterion for the demonstration of the EPR paradox, by way of the argument presented above. Such a prediction for x and p with the average inference variances given, cannot be achieved by any quantum description of the subsystem alone. IV.2.3. The EPR-nonlocality criteria Remark 4.2.16. The principle of locality was a critical component of the EPR argument. Actually the EPR paradox is regarded as a statement of the mutual incompatibility of locality, entanglement, and completeness. Experimental studies of Bell's inequalities have shown that quantum mechanics is complete by ruling out the possibility of hidden variables. Consequently it is usually accepted that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state corresponding with the measurement result, irrespective of how far apart these particles are. In the situation proposed by EPR, the position or momentum of the unmeasured particle becomes a reality if and only if, the corresponding quantity of the other particle is measured. Since only one quantity or the other is measured, the position and the momentum of the unmeasured particle need not be simultaneous realities. In this way the EPR paradox also is resolved. From Eq. (4.2.21) and Eq. (4.2.30) we obtain the EPR-nonlocality criteria  loc.inf. 2 x  nonloc.inf. 2 x   i Px i B  loc.i 2 x  nonloc.i 2 x   0,  loc.inf. 2 p  nonloc.inf. 2 p   j P pj B  loc.j 2 p  nonloc.j 2 p  0, 4.2.32 132 and nonloc.inf. x nonloc.inf. p  loc.inf. x loc.inf. p  0. 4.2.33 These EPR-nonlocality criteria has been achieved experimentally [8], [9], (see subsection IV.5, Remark 4.5.3 Remark 4.5.4). IV.3. Nonlocal Schrödinger equation implies the Postulate of Nonlocality In this subsection we obtain nonlocal Schrödinger equation (NSE) which corresponding to position-momentum entangled pairs A and B (see Fig. 4.2.1) with well correlated position xB   xA   x0 4.3.1 and anti-correlated momentum pB  pA . 4.3.2 Remark 4.3.1. As pointed out in subsection IV.2 it is generally agreed that the assumption of locality is invalid for entangled states: measurement of either particle of an entangled system projects both particles onto a state consistent with the result of measurement, regardless of how far apart the particles are. It allow us to use special nonlocal generalization of the canonical Schrödinger equation. Remark 4.3.2. As pointed out in [10], [11] from nonlocal Schrödinger equation one obtains collapsed wave function corresponding to GRW collapse model. It allow us to use similar nonlocal Schrödinger equation also for entangled states. Remark 4.3.3. The spirit of the original EPR paradox is present, in 133 that the canonical EPR correlations (4.3.1) and (4.3.2) give a boundary conditions for the solutions of the nonlocal Schrödinger equation. Remark 4.3.4. In this subsection we denote (i) xA  x1 ,xB  x2 , (ii) xA  x 1 , xB   x 2   x 1  x0 . Definition 4.3.1. Let us consider the time-dependent canonical Schrödinger equation i x 1 ,x 2 , t t  Hx 1 ,x 2 , t, t  0,T, x 1 ,x 2  2 . 4.3.3 Let x1 ,x2 , t be a classical solution of the time-dependent Schrödinger equation (4.3.3). The time-dependent Schrödinger equation (4.3.3) is a weakly well preserved (in sense of Colombeau generalized functions) by corresponding to x1 ,x2 , t collapsed Colombeau generalized wave function  #x1 ,x2 , t,  0,1, where #x 1 ,x 2 , t  x 1 ,x 2 , t;  x 1t,  x 2t   1,2x 1 ,  x 1t,x 2 ,  x 2t;,x 1 ,x 2 , t 1,2  x 1t,  x 2t;,x 1 ,x 2 , t2  , 1,2x 1 ,  x 1 ,x 2 ,  x 2 ;,  i1 2  ix i,  x i;,,  ix,  x it;,  2  1/4 exp  x i   x it2 22 iff x i   x i  , 0 iff x i   x i  . i  1,2. 4.3.4 in region   2 if there exists a solution x1 ,x2 , t of Schrödinger equation (4.2.1) such that the estimate 134  D i #x 1 ,x 2 , t t  H#x 1 ,x 2 , t dx 1dx 2   O , t  0,T,x 1 ,x 2D, 4.3.5 with 1/2  , is satisfied. Definition 4.3.2. Equation (4.3.5) with a following boundary conditions x B t  x A t   x 0 , x A t    x A #x A ,x B , t 2dx Adx B  , x B t   x B #x A ,x B , t 2dx Adx B  , 4.3.6 that is time-dependent nonlocal Schrödinger equation corresponding to EPR entangled state. Definition 4.3.3. (i) The time-dependent integral equation (4.3.5) with a boundary conditions (4.3.6) is called the time-dependent nonlocal Schrödinger equation of the order  corresponding to EPR entangled state. (ii) Such collapsed wave function #x1 ,x2 , t,  as mentioned in Definition 4.3.2 is called the  solution of the nonlocal Schrödinger equation (4.3.5)-(4.3.6) of the order . Definition 4.3.4. Let us consider the time-independent canonical Schrödinger equation Hx1 ,x2   0,x1 ,x 2  2 . 4.3.7 Let x1 ,x2  be a classical solution of the time-independent Schrödinger equation (4.3.7). The time-independent Schrödinger equation (4.3.7) is a weakly well preserved (in sense of Colombeau generalized functions) by corresponding to x1 ,x2  Colombeau generalized collapsed wave function  #x1 ,x2 ,  0,1, where 135 #x 1 ,x 2 ,  x 1 ,x 2 ;  x 1 ,  x 2 ,   1,2x 1 ,  x 1 ,x 2 ,  x 2 ;,x 1 ,x 2  1,2  x 1 ,  x 2 ;,x 1 ,x 2 2  , 1,2x 1 ,  x 1 ,x 2 ,  x 2 ;,  i1 2  ix i,  x i;,,  ix,  x i;,  2  1/4 exp  x i   x i 2 22 iff x i   x i  , 0 iff x i   x i  . 4.3.8 in region   2 if there exists a solution x1 ,x2  of Schrödinger equation (4.3.7) such that the estimate  D H#x 1 ,x 2 dx 1dx 2   O , x 1 ,x 2 D, 4.3.9 with 1/2  , is satisfied. Definition 4.3.5. Equation (4.3.9) with boundary conditions x B  x A   x 0 , x A    x A #x A ,x B  2dx Adx B  , x B   x B #x A ,x B  2dx Adx B  , 4.3.10 that is time-independent nonlocal Schrödinger equation corresponding to EPR entangled state. Definition 4.3.6. (i) The time-independent integral equation (4.3.9) with a boundary conditions (4.3.10) is called the time-independent nonlocal Schrödinger equation of the order  corresponding to EPR entangled state. 136 (ii) Such collapsed wave function #x1 ,x2  as mentioned in Definition 4.3.5 is called the  solution of the nonlocal Schrödinger equation (4.3.9) - (4.3.10) of the order . Lemma 4.3.1. Let  be a function    0 a x1 expxfxdx, 4.3.11 where   1, 0  a  ,0  ,0  . Assume that fx is continuous on 0,a. Then   1   f0  o1 / 4.3.12 Lemma 4.3.2. Let fx be a function such that f  C2x  x0  and f  C2x  x0 . Then f x  f x xx0  f x0 x  x 0 , f x  f x xx0  f  x0 x  x 0   fx0 x  x 0 , f x0  fx 0  0  fx 0  0, f  x0  f x 0  0  f x 0  0. 4.3.13 Theorem 4.3.1. Assume that there exists a classical solution x1 ,x2  of the Schrödinger equation (4.3.7) such that x1,x2 D sup |x 1 ,x 2 |  O1/2 , x1,x2 D sup |x 1 ,x 2 /x 1 |  O3/2 , x1,x2 D sup |x 1 ,x 2 /x 2 |  O3/2 . 4.3.14 Then any collapsed wave function #x is given by Eq. (4.3.8) with /  , 1/4    1/2 that is  -solution of the time-independent 137 nonlocal Schrödinger equation (4.3.9)-(4.3.10) of the order . Proof. The Schrödinger equation (4.3.7) has the following form Hx 1 ,x 2   2 2x 1 ,x 2  x 1 2  2 2x 1 ,x 2  x 2 2  Vx 1 ,x 2 x 1 ,x 2   0. 4.3.15 Let #x1 ,x2  be a function  #x1 ,x2   Rx1 ,  x 1Rx2 ,  x 2x1 ,x2 , 4.3.16 where Rx i,  x i  2  1/4 exp  x i   x i 2 22 iff x i   x i  , 0 iff x i   x i  . 4.3.17 From Eq. (4.3.17) by using Eq. (4.3.13) we obtain Rx 1 ,  x 1 x 1  1/41x 1   x 1 exp  x 1   x 1 2 2   Rx 1 ,  x 1x 1 x 1   x 1    Rx 1 ,  x 1x x 1   x 1  , 2Rx 1 ,  x 1 x 1 2  1/41 exp  x 1   x 1 2 2  1/42x 1   x 1 2 exp  x 1   x 1 2 2  Rx 1 ,  x 1 x 1 x x 1   x 1    Rx 1 ,  x 1 x 1 x x 1   x 1    Rx 1 ,  x 1 x  x 1   x 1    Rx 1 ,  x 1 x  x 1   x 1   4.3.18 and 138 Rx 2 ,  x 2 x 2  1/41x 2   x 2 exp  x 2   x 2 2 2   Rx 2 ,  x 2x 1 x 2   x 2    Rx 2 ,  x 2x x 2   x 2  , 2Rx 2 ,  x 2 x 2 2  1/41 exp  x 2   x 2 2 2  1/42x 2   x 2 2 exp  x 2   x 2 2 2  Rx 2 ,  x 2  x 2 x x 2   x 2    Rx 2 ,  x 2  x 2 x x 2   x 2    Rx 2 ,  x 2x  x 2   x 2    Rx 2 ,  x 2x  x 2   x 2  . 4.3.19 From Eq. (4.3.16) by differentiation we obtain 2#x 1 ,x 2  x 1 2  2Rx 1 ,  x 1Rx 2 ,  x 2x 1 ,x 2  x 1 2   x 1 x 1 ,x 2 Rx 2 ,  x 2 Rx 1 ,  x 1 x 1  Rx 1 ,  x 1Rx 2 ,  x 2 x 1 ,x 2  x 1  2 x 1 ,x 2  x 1 Rx 2 ,  x 2 Rx 1 ,  x 1 x 1  x 1 ,x 2 Rx 2 ,  x 2 2Rx 1 ,  x 1 x 1 2  Rx 1 ,  x 1Rx 2 ,  x 2 2x 1 ,x 2  x 1 2 4.3.20 and 2#x 1 ,x 2  x 2 2  2Rx 1Rx 2 ,  x 2x 1 ,x 2  x 2 2   x 2 x 1 ,x 2 Rx 1 ,  x 1 Rx 2 ,  x 2 x 2  Rx 1 ,  x 1Rx 2 ,  x 2 x 1 ,x 2  x 2  2 x 1 ,x 2  x 2 Rx 1 ,  x 1 Rx 2 ,  x 2 x 2  x 1 ,x 2 Rx 1 ,  x 1 2Rx 2 ,  x 2 x 2 2  Rx 1 ,  x 1Rx 2 ,  x 2 2x 1 ,x 2  x 2 2 . 4.3.21 139 By substitution Eq. (4.3.15) and Eq. (4.3.20) Eq. (4.3.21) into LHS of the Eq. (4.3.9) we obtain   H #x 1 ,x 2 dx 1dx 2    dx 1dx 2Rx 1 ,  x 1Rx 2 ,  x 2  2 2x 1 ,x 2  x 1 2  2 2x 1 ,x 2  x 2 2  Vx 1 ,x 2 x 1 ,x 2    2   dx 1dx 2  2 x 1 ,x 2  x 1 Rx 2 ,  x 2 Rx 1 ,  x 1 x 1  x 1 ,x 2 Rx 2 ,  x 2 2Rx 1 ,  x 1 x 1 2   2   dx 1dx 2  2 x 1 ,x 2  x 2 Rx 1 ,  x 1 Rx 2 ,  x 2 x 2  x 1 ,x 2 Rx 1 ,  x 1 2Rx 2 ,  x 2 x 2 2   1,  2,. 4.3.22 Now we go to estimate the quantities 1,  2   dx 1dx 2  2 x 1 ,x 2  x 1 Rx 2 ,  x 2 Rx 1 ,  x 1 x 1  x 1 ,x 2 Rx 2 ,  x 2 2Rx 1 ,  x 1 x 1 2 4.3.23 and 2,  2   dx 1dx 2  2 x 1 ,x 2  x 2 Rx 1 ,  x 1 Rx 2 ,  x 2 x 2  x 1 ,x 2 Rx 1 ,  x 1 2Rx 2 ,  x 2 x 2 2 4.3.24 From Eq. (4.3.23) using Eq. (4.3.14) we obtain 140 |1,|  2   dx 1dx 2  2 x 1 ,x 2  x 1 Rx 1 ,  x 1 Rx 2 ,  x 2 x 1  |x 1 ,x 2 |Rx 1 ,  x 1 2Rx 2 ,  x 2 x 1 2  2O1/2    Rx 2 ,  x 2 Rx 1 ,  x 1 x 1 dx 1dx 2   O3/2    Rx 2 ,  x 2 2Rx 1 ,  x 1 x 1 2 dx 1dx 2   2O1/2    Rx 2 ,  x 2dx 2   Rx 1 ,  x 1 x 1 dx 1  O3/2    Rx 2 ,  x 2dx 2   2Rx 1 ,  x 1 x 1 2 dx 1     Rx 2 ,  x 2dx 2 2O1/2    Rx 1 ,  x 1 x 1 dx 1  O3/2    2Rx 1 ,  x 1 x 1 2 dx 1 . 4.3.25 From Eq. (4.3.24) using Eq. (4.3.14) we obtain |2,|  2   dx 1dx 2  2 x 1 ,x 2  x 2 Rx 1 ,  x 1 Rx 2 ,  x 2 x 2  |x 1 ,x 2 |Rx 1 ,  x 1 2Rx 2 ,  x 2 x 2 2  2O1/2    Rx 1 ,  x 1 Rx 2 ,  x 2 x 2 dx 1dx 2  O3/2    Rx 1 ,  x 1 2Rx 2 ,  x 2 x 2 2 dx 1dx 2 2O1/2    Rx 1 ,  x 1dx 1   Rx 2 ,  x 2 x 2 dx 2  O3/2    Rx 1 ,  x 1dx 1   2Rx 2 ,  x 2 x 2 2 dx 2    Rx 1 ,  x 1dx 1 2O1/2    Rx 2 ,  x 2 x 2 dx 2  O3/2    2Rx 2 ,  x 2 x 2 2 dx 2 . 4.3.26 Having substituted Eq. (4.3.18) into Eq. (4.3.25) and Eq. (4.3.19) into Eq. (4.3.26) and having applied Lemma 4.3.1 we have finalized the proof of the Eq. (4.3.9). We assume now that 141     |x1 ,x 2 |2dx 1dx2  1. 4.3.27 From Eq. (4.3.27) and Eq. (4.3.17) by Lemma 4.3.1 we obtain     R 2x 1 ,  x 1R 2x 2 ,  x 2|x 1 ,x 2 |2dx 1dx2  1 4.3.28 From Eq. (4.3.16) Eq. (4.3.17) by Lemma 4.3.1 we obtain x A     x 1#x 1 ,x 2 dx 1dx 2    x 1R2x 1 ,x 1R2x 2 ,x 2|x 1 ,x 2 |2dx 1dx 2  x 1 , x B     x 2#x 1 ,x 2 dx 1dx 2    x 2R2x 1 ,x 1R2x 2 ,x 2|x 1 ,x 2 |2dx 1dx 2  x 2 . 4.3.29 We choose now: x 1  xA,  x 2  xB  xA  x0 , then a boundary condition xB  xA   x0 given by Eq. (4.3.10) is satisfied. IV.4. Position-momentum entangled photon pairs in non-linear wave-guide The physical system where we expect the entangled photon states to appear include: (A) a Kerr-type nonlinear single-mode wave-guide characterized by strong photon-photon coupling [12] or (B) a chain of coupled non-linear resonators. For two photons with momenta k1  k0  δkandk2  k0  δk and dispersion ωk0  δk  ωk0  vδk  βδk2 /2, 4.4.1 where v is the photon group velocity, the variation of the energy of a photon pair is 142 2ω  ωk0  δk  ωk0  δk  2ωk0  βδk2 . 4.4.2 As the photon-photon interaction conserves both energy and longitudinal momentum, the two-photon states propagating along the non-linear transmission line can be described by the Fock function |ψ2k0   dk1dk2δk1  k2  2k0fk1  k2|k1 ,k2  4.4.3 Fig. 4.4.1. Entangled two-photon states in non-linear wave guides. Adapted from [12] (a) Spectrum of a two-photon state, E  E  2ωk0|β|/κ2 , with total momentum 2k 0 in a wave-guide with quadratic dispersion (4.3.1) for β  0,κ  0 (left) and β  0,κ  0 (right). Solid line corresponds to the continuous spectrum, while the single eigenvalue corresponding to the entangled state is shown by dashed line. (b) Wigner function of the two-photon entangled state. It takes negative values, which is a hallmark of non-Gaussian entangled states. 143 To demonstrate the principle of position-momentum entanglement of photons in Kerr-nonlinear systems, we, first, consider the entangled photon pairs in non-linear optical wave-guides. Classically, Kerr nonlinearity in an isotopic medium manifests itself in the third-order polarisation P3  3E EE  αE EE, where + and correspond to positive and negative frequency parts, E is the electric field, 3 is the susceptibility of the medium 3=xyxy 3 , α xxyy 3 /23. Quantizing electromagnetic field, integrating over transverse degrees of freedom, and neglecting magneto-optical effects (α  0) leading to entanglement over polarization degrees of freedom, one obtains the following Hamiltonian (  c  1): H  H0  Hint,H0  k kωkak  ak, Hint  k L  k1,k2,k3,k4 δk 1  k 2 ,k 3  k 4ak4  ak3  ak1 ak2 , 4.4.4 where akak   is the annihilation (creation) operator of a photon with longitudinal momentum k and energy ω k , L is the length of the system. The non-linear term Hint in Eq. (4.4.3) describes photonphoton interaction with coupling κ  πω2χ3/2nr4A0 , where n r is the refraction index, A is the area occupied by the wave-guide mode and 0 is the vacuum permittivity. Hamiltonian (4.4.3) can be diagonalized exactly in the case of 2ω  δk2 . We consider a sector of the Hilbert space, which consists of all the two-photon states with the total pair momentum 2k 0 and assume the effective mass approximation for the wave-guide dispersion given by Eq. (4.4.1). In the coordinate domain, ax  1/ L k ak expik  k0x, the Hamilton Eq.(4.4.3) takes the form H  dx ωk0 axax  ivaxxax  12 βax 2ax  1 2 dx 1dx 2ax1 ax2 Ux1  x 2ax1ax2 , 4.4.5 where Ux1  x2  2κδx1  x2. For a two-photon state, described by the wave-function 144 t A/Bx 1 ,x 2   ψ  dx 1dx 2 fx 1 ,x 2ax1 ax2 |0, 4.4.6.a one obtains the following Schrödinger equation: 2ωk0  ivx1  x2  1 2 βx1 2  x2 2   2κδx 1  x 2fx1 ,x 2  Efx 1 ,x 2, 4.4.6.b where E is the energy of a two-photon state. Equation (4.4.6b) has scattering state solutions, which correspond to the continuous spectrum of non-interacting photons with energies given by Eq.(4.4.2) (See Fig. 4.4.1(a)). When the curvature of the wave-guide dispersion and the photon coupling constant κ are of opposite signs, βκ  0, there exists a bound state solution with fx 1 ,x 2  ξ 2L exp|x 1  x 2 |ξ,ξ  |κ/β| 4.4.7 The energy of this state is split from the continuum of weakly correlated scattering states, as we show in Fig. 4.4.1(a), and it is given by Eb  2ωk0  κ2 /β, 4.4.8 as expected from binding of a one-dimensional massive particle to an attractive -functional potential well [13]. In the momentum domain, the two-photon bound state wave-function is given by Eq.(4.4.3) with fk 1  k 2  8ξ3/2 2L k 1  k 22  4ξ2 4.4.9 The state (4.4.9) can be characterised by the Wigner function defined as the expectation value β δ 145 Wx1 ,k1 ;x2 ,k2  π2ψ|Πx1 ,k1 Πx2 ,k2|ψ of the parity operator Πx,k   dζe2ixζakζ  |00|akζ. After straightforward calculations, one obtains Wx 1 ,k 1 ;x 2 ,k 2  ξ2e2ξ|δx| 2π2δk 2  ξ2 cos2δk|δx|  ξ δk sin2δk|δx|δk 1  k 2 ;2k 0, 4.4.10 where δx  x1  x2 . This function is negative for cos2δk|δx|  ξ/δksin2δk|δx|  0, as shown in Fig. 4.4.1(b), which implies that the state (4.4.9) is entangled in position-momentum degrees of freedom. Moreover, for ξ  , the two-photon wavefunction approaches the ideal Einstein-Podolsky-Rosen state in which position and momenta are perfectly (anti-) correlated: ψ  dδk|k0  δk,k0  δk  dxe2ik0x |x,x. Alternatively, to demonstrate that the state (4.4.9) is entangled in position-momentum degrees of freedom, one can find the uncertainties x1  x2 and k1  k2 calculated over the joint probability distributions Px1 ,x2 and Pk1 ,k2 respectively, for which, the separability criterion: x2  x12k2  k2  1, 4.4.11 can be applied. Although, the states for which the inequality (4.4.11) is violated are inseparable, they do not necessarily lead to EPR paradox. In order for an EPR "paradox" to arise, correlations must violate a more strict inequality: 146 x2  x12k1  k22  1/4, 4.4.12 which can be accessible experimentally. Nonlocal Schrödinger equation (4.3.9) corresponding to Schrödinger equation (4.4.6) (see subsection IV.3) is   dx 1d x 2 2ωk0  ivx1  x2  12 βx1 2  x2 2   2κδx 1  x 2f #x 1 ,x 2  Efx 1 ,x 2  0, 4.4.13 where f #x1 ,x2 is given by Eq. (4.3.8). Remark 4.4.1. Note that. We assume now the canonical postulate of locality. Then: (a) Whenever a measurement of the coordinate x 2 of a particle B is performed at instant t with the result x 2 B  x2  ,x2  ,  1 according to quantum mechanics a state vector | t x2  B collapses at instant t to the state vector  t,,,xB x2 B ~  LxB B ,|t x2  B , 4.4.14 see Chapter I. (b) Under conditions given by Eq. (4.4.14) two-particle wave function t A/Bx1 ,x2  given by Eq. (4.4.6b), collapses at instant t by the law t A/Bx1 ,x2  col lapse   LxB B ,t A/Bx 1 ,x2 . 4.4.15 Remark 4.4.2. Note that. We assume now the postulate of nonlocality. Then: (i) Whenever a measurement of the coordinate x 1 of a particle B is performed at instant t with the result xB  x1  ,x1  ,  1. Then: (a) According to quantum mechanics a state vector | t x  B collapses at instant t to the state vector  t,,,xB x1 B ~  LxB B ,|t x1  B , 4.4.16 147 where  LxB B , is a norm-reducing, positive, self-adjoint, linear operator in the 2 -particle non projective Hilbert space H, representing the localization of particle B around the point x B , see Chapter I. (b) According postulate of nonlocality (see Chapter I) a state vector | t x2  A immediately collapses at instant to the state vector  t,,,xA x1 A ~  LxBx0 A ,|t x1  A 4.4.17 and this is true independent of the distance in Minkovski spacetime M4  1,3 that separates the particles. Thus |t x  B col lapse   t,,,xB x B  |t x  A col lapse   t,,,xBx0 x A . 4.4.18 (ii) Under conditions given by Eq. (4.4.16) Eq. (4.4.18) two-particle wave function t A/Bx1 ,x2  given by Eq. (4.4.6b) collapses at instant t by the law t A/Bx1 ,x 2  col lapse   LxBx0 A  LxB B ,t A/Bx1 ,x2 . 4.4.19 IV.5. Position-momentum entangled photon pairs and the experimental verification of the postulate of nonlocality In paper [14] it is reported on a demonstration of the EPR paradox using position-entangled and momentum-entangled photon pairs produced by spontaneous parametric down conversion. Transverse correlations from parametric down conversion have been studied both theoretically and experimentally. It was find experimentally that the position and momentum correlations are strong enough to allow the position or momentum of a photon to be inferred from that of its partner with a product of variances  0.012 , which violates the 148 separability bound by 2 orders of magnitude. In the idealized entangled state proposed by EPR, the positions and momenta of the two particles are perfectly correlated. However such idealized entangled state is non-normalizable and cannot be realized in the laboratory. However, the state of the light produced in parametric down conversion can be made to approximate the EPR state under suitable conditions. In parametric down conversion, a pump photon is absorbed by a nonlinear medium and reemitted as two photons (conventionally called signal and idler photons), each with approximately half the energy of the pump photon. Considering only the transverse components, the momentum conservation of the down conversion process requires p1  p2  pp , where 1,2, and p refers to the signal, idler, and pump photons, respectively. Provided the uncertainty in the pump transverse momentum is small, the transverse momenta of the signal and idler photons are highly anticorrelated. The exact degree of correlation depends on the structure of the signal idler state. In the regime of weak generation, this state has the form |1,2  |vac  dp1dp2Ap1 ,p2 |p1 ,p2 , 4.5.1 where vac denotes the vacuum state and the two-photon amplitude Ap 1 ,p 2  is Ap 1 ,p 2   Epp1 ,p2  expikzL  1 ikz . 4.5.2 Here is the coefficient of the nonlinear interaction, Ep is the amplitude of the plane-wave component of the pump with transverse momentum p1p2 ,L is the length of the nonlinear medium, and 149 kz  kp,z  k1,z  k2,z (where k  p/) is the longitudinal wave vector mismatch, which generally increases with transverse momentum and limits the angular spread of signal and idler photons. The vacuum component of the state makes no contribution to photon counting measurements and may be ignored. Also, there is no inherent difference between different transverse components; so without loss of generality, we consider the scalar position and momentum. The narrower the angular spectrum of the pump field and the wider the angular spectrum of the generated light, the more closely the integral (4.4.1) approximates dp1dp2p1  p2 |p1 ,p2   |EPR and the stronger the correlations in the position and momentum become. The experimental setup used to determine position and momentum correlations is portrayed in Fig. 4.5.1(a)-(b). The idea is to measure the positions and momenta by measuring the down converted photons in the near and far fields, respectively [15]. The source of entangled photons is spontaneous parametric down conversion generated by pumping a 2 mm thick type-II-bariumborate (BBO) crystal with a 30 mW, cw, 390 nm laser beam. A prism separates the pump light from the down converted light. The signal and idler photons have orthogonal polarizations and are separated by a polarizing beam splitter. In each arm, the light passes through a narrow 40 m vertical slit, a 10 nm spectral filter, and a microscope objective. The objective focuses the transmitted light onto a multimode fiber which is coupled to an avalanche photodiode singlephoton counting module. The spectral filter ensures that only photons with nearly equal energies are detected. 150 Fig. 4.5.1(a). Experimental setup for measuring position photon correlations. Position correlations are obtained by imaging the birth place of each photon of a pair onto a separate detector. Adapted from [14] Fig. 4.5.1(b). Experimental setup for measuring correlations in transverse momentum. Correlations in transverse momentum are obtained by imaging the propagation direction of each photon of a pair onto a separate detector. Adapted from [14] 151 To measure correlations in the positions of the photons, a lens of focal length 100 mm (placed prior to the beam splitter) is used to image the exit face of the crystal onto the planes of the two slits [Fig.4.5.1(a)]. One slit is fixed at the location of peak signal intensity. The other slit is mounted on a translation stage. The photon coincidence rate is then recorded as a function of the displacement of the second slit. To measure correlations in the transverse momenta of the photons, the imaging lens is replaced by two lenses of focal length 100 mm, one in each arm, at distance f from the planes of the two slits [Fig.4.5.1(b)]. These lenses map transverse momenta to transverse positions, such that a photon with transverse momentum k comes to a focus at the point x  fk/k in the plane of the slit. Again, one slit is fixed at the location of the peak count rate while the other is translated to obtain the coincidence distribution. By normalizing the coincidence distributions, the conditional probability density functions px2 |x1  and pp2 |p1  were obtained (see Fig. 4.5.2-4.5.3). These probability densities are then used to calculate the uncertainty in the inferred position or momentum of photon 2 given the position or momentum of photon 1: x 2 2x 1    x 22px 2 |x 1 dx 2   x 2px 2 |x 1 dx 2 2 , p2 2p1    p22pp2 |p1 dx 2   p2pp2 |p1 dx 2 2 . 4.5.3 Because of the finite width of the slits, the raw data in Fig. 4.5.24.5.3 describe a slightly broader distribution than is associated with the down conversion process itself. By adjusting the computed values of x 2 and p2 to account for this broadening (an adjustment smaller than 10%), we obtain the correlation uncertainties x2  0.027 mm and Δp2 3.7 mm 1 . 152 Fig. 4.5.2. The conditional probability distribution of the relative birthplace of the entangled photons. The solid line is the theoretical prediction and the dots are the experimented data. Adapted from [14] Fig. 4.5.3. The conditional probability distribution of the relative transverse momentum of the entangled photons. The solid line is the theoretical prediction and the dots are the experimented data. Adapted from [14] 153 The widths of the distributions determine the uncertainties in inferring the position or the momentum of one photon from that of the other. The experimentally measured variance product is then [14] expx2 2x1 expp2 2p1   0.012 . 4.5.4 Also shown in Fig. 4.5.2-4.5.3 are the predicted probability densities. These curves contain no free parameters and are obtained directly from the two-photon amplitude Ap1 ,p2  [14], which is determined by the optical properties of BBO and the measured profile of the pump beam. Figure 4.5.2 indicates that the correlation widths obtained are intrinsic to the down conversion process and are limited only by the degree to which it deviates from the idealized EPR state (4.5.1). The value of p2  p1  is limited by the finite width of the pump beam. The pump photons in a Gaussian beam of width w have an uncertainty /2w in transverse momentum which, due to conservation of momentum, is imparted to the total momentum p1  p2 of the signal and idler photons. The value of x2  x1  is limited by the range of angles over which the crystal generates signal and idler photons. If the angular width of emission is  , then the principle of diffraction indicates that the photons cannot have a smaller transverse dimension than ~k s,i1 . Careful analysis based on the angular distribution of emission yields x2  x1   1.88k s,i1 [14]. With the measured beam width of w  0.17 mm and predicted angular width 0.012 rad, the theory predicts [14]: th x2 2x1 th p2 2p1   0.00362 . 4.5.5 Remark 4.5.1. This is somewhat smaller than the experimentally calculated value of 0.012 , even though the data appear to closely match the theoretical curves expx2 2x1 expp2 2p1   th x2 2x1 th p2 2p1   0.012  0.00362  0.00642 . 4.5.6 Remark 4.5.2. The reason for this discrepancy is that the experimental distributions have small (1% of the peak) but very broad wings. 154 Remark 4.5.3. The origin of these uncoincidence counts is unknown [14]. Remark 4.5.4. In paper [14] it was assumed that these counts are perhaps due to scattering from optical components. If these counts are treated as a noise background and subtracted, the experimentally obtained uncertainties come into somewhat better agreement with the theoretically predicted values, yielding an uncertainty product of 0.0042 : EPR nonloc.x2  x1 ,p2  p1   expx2 2x1 expp2 2p1   th x2 2x1 th p2 2p1   0.0062 . 4.5.7 Thus final value of uncoincidence counts is EPR nonloc.x2  x1 ,p2  p1   0.0062 . 4.5.8 Remark 4.5.5. Note that the separability criterion derived by Mancini et al. [16] is more useful here. We remind that it states that separable systems satisfy the joint uncertainty product x2  x1 p2  p1   2 , 4.5.9 where the uncertainties are calculated over the joint probability distributions Px1 ,x2  and Pp1 ,p2 , respectively. In these experiments the widths of the conditional probability distributions are P. Therefore the results of [14] constitute a 2-order-of-magnitude violation of Mancini's separability criterion as well as a strong violation of EPR criterion. IV.6. The EPR Paradox Resolution by using quantum mechanical formalism based on the probability representation of quantum states IV.6.1. Preleminaries We remind that any given n -dimensional quantum system is identified by a set Q : 155 Q  H,,,2,1 ,H,G, |t  4.6.1 where: (i) H that is some infinite-dimensional complex Hilbert space, (ii)   ,,P that is complete probability space , (iii)   n , that is measurable space , (iv) 2,1 that is complete space of complex valued random variables X :   n such that   XdP  ,   X2dP  , 4.6.2 see Chapter II subsection II.1 postulate Q.I.1. Remark 4.6.1. Let BM4 be a Boolean algebra of physical events in Minkowski spacetime and let M4 ph  BM4 ,P be a measure algebra of physical events in Minkowski spacetime, i.e., M4 ph that is a Boolean algebra BM4 with a probability measure P, see Chapter III subsection III.2, Definition 3.2.3. We remind that we denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc., and we write for a short AOcx,BOcx, . . . iff there physical events Ax,Bx. . . were occured. Remark 4.6.2. We assume that particle A is initially in the state |A   H. Let Aq, t  A |A ,Q,q,q, t  BM4 be a physical event which consists of the performing a measurement of the observable Q   q 1 q 2 q|qq|dq with the accuracy q, and the result is obtained in the range q  q,q  q at the instant t. We assume that A |A ,Q,q,q, t  M4 ph . Remark 4.6.3. Note that: if the physical event was occurred then immediately after the measurement at the instant A |A ,Q,q,q, t t 156 unconditional measure collapses to conditional measure P X A |A ,Q,q,q, t , where X  M4 ph : P X A |A ,Q,q,q, t  P X  A |A ,Q,q,q, t P A |A ,Q,q,q, t . 4.6.3 Remark 4.6.4. Remind if we are to suppose that a particle at a definite position x is to be assigned a state vector |x  H, and if further we are to suppose that the possible positions are continuous over the range , and that the associated states are complete, then we are lead to requiring that any state |A  of the particle must be expressible as |A      |x x|A dx 4.6.4 with the states |x  by  -function normalised, i.e. x|x    x  x . Definition 4.6.1. Let B   a,b  a,b where a,b  Ba,b is the Borel algebra on a set a,b. Let |  H. We define now a signed measure P|A  : B    by formula P|A A   A xp |A xdx, 4.6.5 where p|A x  |x|A | 2 . Remark 4.6.5. We assume now that ,,P  ,B,PB  and P|  PB , i.e. P | is absolutely continuous with respect to P .By RadonNicodym theorem we obtain for any A  a,b : P|A   A X |dP, 4.6.6 P 157 i.e. X |  dP| dP . 4.6.7 Remark 4.6.6. We assume now that: (i) a measure algebra M4 ph  BM4 ,P admits a representation  : M4 ph  ,B,PB  of the measure algebra M4 ph  BM4 ,P in the measure algebra B  ,B,PB , such that (ii) PBX  P 1X for any X  B and (iii) for any physical event such that A |A ,Q,q,q, t  M4 ph (see Remark 4.6.2) the following condition holds  A |A ,Q,q,q, t  |q  q  X |A   q  q, 4.6.8 where |q  q  X |A   q  q  B . IV.6.2.The EPR Paradox Resolution The classical weak EPR argument We briefly remind now the EPR argument [1]. Suppose that a system of two identical particles is prepared in a state such that their relative distance is large and constant |r1  r2 |  L  x0 , i.e., they are space-like separated, and the total momentum is zero p 1  p 2  0 (see Fig. 4.6.1). This preparation is, in principle, possible because the two observables, say x 1  x 2 and p 1  p 2 , are compatible, i.e., both of them can be set to certain values with certainty on the same state. Correspondingly according to quantum mechanics they are in fact represented by commuting operators [1]. 158 Fig. 4.6.1. Schematic representation of EPR thought experiment Remark 4.6.7. Then one can measure the value of either of the two incompatible single particle observables, say x 1 or p1 and correspondingly deduce the value of either x2  x0  x1 or p2  p1 without interacting with particle 2. Because of this they correspond, according to the EPR argument, to elements of reality of the state of particle 2 that are independent of measurements and should be predictable by the theory [1]. On the other hand, quantum mechanics cannot predict the value of both x 2 and p2 on the same state, because they are incompatible observables and this would be in contrast to Heisenberg uncertainty principle. Remark 4.6.8. Thus, to conclude EPR, there are elements of reality of a state that cannot be predicted by the theory and therefore the theory is incomplete [1]. The strong EPR argument Remark 4.6.9. Note then in additional to canonical EPR thought experiment: (i) one can measure at instant t the value of single particle 1 observable, say x 1 t and deduce the value x2 t  x0  x1 t of particle 2 at instant t without interacting with particle 2 which at instant t is in a state, say 2 t . Such a measurement however is not disturbed by the particle 2 and thus is not altered by its state 2 t and therefore the value of single particle 2 observable, say p2 is the same as before the measurement on particle 1. Therefore one can measure the value in the state exactly without any p2 t 2 t 159 uncertainty. On the other hand, Heisenberg uncertainty principle predicts that the position and the momentum of any particle cannot both be measured or predicted exactly, at the same time , even in the theory. Let A and B be two particles A and B with a state vector |A  |A      |x x|A dx 4.6.9 and with a state vector |B  |B      |x x|B dx 4.6.10 respectively, and with perfectly correlated position xB  xA  x0 4.6.11 and perfectly anti-correlated momentum pB  pA. 4.6.12 We define now a signed measures P|A  : B    and P|B  : B    by formulas P|A A   A xp |A xdx, 4.6.13 and P|B A   A xp |B xdx, 4.6.14 where p|A x  |x|A | 2 and p|B x  |x|B | 2 respectively. Remark 4.6.10. We assume now that ,,P  ,B,P and (i) P|A   P, (ii) P|B   P. We define now random variables X |A  and X |B  by formulas x 2 t p2 t t 160 X |A   dP|A  dP ,X |B   dP|B  dP 4.6.15 respectively. Notice that from Eq. (4.6.11), Eq. (4.6.13)-(4.6.14) and Eqs.(4.6.15) it follows that X |B   X |A   x0 ,a.s. 4.6.16 Let B |B ,X,xB ,x, t  BM4 be a physical event which consists of performing a measurement of the observable X  x1 x2 x|x x |dx with an accuracy x, and the result is obtained in the range xB  x,xB  x at instant t. Remark 4.6.11. Note that: if the physical event was occurred then immediately after the measurement at the instant unconditional measure collapses to conditional measure P X B |B ,X,xB ,x, t , where X  M4 ph : P X B |B ,X,x B ,x, t  P X  B |B ,X,x B ,x, t P B |B ,X,x B ,x, t , 4.6.17 see Remark 4.6.3. Notice that: (i) from Eq. (4.6.8) it follows that  B |B ,X,xB ,x, t  X B xB ,x, 4.6.18 where we write for short X B xB ,x instead |xB  x  X |B   xB  x, i.e. B |B ,X,xB ,x, t t P 161 X B xB ,x  |xB  x  X |B   xB  x, 4.6.19 see Remark 4.6.2; (ii) from Eq. (4.6.11), Eq. (4.6.16) and Eq. (4.6.19) it follows that X B x B ,x  |x B  x  X |B   x B  x  |x B  x 0   x  X |B   x 0  x B  x 0   x  |x A  x  X |A   x A  x  X B x A,x, 4.6.20 and thus X B xB ,x  X B xA,x 4.6.21 (iii) from Eq. (4.6.17) - (4.6.19) it follows that: (i) unconditional measure PB immediately after the measurement at instant t collapses to conditional measure PBX||B xB ,x, where X  B : PB X |B xB ,x  PB X  X B x B ,x PB X B xB ,x . 4.6.22 Remark 4.6.12. (i) From Eq. (4.6.22) it follows that the unconditional probability density function pBx  |x|B |2 immediately after the measurement at instant t collapses to the following conditional probability density function as pB x|X B x B ,x  pBx PB X B x B ,x  x  X B x B ,x 0  x  X B x B ,x 4.6.23 see Appendix B. (ii) From Eq. (4.6.21) and Eq. (4.6.22) it follows that the unconditional probability density function pAx  |x|A |2 immediately after the measurement at instant t collapses to the following conditional probability density function as 162 pA x|X  A x A,x  pAx PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x 4.6.24 From Eq. (4.6.23) it follows that a wave function Bx  x|B  immediately after the measurement at instant t collapses to the following wave function B coll x  Bx PB X B x B ,x  x  X B x B ,x 0  x  X B x B ,x 4.6.25 From Eq. (4.6.24) it follows that immediately after the measurement on particle B at instant t a wave function Ax  x|A  collapses to the following wave function A coll x  Ax PB X  A x A,x  x  X  A x A,x 0  x  X  A x A,x 4.6.26 Thus the measurement on particle B alters a wave function Ax even if particles A and B are space-like separated and therefore EPR paradox disappears. 163 Chapter V ERP-B PARADOX RESOLUTION V.1. EPR-B experiment The EPR-B, the spin version of the Einstein-Podolsky-Rosen experiment proposed by Bohm, see [17], [18] Bohm: "We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore   1/ 2 12  12 where 1 refers to the wave function of the atomic state in which one particle (A) has spin /2, etc. The two atoms are then separated by a method that does not influence the total spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle A is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle B is opposite to that of A. If this were a classical system, there would be no difficulty in interpreting the above results, because all components of the spin of each particle are well defined at each instant of time. Thus, in the molecule, each component of the spin of particle A has, from the very beginning, a value opposite to that of the same component of B; and this relationship does not change when the atom disintegrates. In other words, the two spin vectors are correlated. Hence, the measurement of any component of the spin of A permits us to conclude also that the same component of B is opposite in value. The possibility of obtaining knowledge of the spin of particle B in this way evidently does not imply any interaction of the apparatus with particle B or any interaction between A and B. In quantum theory, a difficulty arises, in the interpretation of the 164 above experiment, because only one component of the spin of each particle can have a definite value at a given time. Thus, if the x component is definite, then the y and z components are indeterminate and we may regard them more or less as in a kind of random fluctuation. In spite of the effective fluctuation described above, however, the quantum theory still implies that no matter which component of the spin of A may be measured the same component of the spin of B will have a definite and opposite value when the measurement is over. Of course, the wave function then reduces to 12 or 12, in accordance with the result of the measurement. Hence, there will then be no correlations between the remaining components of the spins of the two atoms. Nevertheless, before the measurement has taken place (even while the atoms are still in flight) we are free to choose any direction as the one in which the spin of particle A (and therefore of particle B) will become definite. In order to bring out the difficulty of interpreting the result, let us recall that originally, the indeterminacy principle was regarded as representing the effects of the disturbance of the observed system by the indivisible quanta connecting it with the measuring apparatus. This interpretation leads to no serious difficulties for the case of a single particle. For example, we could say that on measuring the z component of the spin of particle A, we disturb the x and y components and make them fluctuate. This point of view more generally implies that the definiteness of any desired component of the spin is (along with the indefiniteness of the other two components) a potentiality which can be realized with the aid of a suitably oriented spinmeasuring apparatus. In the case of complementary pairs of continuous variables, such as position and momentum, one obtains from this point of view the well known wave-particle duality. In other words, the electron, for example, has potentialities for mutually incompatible wave-like and particle-like behavior, which are realized under suitable external conditions. In the laboratory those conditions are generally determined by the measuring apparatus although, more generally, 165 they may be determined by any arrangement of matter with which the electron interacts. But in any case, it is essential that there must be an external interaction, which disturbs the observed system in such a way as to bring about the realization of one of its various mutually incompatible potentialities. As a result of this disturbance, when any one variable is made definite, other (noncommuting) variables must necessarily become indefinite and undergo fluctuation. Evidently, the foregoing interpretation is not satisfactory when applied to the experiment of ERP. It is of course acceptable for particle A alone (the particle whose spin is measured directly). But it does not explain why particle B (which does not interact with A or with the measuring apparatus) realizes its potentiality for a definite spin in precisely the same direction as that of A. Moreover, it cannot explain the fluctuations of the other two components of the spin of particle B as the result of disturbances due to the measuring apparatus. In this subsection we explain EPR-B experiment using reduction to a sort of generic EPR correlations for two particles A and B with maximally correlated position zA and zB . This explanation avoids the EPR-Bohm paradox. Fig. 5.1.1. Einstein-Podolsky-Rosen-Bohm experiment 166 Fig. 5.1.1 presents the Einstein-Podolsky-Rosen-Bohm experiment. A source S created in O pairs of identical atoms A and B, but with opposite spins. The atoms A and B split following the y -axis in opposite directions, and head towards two identical Stern-Gerlach apparatus EA and EB . The electromagnet EA "measures" the spin of A along the z-axis and the electromagnet EB "measures" the spin of B along the z -axis, which is obtained after a rotation of an angle  around the y -axis. Remark 5.1.1. So far we have consistently made use of the idea that if we know something definite about the state of such a physical system, say that we know z component of the spin of a particle is Sz   1 2 , then we assign to the system the state |Sz    1 2  , or, more simply, |. Remark 5.1.2. We can also note that these two states | and | are mutually exclusive, i.e. if an atom is in the state |, then the result Sz   1 2  is never observed, and furthermore, we note that the two states | and | cover all possible values for Sz. Remark 5.1.3. When we say that we `know' the value of some physical observable of a quantum system, we are presumably implying that some kind of measurement has been made that provided us with this knowledge. Furthermore, it is assumed that in the process of acquiring this knowledge, the system, after the measurement has been performed, survives the measurement, and moreover if we immediately remeasured the same quantity, we would get the same result. This is certainly the situation with the measurement of spin in the Stern-Gerlach experiment. If an atom emerges from one such a set of apparatus in a beam which indicates Sz  1 2  for that atom, and we passed the atom through a second apparatus, also with its magnetic field oriented in the z direction, we would find the atom emerging in the Sz  1 2  beam once again. Under such circumstances, we would be justified in saying that the atom has been prepared in the state 167 Sz  1 2  , etc. Definition 5.1.1. Assume that atom A has been prepared in the state Sz  1 2  , Sz   1 2  , etc. Then we will say that these events Sz  1 2  , Sz   1 2  , etc. occur. We will denote these events by symbols Sz  1 2  A , Sz   1 2  A , etc., or 1 2  A ,  1 2  A , etc. Definition 5.1.2. Assume that we know exactly that atom A is in the state 1 2  ,  1 2  , etc. Then we will say that these events 1 2  ,  1 2  , etc. occur and we will denote these events again by symbols 1 2  A ,  1 2  A , etc. Definition 5.1.3. Assume that these events 1 2  A ,  1 2  A , etc. occur in the point x  t,x1 ,x2 ,x3   t,r  M4 of Minkowski spacetime M4 . Then we will denote these events by symbols 1 2  x A ,  1 2  x A , etc. or 1 2  tA,zA  A ,  1 2  tA,zA  A , etc. Assumption 5.1.1. We claim for any x  M4 that: 1 2  x A  M4 ,  12  x A  M4 ,etc. 5.1.1 Here M4 is a measure algebra M4  BM4 ,P with a probability measure P, see Chapt. III, subsection III.2, Definition 3.2.3. Remark 5.1.4. Note that for any x  M4 and for any atom A these events 1 2  x A ,  1 2  x A are mutually exclusive, see Remark 5.1.2, and therefore for any x  M4 P 1 2  x A   12  x A  0. 5.1.2 Remark 5.1.5. We remind that if an atom is prepared in an arbitrary initial state |S, then the probability amplitude of finding it in some other state |S   is given by S|S  S||S  S||S 5.1.3 which leads, by the cancellation trick to 168 |S  ||S  ||S 5.1.4 and therefore the states | form a complete set of orthonormal basis states for the state space of the system. Suppose we have an n -dimensional quantum system which contains only a quantum observable with discrete values such as Sz, etc. Then we claim the following: Q d .V.1. Any given n -dimensional quantum system which contains only a quantum observable with discrete values such that mentioned above is identified by a set Qd: Q d  Hd,d,d,2,1 d ,Gd, |t  , 5.1.5 where: (i) Hd that is some finite-dimensional complex Hilbert space, (ii) d  d,d,Pd that is complete probability space , (iii) d  n ,d that is measurable space , (iv) 2,1 d d that is complete space of discrete complex valued random variables Xd : d  n such that  d XddPd  ,  d Xd 2dPd   5.1.6 (v) Gd : Hd 2,1d that is one to one correspondence such that |Qd|   d Gd Qd|  dPd  Ed Gd Qd|  5.1.7 for any |  Hd and for any Hermitian operator with discrete spectrum Qd : Hd  Hd, where Qd   Hd  CHd,CHd is C algebra of the Hermitian adjoint operators in Hd and  Hd is commutative subalgebra of CHd. (vi) | t  is a continuous vector function |t  :   Hd which represented the evolution of the quantum system Qd. Q d .V.2. For any |1 , |2   Hd and for any Hermitian operator 169 Q d : Hd  Hd such that 1 Qd 2  2 Qd 1  0 5.1.8 the equality is valid Gd Qd|1   |2    Gd Qd|1    Gd Qd|2  . 5.1.9 Remark 5.1.6. Let Sz  and Sz  be discrete random variables Sz  : d  1,1, Sz  : d  1,1 correspondingly such that: (i) Sz   G|, (ii) Pd1   1,where 1  |Sz   1 , (iii) Pd1   0,where 1  |Sz   1 and (i) Sz   G|, (ii) Pd1   1,where 1  |Sz   1 , (iii) Pd1   0,where 1  |Sz   1 . 5.1.10 Let Qc be any n -dimensional quantum system which contains only a quantum observable with continuous values. We remind that such a quantum system is identified with a set Q Q  H,,,2,1 ,G, |t . 5.1.11 Definition 5.1.4. We define now a composite quantum system Qc,d which contains both sort of quantum observables by a set Qc,d Q c,d  Hc,d,c,d,c,d,2,1 c,d ,Gc,d, |t  5.1.12 where: (i) Hc,d  Hc  Hd that is composite complex Hilbert space, (ii) c,d  c,d,c,d,Pd that is complete probability space , with c,d  c d,c,d  c d,c,d  c d,2,1 c,d  2,1 c  2,1 d ,Gc,d  Gc  Gd, (iii) c,d  n ,c,d that is measurable space with с,d  с  d, 170 (iv) 2,1 c,dd that is complete space of random variables Xc,d : c,d  n such that  c,d Xc,ddPc dPd  ,  c,d Xc,d 2dPc dPd  ,  c,d 5.1.13 (v) Gc,d : Hc,d 2,1 c,dd that is one to one correspondence such that |Qc,d|   c,d Gc,d Qc,d|  dPc dPd  Ec,d Gc,d Qc,d|  5.1.14 for any |  Hc,d and for any Hermitian operator Qc,d : Hc,d  Hc,d, where Qc,d  Hc,d  CHc,d,CHc,d is C -algebra of the Hermitian adjoint operators in Hc,d and Hc,d is commutative subalgebra of CHc,d. (vi) | t  is a continuous vector function |t  :   Hd which represented the evolution of the quantum system Qc,d. V.2. EPR-B paradox resolution The usual conclusion of EPR-B experiment is to reject the non-local realism for two reasons: the impossibility of decomposing a pair of entangled atoms into two states, one for each atom, and the impossibility of interaction faster than the speed of light. Remark 5.2.1. We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle, that is the "relaxed locality principle" introduced in Chapter IV.1. Remark 5.2.2. The solution to the entangled state is obtained by resolving the Pauli equation from an initial singlet wave function with a spatial extension as: 171 0rA ,rB  1 2 frAfrB|A  |B   |A  |B , 5.2.1 The initial wave function of the entangled state is the singlet state (5.2.1) with fr  20 2 1 2 e  x 2  y2  z2 40 2 iff r  , 0 iff r   r  x,y, z,0  1,  1 5.2.2 and where |A  and |B  are the eigenvectors of the operators  zA and zB : zA |A   |A ,zB |B   |B . 5.2.3 Remark 5.2.3. We treat the dependence with y strictly quasiclassically, i.e., with speed vyA v0 ,0 A  for A and vy B v0 ,0 B  for B such that P y  vyA v0 ,0 A t    1, P y  vyA v0 ,0 A t    0, P y  vyB v0 ,0 B t    1, P y  vyB v0 ,0 B t    0,   1, 5.2.4 where vy A v0 ,0 A   0A  v0 ,vy B v0 ,0 B   0B  v0 , 0A   cos2 0 A 2 ,0A   sin2 0 A 2 , 0B   cos2 0 B 2 ,0B   sin2 0 B 2 . 5.2.5 The wave function rA,rB, t of the two identical particles A and B, electrically neutral and with magnetic moments 0 , subject to magnetic fields EA and EB , admits on the basis of |A  and |B  four components a,brA,rB, t and satisfies the two-body Pauli equation 172 i a,bt t    2 2m A   2 2m B a,bt  Bj EA jcac,bt  Bj EB  jd ba,dt 5.2.6 with the initial conditions: a,b0,rA,rB  0 a,brA,rB, 5.2.7 where 0 a,brA,rB corresponds to the singlet state (5.2.1). Below we explain the EPR-B experiment by using nonlocal two-body Pauli equation  drAdrB  dt i  #a,bt, t ,rA ,rB  t    2 2m A   2 2m B #a,bt, t ,rA ,rB  Bj EA jca#c,bt, t ,rA ,rB   Bj EB  jd b#a,dt, t ,rA ,rB   O , drA  dx AdyAzA ,drB  dx BdyBzB 5.2.8 with a boundary condition drAdrBzAt1  #t1 , t ,rA ,rB 2  drAdrBzBt2  #t2 , t ,rA ,rB 2 . 5.2.9 One of the difficulties of the canonical interpretation of the EPR-B experiment is the existence of two simultaneous measurements. By doing these measurements one after the other, the interpretation of the experiment will be facilitated. That is the purpose of the two-step version of the experiment EPR-B studied below. V.2.1. First step EPR-B: Spin measurement of A Consider that at time t0 the particle A arrives at the entrance of electromagnet EA . Remark 5.2.4. We assume that a particle A collapses in a magnetic field EA at some instant t  into two particles A  and A  , i.e. the spinor z,y, t collapses in a magnetic field EA at some instant t  into 173 two spinors z,y, t, t , and z,y, t, t , given by Eq. (6.1.9a)- (6.1.9b), see Assumption 5.1.1. Remark 5.2.5. The particles A  and A stay within the magnetic field for a time t   t  lv0 . Thus after exit of the magnetic field EA , at time t1  t0  t  t, the wave functions z,y, t0  t  t, and z,y, t0  t  t, become rA,rB , t0  t  t  frB   f rA, t|A  |B  5.2.10.a and rA,rB , t0  t  t  frB   f rA, t|A  |B  5.2.10.b respectively, with f r, t  cos 0 2 fx, z  z  utexp i muz  t f r, t  sin 0 2 fx, z  z  utexp i muz  t 5.2.11 wherez and u are given by z  BB0  t2 2m  105m, u  BB0  t m  1m/s. 5.2.12 Remark 5.2.6. We deduce that: the beam of particle A is divided into two A  and A  , and the beam of particle B is divided into two B  and B  . Remark 5.2.7. Our first conclusion is: the position of B  and B  does not depend on the spin measurement of A  and A  , only the spins are involved. We conclude from equation (5.2.10) that the spins of A  and B  (A  and B  ) remain opposite throughout the experiment. These are the two properties used in the relaxed causal interpretation. Remark 5.2.8. By "relaxed locality principle" and decoherence it follows that the interaction between A , A  , B , and B  is absent, we assume the existence of wave functions 174 0 ArA,0 A,0 A,0 ArA,0 A,0 A,0 BrB ,0 B,0 B,0 BrB ,0 B,0 B. 5.2.13 V.2.2. Second step EPR-B: Spin measurement of B The second step is a continuation of the first one and corresponds to the EPR-B experiment broken down into two steps. On a pairs of particles A , B  and A  , B  in a singlet state, first we made the Stern and Gerlach measurement on the A  and A  atom at instant t1 between t0 and t0  t  tD : t0  t1  t0  t  tD. 5.2.14 Secondly, we make the Stern and Gerlach measurement on the B  and B  atom with an electromagnet EB forming an angle  with EA at instant t2 between t0  t  tD and t0  2t  tD : t0  t  tD  t2  t0  2t  tD 5.2.15 At the exit of magnetic field EA , at time t0  t  tD, the pair of particles wave functions is given by Eq. (5.2.10a) and Eq. (5.2.10b) respectively. Immediately after the measurements of A  and A  , still at time t0  t  tD, the wave functions of B  and B  depend on the measurements  of A respectively such that: B/ArB, t0  t  t1  frB|B , 5.2.16.a and B/ArB, t0  t  t1  frB|B . 5.2.16.b Then, the measurement of B  and B  at time t2  t0  2t  tD yields, in this two-step version of the EPR-B experiment, the same results for spatial quantization and correlations of spins as in the EPR-B experiment. 175 V.2.3. Resolution of the EPR-B experiment in de Broglie-Bohm interpretation by the "relaxed locality principle" We assume, at the creation of the two entangled particles A and B, that each of the two particles A and B has an initial wave function with opposite spins: 0 ArA ,0 A ,0 A  frA cos 0 A 2 |A   sin 0 A 2 ei0 A |A  5.2.17 and 0 BrB ,0 B ,0 B  frB cos 0 B 2 |B   sin 0 B 2 ei0 B |B   frB cos  2  0 A 2 |B   sin  2  0 A 2 ei 0 A |B   0 BrB ,0 B ,0 B  frB sin 0 A 2 |B   cos 0 A 2 ei0 A |B  5.2.18 with 0 B    0 A and 0 B  0 A  . The two particles A and B are statistically prepared as in the Stern and Gerlach experiment. Then the Pauli principle tells us that the two-body wave function must be antisymmetric; after calculation we find the same singlet state (5.2.1): 0rA,A,A,rB ,B ,B  ei A frAfrB  |A  |B   |A  |B . 5.2.19 Thus, we can consider that the singlet wave function is the wave function of a family of two fermions A and B with opposite spins: the direction of initial spin A and B exists, but is not known. It is a local hidden variable which is therefore necessary to add in the initial conditions of the model. Here, we assume that at the initial time we know the spin of each particle (given by each initial wave function) and the initial position of each particle. 176 V.2.3.1. Step 1: spin measurement of A in de BroglieBohm interpretation In Eq. (5.2.19) particle A can be considered independent on B. We can therefore give it the wave function ArA , t0  t  t  cos 0 A 2 f rA , t|A   sin 0 A 2 ei0 A f rA , t|A  5.2.20 which is the wave function of a free particle in a Stern Gerlach apparatus and whose initial spin is given by 0 A ,0 A . For an initial polarization 0 A ,0 A  and an initial position z0 A , we obtain, in the de Broglie-Bohm interpretation [17] of the Stern and Gerlach experiment, an evolution of the position zAt and of the spin orientation of A ,AzAt, t, see [19]. The case of particles B  is different. B  follows a rectilinear trajectories with yBt  vy v0 ,0 t, zBt  z0 B and xBt  x0 B . By contrast, the orientation of its spin moves with the orientation of the spin of A  : Bt    AzAt, t 5.2.21 and Bt  AzAt, t  . 5.2.22 Remark 5.2.9. Let At,rAt, AzAt, t, AzAt, t denote events such that: "at instant t particles A  obtain the position coordinates rAt  xAt,yAt,zAt and spin orientation  At  AzAt, t and At  AzAt, t. Let Bt,rBt, BzBt, t denote events such that: "at instant t particles B obtain the position coordinates rBt  xBt,yBt,zBt and spin orientation  B  BzBt, t and BzBt, t. Then in accordance with the relaxed principle of locality (see 177 subsection IV.1) we assume that At1 ,rAt, A,At,Bt,rB t, B t,B t s.l .s .   M4 # ,t1 ,r1 , t2 ,r2  s .l .s.  , 5.2.23 see subsection IV.1, Definition 4.1.2. We can then associate the wave functions: BrB, t0  t  t  frBcos Bt 2 |B  5.2.24 and BrB, t0  t  t  frBsin Bt 2 ei Bt|B  5.2.25 These wave functions are specific, because they depend upon initial conditions of A (position and spin). The orientation of spin of the particles B  is driven by the particles A  respectively through the singlet wave functions. V.2.3.2. Step 2: spin measurement of B  in de BroglieBohm interpretation V.2.3.2.1. The prediction of the result of the spin measurement of B  under assumption of canonical postulate of locality At the time t0  t  tD , immediately after the measurement of A, Bt0  t  tD   or 0 in accordance with the value of AzAt, t and the wave functions of B  are given by Eq. (5.2.16a) and Eq.(5.2.16b) respectively. The frame Ox yz corresponds to the frame Oxyz after a rotation of an angle  around the y -axis (see Fig. 5.1.1). B corresponds to the 178 B  -spin angle with the z -axis, and  B to the B-spin angle with the z -axis, then  Bt0  t  tD     or  . In this second step, we are exactly in the case of a particle in a simple Stern and Gerlach experiment (with magnet EB ) with a specific initial polarization equal to    or  and not random like in step 1. Then, the measurement of B, at time t0  2t  tD), gives, in this interpretation of the two-step version of the EPR-B experiment, the same results as in the EPR-B experiment above. Thus we obtain EPR-B paradox again in de Broglie-Bohm interpretation. Remark 5.2.10. Note that the derivation EPR-B paradox in the de Broglie-Bohm interpretation completely based on the canonical postulate of locality. V.2.3.2.2. The prediction of the result of the spin measurement of B  under assumption of postulate of nonlocality We assume now a weak or strong postulate of nonlocality, see subsections I.4.1-I.4.2. At the time t1  t0  t  tD , immediately after the spin measurement of A  , Bt0  t  tD   or 0 in accordance with the value of AzAt, t and the wave functions of B  are given by Eq. (5.2.16a) and Eq.(5.2.16b) respectively. Remark 5.2.11. In accordance with the postulate of nonlocality it follows: (i) Whenever a measurement of the spin of a particle A  is performed at instant t1 and particle A  is found in the state |z, i.e., a state | t1 A collapses at instant t1 to the state |z,A with respect of the Heisenberg spin uncertainty relations, then a state | t1 B  179 immediately collapses at instant t1 to the state |z,B  with respect of the Heisenberg spin uncertainty relations, and this is true independent of the distance in Minkowski spacetime that separates the particles, e.g., |t1 A col lapse  |z,A  |t1 B  col lapse  |z,B  5.2.26 In accordance with Heisenberg spin uncertainty relations (1.4.5) spin of a particle B obtains an uncertainty along direction Oz (see Fig. 5.1.1) and therefore EPR-B paradox disappears. (ii) Whenever a measurement of the spin of a particle A is performed at instant t1 and particle A  is found in the state |z, i.e., a state | t1 A collapses at instant t1 to the state |z,A with respect of the Heisenberg spin uncertainty relations (1.4.5), then a state | t1 B  immediately collapses at instant t1 to the state |z,B  with respect of the Heisenberg spin uncertainty relations (1.4.5), and this is true independent of the distance in Minkowski spacetime that separates the particles, e.g., |t1 A col lapse  |z,A  |t1 B col lapse  |z,B. 5.2.27 In accordance with Heisenberg spin uncertainty relations (1.4.5) spin of a particle B obtains an uncertainty along direction Oz (see Fig. 5.1.1) and therefore EPR-B paradox disappears. V.2.4. Physical explanation of non-local influences using the relaxed principle of locality From the wave function of two entangled particles, we find spins, trajectories and also a wave function for each of the two particles. In this interpretation, the quantum particle has a local position like a classical particle, but it has also a non-local behavior through the wave function. So, it is the wave function that creates the non 180 classical properties. We can keep a view of a local realist world for the particle, but we should add a non-local vision through the wave function. As we saw in step 1, the non-local influences in the EPR-B experiment only concern the spin orientation, not the motion of the particles themselves. Indeed only spins are entangled in the wave function but not positions and motions like in the initial EPR experiment. This is a key point in the search for a physical explanation of non-local influences. V.3. EPR-B paradox resolution by using quantum mechanical formalism based on the probability representation of quantum states V.3.1. Preleminaries We remind now EPR-B argument [17] in original D. Bohm formulation: "The Hypothetical Experiment of Einstein, Rosen, and Podolsky. We shall now describe the hypothetical experiment of Einstein, Rosen, and Podolsky. We have modified the experiment somewhat, but the form is conceptually equivalent to that suggested by them, and considerably easier to treat mathematically. Suppose that we have a molecule containing two atoms in a state in which the total spin is zero and that the spin of each atom is /2. Roughly speaking, this means that the spin of each particle points in a direction exactly opposite to that of the other, insofar as the spin may be said to have any definite direction at all. Now suppose that the molecule is disintegrated by some process that does not change the total angular momentum. The two atoms will begin to separate and will soon cease to interact appreciably. Their combined spin angular momentum, however, remains equal to zero, because by hypothesis, no torques have acted on the system. Now, if the spin was a classical angular momentum variable, the interpretation of this process would be as follows: while the two atoms were together in the form of a molecule, each component of 181 the angular momentum of each atom would have a definite value that was always opposite to that of the other, thus making the total angular momentum equal to zero. When the atoms separated, each atom would continue to have every component of its spin angular momentum opposite to that of the other. The two spin-angularmomentum vectors would therefore be correlated. These correlations were originally produced when the atoms interacted in such a way as to form a molecule of zero total spin, but after the atoms separate, the correlations are maintained by the deterministic equations of motion of each spin vector separately, which bring about conservation of each component of the separate spin-angularmomentum vectors. Suppose now that one measures the spin angular momentum of any one of the particles, say No. 1. Because of the existence of correlations, one can immediately conclude that the angularmomentum vector of the other particle (No. 2) is equal and opposite to that of No. 1. In this way, one can measure the angular momentum of particle No. 2 indirectly by measuring the corresponding vector of particle No. 1. Let us now consider how this experiment is to be described in the quantum theory. Here, the investigator can measure either the x,y, or z component of the spin of particle No. 1, but not more than one of these components, in any one experiment. Nevertheless, it still turns out as we shall see that whichever component is measured, the results are correlated, so that if the same component of the spin of atom No. 2 is measured, it will always turn out to have the opposite value. This means that a measurement of any component of the spin of atom No. 1 provides, as in classical theory, an indirect measurement of the same component of the spin of atom No. 2. Since, by hypothesis, the two particles no longer interact, we have obtained a way of measuring an arbitrary component of the spin of particle No. 2 without in any way disturbing that particle. If we accept the definition of an element of reality suggested by ERP, it is clear that after we have measured  z for particle 1, then  z for particle 2 must be regarded as an element of reality; existing separately in 182 particle No. 2 alone. If this is true, however, this element of reality must have existed in particle No. 2 even before the measurement of  z at for particle No. 1 took place. For since there is no interaction with particle No. 2, the process of measurement cannot have affected this particle in any way. But now let us remember that, in each case, the observer is always free to reorient the apparatus in an arbitrary direction while the atoms are still in flight, and thus to obtain a definite (but unpredictable) value of the spin component in any direction that he chooses. Since this can be accomplished without in any way disturbing the second atom, we conclude that if criterion of ERP is applicable, precisely defined elements of reality must exist in the second atom, corresponding to the simultaneous definition of all three components of its spin. Because the wave function can specify, at most, only one of these components at a time with complete precision, we are then led to the conclusion that the wave function does not provide a complete description of all elements of reality existing in the second atom." Actually, most experiments have been performed using polarization of photons. The quantum state of the pair of entangled photons is not the singlet state.The polarization of a photon is measured in a pair of perpendicular directions. Relative to a given orientation, polarization is either vertical (denoted by V or by  ) or horizontal (denoted by H or by  ). The photon pairs are generated in the quantum state |EPRB   1 2 |Vs  |Vi  |Hs  |Hi 5.3.1 where |V and |H denotes the state of a single vertically or horizontally polarized photon, respectively, relative to a fixed and common reference direction for both particles.This state cannot be factored into a simple product of signal and idler states: |EPRB  |As  |Bi 5.3.2 for any choice of |As and |Bi . This means the state of one particle 183 cannot be specified without making reference to the other particle. Such particles are said to be entangled and |EPRB is an entangled state. If we measure the polarizations of signal and idler photons in the H,V basis there are two possible outcomes: both vertical or both horizontal. Each occurs half of the time. We could instead measure the polarizations with polarizers rotated by an angle  . We use the rotated polarization basis |V   cos|V  sin|H, |H   sin|V  cos|H, 5.3.3 where |V  describes a state with polarization rotated by  from the vertical, while |H  is  from the horizontal. In this basis the state is |EPRB   1 2 |V  |V   |H  |H . 5.3.4 Remark 5.3.1. After the signal photon is measured the idler is equally likely to be V or H . A measurement of its polarization, at any angle , finds V or H with probability PV ,  1 2 |V|V |2  1 2 |V|H |2   1 2 cos2    sin2    1 2 , PH ,  1 2 |H|V |2  1 2 |H|H |2  1 2 . 5.3.5 Remark 5.3.2. Let ,, be a measure space and f be a Borel function. Note that A   A fd,A   5.3.6 is a signed measure satisfying A  0  A. We say is absolutely continuous with respect to (w.r.t.) and write  ν . Theorem (Radon-Nikodym). Let X be a set, let  be a  -algebra of λ ν λ 184 subsets of X , let  be a  -nite measure defined on , and let  be a signed defined on . Suppose that   . Then there exists a function f : X   on X that is integrable w.r.t. the measure  and that satisfies Eq. (5.3.6) for all A  . Moreover any two functions with this property are equal almost everywhere (a.e.) on X. Remark 5.3.3. If X fdν  1 for any f ≥ 0 a. e. ν, then is a probability measure and f is called its probability density function (p.d.f.) w.r.t. . Remark 5.3.4. Remind that: (i) A random variable X is a measurable function from a probability space ,,P to the reals , i.e., it is a function X :    such that for every Borel set B  B : X1B  X  B  , where we use the shorthand notation X  B    |X  B and where B is Borel algebra of the all Borel subset B of . (ii) If X is a random variable, then for every Borel subset B of of ,X1B   and we can define a function on Borel sets by PXB  PX1B. 5.3.7 (iii) This function PX : B   is in fact a probability measure, and and ,B,X is a probability space. (iv) The measure PX is called the distribution of the random variable X . If X gives measure one to a countable set of reals, then X is called a discrete random variable. (v) Let X be a discrete random variable.Then the probability mass function fX : AA    0,1 for X is defined as fXx  P  |X  x. 5.3.8 The total probability for all hypothetical outcomes x : xA fXx  1. (vi) Let ,,P be a probability space mentioned above, see Remark 5.3.4. The distribution PX is often given in terms of the λ ν 185 cumulative distribution function (c.d.f.) FX defined by FXx  PX  x. Remark 5.3.5. For any probability measure PX on ,B  corresponding to a random variable X (or to a c.d.f. FX ) if PX has a p.d.f. pX w.r.t. a measure P, then pX is also called the p.d.f. of FX or X w.r.t. P. Remark 5.3.6. (Discrete c.d.f. and p.d.f.). Let a1  a2 . . . an be a sequence of real numbers and let pi, i  1,2, . . . ,n be a sequence of positive numbers such that i1 n pi  1. Then Fx   i1 m pi ai  x  ai1 , i  1,2, . . . ,n  1 0   x  a1 5.3.9 is a stepwise c.d.f. It has a jump of size p i at each a i and is flat between a i and ai1 . Such a c.d.f. is called a discrete c.d.f. The corresponding probability measure PF is PFA  i|a iA pi,A  . 5.3.10 Remark 5.3.7. Remind that the counting measure  on a measurable space , is the positive measure A  |A| if A is finite  if A is infinite 5.3.11 for all A  Σ, where |A| denotes the cardinality of the set A. Let ν be the counting measure on . Then PFA   A fFdν  a iA fFai ,A  , 5.3.12 where fFai  pi, i  1,2, . . . . That is, fF is the p.d.f. of PF or F w.r.t. ν. Hence, any discrete c.d.f. has a p.d.f. w.r.t. counting measure. A p.d.f. w.r.t. counting measure is called a discrete p.d.f. 186 Definition 5.3.1. Let ,,P be a probability space and X is a discrete random variable X :   . The conditional probability of event A   given X,y is defined as PA|X  y  PA  X  y PX  y  P A  X 1y P X 1y , 5.3.13 where P X 1y  0. Definition 5.3.2. (I) Let X |V be a discrete random variable X |V :    with the probability mass function (see Remark 5.3.4.v) fX |V defined by fX |V x  1 2 if x  1 1 2 if x  0 5.3.14 Thus there exist: (i) 1,   such that 1,  XX |V 1 1 and P1,  1/2, (ii) 2,   such that 2  XX |V 1 0 and P2,  1/2, (iii) 1,  \2, mod,P  0. (II) Let X |H be a discrete random variable X |H :    such that   1,X |H  1  X |V. 5.3.15 Therefore 1,  XX |H 1 0,2,  XX |H 1 1 and the probability mass function f X |H is f X |H x  1 2 if x  1 1 2 if x  0 5.3.16 Remark 5.3.8. Let BM4 be a Boolean algebra of physical events in Minkowski spacetime and let M4 ph  BM4 ,P be measure algebra of 187 physical events in Minkowski spacetime, i.e., M4 ph that is a Boolean algebra BM4 with a probability measure P, see Chapter III subsection III.2, Definition 3.2.3. We remind that we denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc., and we write for a short AOcx,BOcx, . . . iff there physical events Ax,Bx. . . were occured. Definition 5.3.3. Let  be the measurement operator corresponding to measurments the photon polarization (see Appendix A) in polarization basis |V , |H , see Eq. (5.3.3). Let A|A  ,, t  BM4 be a physical event which consists on performing a measurement with absolute certainty of the observable  at instant t. Remark 5.3.9. We assume that: (i) particle A is initially in the state |A    1 2 |V   |H , (ii) A|A  ,, t  M4 ph . Remark 5.3.10. Note that: (i) if there physical event A|A  ,, t was occurred then immediately after the measurement at the instant t a particle A will be in the state V A   |V  or in the state H A   |H , (ii) immediately after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X AOc|A  ,, t  PX  AOc|A  ,, t PAOc|A  ,, t , 5.3.17 where X  M4 ph . Remark 5.3.11. Let A|A , t be a physical event which consists that a particle A at the instant t is in the state |A . Note that: (i) A Oc|A  ,, t  AOc VA , t  AOc HA , t, P X AOc|A  ,, t : 188 (ii) A Oc VA , t  AOc HA , t  M4 ph , (iii) from (i), (ii) and (5.3.17) it follows that: P X AOc|A  ,, t  P X A VA , t  PX  A VA , t PA VA , t 5.3.18 or P X AOc|A  ,, t  P X A HA , t  PX  A HA , t PA HA , t , 5.3.19 where X  M4 ph . Remark 5.3.12. We assume now that: (i) a measure algebra M4 ph  BM4 ,P admits a representation  : M4 ph  ,B,PB  of the measure algebra M4 ph  BM4 ,P in the measure algebra B  ,B,PB , such that (ii) PBX  P 1X for any X  B and (iii) for physical events A V A , t  M4 ph and A H A , t  M4 ph (see Remark 5.3.10) the following conditions hold AOc VA , t  1, 5.3.20 and AOc HA , t  2,, 5.3.21 where 1,  XX |V 1 1 and 2,  XX |H 1 1, see Definition 5.3.2. Remark 5.3.13. We note that the product of the noise in a polarization measurement in the polarization basis  V A , HA  and the polarization disturbance in the polarization basis  V A , H A  caused by that measurement should be no less than , : ,  1 2 |A  |,|A  |  0, 5.3.22 see Appendix A. Note that Heisenberg's noise-disturbance 189 uncertainty relation for the case of the polarization measurement is: for any apparatus A to measure an observable , the relation , |A  ,A, |A  ,A  1 2 |A  |,|A  |  , 5.3.23 holds for any input state |A   and any observable  , where ,A  ,A stands for the noise of the  measurement in the state A  using apparatus A and ,A  ,A stands for the disturbance of  in the state A  caused by apparatus A. We refer to the above relation as the Heisenberg noise-disturbance uncertainty relation in polarization measurements, see Appendix A. Remark 5.3.14. We use the rotated polarization basis |V , |H  with uncertainty   , |A  ,A |V   |V |        , |V   cos |V  sin|H, |H   |H |        , |H   sin |V  cos |H 5.3.24 where |V  describes states with polarization rotated by any     ,    from the vertical, while |H  is  from the horizontal. We abbreviate for short |V   cos   |V  sin   |H, |H   sin   |V  cos   |H, Remark 5.3.15. After the signal photon is measured the idler is equally likely to be V or H . A measurement of its polarization, at any angle   :           , finds V or H with probability 190 PV ,    1 2 |V |V |2  1 2 |V |H |2   1 2 cos2       sin2       1 2 , PH ,    1 2 |H |V |2  1 2 |H |H |2  1 2 . 5.3.25 where   , |A  ,A,   , A  ,A and A   1 2 |V   |H . Definition 5.3.4. (I) Let X |V  be a discrete random variable X |V  :    with the probability mass function (see Remark 5.3.4.v) fX V defined by fX V x  1 2 if x  1 1 2 if x  0 5.3.26 Thus there exist: (i) 1,   such that 1,  XX V 1 1 and P1,   1/2, (ii) 1,   such that 2,  XX V 1 0 and P2,   1/2, (iii) 1,  \2, mod,P  0. Thus there exist: (i) 1,   such that 1,  XX |V 1 1 and P1,   1/2, (ii) 2,   such that 2,  XX |V 1 0 and P2,   1/2, (iii) 1,  \2, mod,P  0. (III) Let X |H  be a discrete random variable X |H  :    such that   1, X |H   1  X |V . 5.3.27 Therefore 1,  XX H 1 0,2,  XX H 1 1 and the probability mass function f X H is 191 f X H x  1 2 if x  1 1 2 if x  0 5.3.28 Remark 5.3.16. Let BM4 be a Boolean algebra of physical events in Minkowski spacetime and let M4 ph  BM4 ,P be a measure algebra of physical events in Minkowski spacetime, i.e., M4 ph that is a Boolean algebra BM4 with a probability measure P, see Chapter III subsection III.2, Definition 3.2.3. We remind that we denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc., and we write for a short AOcx,BOcx, . . . iff there physical events Ax,Bx. . . were occured. Definition 5.3.5. Let  be the measurement operator corresponding to measurements of the photon polarization (see Appendix A) in the polarization basis |V , |H , see Eq. (5.3.3). Let A|A  ,,, t  BM4 be a physical event which consists of the performing a measurement (on the particle A at the instant t ) of the observable  with accuracy   , |A  ,A, where the particle A is initially in the state A  at the instant t. Here   ,,A stands for the noise of the A measurement in the state  using apparatus A and   ,,A stands for the disturbance of  in the state  caused by apparatus A. Remark 5.3.17. We assume that: (i) the particle A is initially in the state |A    1 2 |V   |H , (ii) A|A  ,,, t  M4 ph . Remark 5.3.18. Note that: (i) if the physical event A|A  ,,, t was occurred then immediately after the measurement at the instant t the particle A would be in the state V A   |V  or in the state 192 H A   |H , (ii) immediately after the measurement at the instant t unconditional measure P collapses to conditional measure P X A Oc|A  ,,, t : P X AOc|A  ,,, t  PX  AOc|A  ,,, t PAOc|A  ,,, t , 5.3.29 where X  M4 ph . Remark 5.3.19. Let A|A , t be a physical event which consists of that a particle A at the instant t is in the state |A . Note that: (i) A Oc|A  ,,, t  AOc V A , t  AOc H A , t, (ii) A Oc V A , t  AOc H A , t  M4 ph , (iii) from (i), (ii) and (5.3.17) it follows that: P X AOc|A  ,,, t  PX  A V A , t PA V A , t 5.3.30 or P X AOc|A  ,,, t  PX  A H A , t PA H A , t , 5.3.31 where X  M4 ph . Remark 5.3.20. We assume now that: (i) a measure algebra M4 ph  BM4 ,P admits a representation  : M4 ph  ,B,PB  of the measure algebra M4 ph  BM4 ,P in the measure algebra B  ,B,PB , such that (ii) PBX  P 1X for any X  B and (iii) for physical events A V A , t  M4 ph and A H A , t  M4 ph (see Remark 5.3.18) the following conditions hold 193 AOc V A , t  X |V A 1 1  1, 5.3.32 and AOc H A , t  X |H A 1 1  2,, 5.3.33 where 1,  XX V 1 1 and 2,  XX |H 1 1. V.3.2. The EPR-B Paradox Resolution Remind that in the well-known Einstein-Podolsky-Rosen-Bohm gedanken experiment with photons (Fig. 5.3.1), a source emits pairs of photons in a nonfactorizing state: |EPRB   1 2 |V 1  |V 2  |H 1  |H 2 . 5.3.34 After the particles have been space-like separated, one performs correlated measurements of their polarizations along arbitrary directions a and b . Two photons in a singlet state are space-like separated. The linear polarizations of photon 1 and photon 2 are measured along a and b . Quantum mechanics predicts strong correlations between these measurements. Fig. 5.3.1. Einstein-Podolsky-Rosen-Bohm gedanken experiment with photons 194 Quantum mechanics predicts that by a measurement of the linear polarization of photon 1 using a noiseless measuring apparatus A one obtais definite outcome exactly only |V 1 or |H 1 . In the canonical Copenhagen interpretation QM predicts that the state |EPRB has collapsed, at the moment of measurement, from |EPRB to either |V 1  |V 2 or |H 1  |H 2 . Remark 5.3.21. The process described above seems to be nonlocal: the state changes instantly even though the particles could be space-like separated. We are accustomed to saying that this sort of instantaneous action at a distance is forbidden by relativity. Assume that the state |EPRB has collapsed to |V 1  |V 2 . Thus in the canonical Copenhagen interpretation result of the measurement of the polarizations of photon 1 predicts exactly the polarization of photon 2. This means that if we measure the linear polarization of photon 1 (using a noiseless measuring apparatus A) in any basis |V1 , |H2 the result will be completely random ( |V1 or |H1 with equal probability 1/2). Remark 5.3.22. However, there is a perfect correlation: whenever we measure with certainty the linear polarization of photon 1 with outcome say |V1 (using a noiseless measuring apparatus A) then we will measure with certainty (using a noiseless measuring apparatus B) the linear polarization of photon 2 exactly with outcome |V2 . Remark 5.3.23. We note that such a perfect correlation implies that the corresponding probability mass functions: f X |V1 , f X |H1 f X |V2 and f X |H2 (see Definition 5.3.2) are perfectly contracted for any , by the following equations: f X |V1 x  f X |V2 x, 5.3.35 and f X |H1 x  f X |H2 x, 5.3.36 195 where f X |H1 x  1  f X |V1 x. Now we go to prove that Eqs. (5.3.35) - (5.3.36) hold without any instantaneous action at a distance. Remark 5.3.24. We assume now that: (i) photon 1 is initially in the state |1    1 2 |V 1  |H 1 , (ii) A|1  ,, t  M4 ph , see Remark 5.3.9. If the measurement of the linear polarization of photon 1 (using a noiseless measuring apparatus A) was performed at the instant t , i.e., A Oc|A  ,, t after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X AOc|1  ,, t  P X AOc|V 1 , t  PX  AOc|V 1 , t PA|V 1 , t 5.3.37 or collapses to the conditional measure P X AOc|1  ,, t  P X AOc|H 1 , t  P X  AOc HA 1 , t PA|H 1 , t , 5.3.38 where X  M4 ph , see Remark 5.3.19. Remark 5.3.25. We assume now that: (i) immediately after the measurement at the instant t the particle 1 is in the state |V 1 . In this case immediately after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X A|V 1 , t which is given by P X AOc|V 1 , t  PX  AOc|V 1 , t PA|V 1 , t 5.3.39 where X  M4 ph , (ii) immediately after the measurement at the instant t the particle 1 is in the state |H 1 . In this case immediately after the measurement at the instant t the unconditional measure P collapses to the 196 conditional measure P X A Oc|V 1 , t given by P X AOc|H 1 , t  PX  AOc|H 1 , t PA|H 1 , t , 5.3.40 where X  M4 ph . (1) From Eq. (5.3.39) and Eq. (5.3.20) Eq. (5.3.21) we obtain P A|H 2 , t A Oc|V 1 , t  PA|H 2 , t  A Oc|V 1 , t PA|V 1 , t  PBA|H 2 , t  A Oc|V 1 , t PBA|V 1 , t  PB2,  1,  PB1,   PB PB1,   0. 5.3.41 Therefore A Oc|H 2 , t and A Oc|V 1 , t are mutually exclusive (disjoint) physical events, i.e., they cannot both occur simultaneously: AOc|H 2 , t  A Oc|V 1 , t  M4 ph . 5.3.42 (2) From Eq. (5.3.39) and Eq. (5.3.20) Eq. (5.3.21) we obtain P A|V 2 , t A Oc|V 1 , t  PA|V 2 , t  A Oc|V 1 , t PA|V 1 , t  PBA|V 2 , t  A Oc|V 1 , t PBA|V 1 , t  PB1,  1,  PB1,   PB1,  PB1,   1. 5.3.43 Therefore physical events A Oc|V 2 , t and A Oc|V 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc|V 2 , t  A Oc|V 1 , t. 5.3.44 (3) From Eq. (5.3.40) and Eq. (5.3.20) Eq.(5.3.21) we obtain 197 P A|H 2 , t A Oc|H 1 , t  PA|H 2 , t  A Oc|H 1 , t PA|H 1 , t  PBA|H 2 , t  A Oc|H 1 , t PBA|H 1 , t  PB2,  2,  PB1,   PB2,  PB2,   1. 5.3.45 Therefore physical events A Oc|H 2 , t and A Oc|H 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc|H 2 , t  A Oc|H 1 , t. 5.3.46 (4) From Eq. (5.3.40) and Eq. (5.3.20) Eq.(5.3.21) we obtain P A|V 2 , t A Oc|H 1 , t  PA|V 2 , t  A Oc|H 1 , t PA|H 1 , t  PBA|V 2 , t  A Oc|H 1 , t PBA|H 1 , t  PB1,  2,  PB2,   PB PB2,   0. 5.3.47 Therefore A Oc|H 2 , t and A Oc|V 1 , t are mutually exclusive (disjoint) physical events, i.e., they cannot both occur simultaneously: AOc|V 2 , t  A Oc|H 1 , t  M4 ph . 5.3.48 Remark 5.3.26. Under rigorous consideration using Heisenberg noise-disturbance uncertainty relation (see Appendix A) quantum mechanics predicts that by a measurement of the linear polarization of photon 1 using measuring apparatus A one obtains definite outcome exactly only |V 1 or |H 1 , where   ,1 ,A stands for the noise of the  measurement in state 1  using apparatus A, see Appendix A, Eq. (A.13). In the canonical Copenhagen interpretation QM predicts that the state |EPRB has collapsed, at the moment of the measurement, from |EPRB to either |V 1  |V 2 or |H 1  |H 2 . 198 Remark 5.3.27. However, there is a perfect correlation: whenever we measure with uncertainty  the linear polarization of photon 1 with outcome say |V 1 using measuring apparatus A then we will measure with uncertainty  (using a similar measuring apparatus B with ,1 ,A   ) the linear polarization of photon 2 exactly with outcome |V 2 . Remark 5.3.28. We note that such a perfect correlation implies that the corresponding probability mass functions: f X V 1 , f X H 1 f X V 2 and f X H 2 (see Definition 5.3.4) are perfectly contracted for any , by the following equations: f X V 1 x  f X V 2 x, 5.3.49 and f X H 1 x  f X H 2 x, 5.3.50 where f X H 1 x  1  f X V 1 x. Now we go to prove that Eqs. (5.3.49) - (5.3.50) hold without any instantaneous action at a distance. Remark 5.3.29. We assume now that: (i) photon 1 is initially in the state |1    1 2 |V 1  |H 1 , (ii) A|1  ,,, t  M4 ph , where   ,1 ,A stands for the noise of the  measurement in state 1  using apparatus A. If the measurement of the linear polarization of photon 1 using measuring apparatus A was performed at the instant t , i.e., AOc|A  ,,, t after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X AOc|1  ,, t  P X AOc|V 1 , t  PX  AOc|V 1 , t PA|V 1 , t 5.3.51 199 or collapses to the conditional measure P X AOc|1  ,, t  P X AOc|H 1 , t  P X  AOc H A  1 , t PA|H 1 , t , 5.3.52 where X  M4 ph , see Remark 5.3.19. Remark 5.3.30. We assume now that: (i) immediately after the measurement at the instant t the particle 1 is in the state |V 1 . In this case immediately after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X A|V 1 , t given by P X AOc|V 1 , t  PX  AOc|V 1 , t PA|V 1 , t 5.3.53 where X  M4 ph , (ii) immediately after the measurement at the instant t the particle 1 is in the state |H 1 . In this case immediately after the measurement at the instant t the unconditional measure P collapses to the conditional measure P X A Oc|H 1 , t given by P X AOc|H 1 , t  PX  AOc|H 1 , t PA|H 1 , t , 5.3.54 where X  M4 ph . (1) From Eq. (5.3.53) and Eq. (5.3.32) Eq.(5.3.33) we obtain P A|H 2 , t A Oc|V 1 , t  PA|H 2 , t  A Oc|V 1 , t PA|V 1 , t  PBA|H 2 , t  A Oc|V 1 , t PBA|V 1 , t  PB2,  1,  PB1,   PB PB1,   0. 5.3.55 Therefore A Oc|H 2 , t and A Oc|V 1 , t are mutually exclusive 200 (disjoint) physical events, i.e., they cannot both occur simultaneously: AOc|H 2 , t  A Oc|V 1 , t  M4 ph . 5.6.56 (2) From Eq. (5.3.39) and Eq. (5.3.20) Eq. (5.3.21) we obtain P A|V 2 , t A Oc|V 1 , t  PA|V 2 , t  A Oc|V 1 , t PA|V 1 , t  PBA|V 2 , t  A Oc|V 1 , t PBA|V 1 , t  PB1,  1,  PB1,    PB1,  PB1,   1. 5.3.57 Therefore physical events A Oc|V 2 , t and A Oc|V 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc|V 2 , t  A Oc|V 1 , t. 5.3.58 (3) From Eq. (5.3.40) and Eq. (5.3.20) Eq.(5.3.21) we obtain P A|H 2 , t A Oc|H 1 , t  PA|H 2 , t  A Oc|H 1 , t PA|H 1 , t  PBA|H 2 , t  A Oc|H 1 , t PBA|H 1 , t  PB2,  2,  PB1,   PB2,  PB2,   1. 5.3.59 Therefore physical events A Oc|H 2 , t and A Oc|H 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc|H 2 , t  A Oc|H 1 , t. 5.3.60 (4) From Eq. (5.3.40) and Eq. (5.3.20) Eq. (5.3.21) we obtain 201 P A|V 2 , t A Oc|H 1 , t  PA|V 2 , t  A Oc|H 1 , t PA|H 1 , t  PBA|V 2 , t  A Oc|H 1 , t PBA|H 1 , t  PB1,  2,  PB2,   PB PB2,   0. 5.3.61 Therefore A Oc|H 2 , t and A Oc|V 1 , t are mutually exclusive (disjoint) physical events, i.e., they cannot both occur simultaneously: AOc|V 2 , t  A Oc|H 1 , t  M4 ph . 5.3.62 Definition 5.3.6. Let  be the measurement operator corresponding to measurements of the photon polarization (see Appendix A) in the polarization basis |V , |H  and let  be the measurement operator corresponding to measurements of the photon polarization in the polarization basis |V, |H see Eq.(1.3.3). We assume that particle 1 is initially in the state |1  . Let A|1  ,,; 1  ,,, t  BM4 be a physical event which consists of: (i) performing a measurement (on particle A at instant t ) of the observable  with accuracy   , |A  ,A, where particle A is initially in the state A  at the instant t, (ii) immediately after the measurement on particle 1 at the instant t the particle 1 is in the state 1  , (iii) performing a measurement (on particle 1 at the instant t ) of the observable  with accuracy   , |A  ,A particle A obtains disturbance   , 1  ,A of the observable  in the state 1   caused by apparatus A. Definition 5.3.7. (I) Let X V,  be a discrete random variable X V, :    with the probability mass function (see Remark 5.3.4.v) fX V, defined by 202 fX V, x  1 2 if x  1 1 2 if x  0 5.3.63 Thus there exist: (i) 1,,   such that 1,,  XX V, 1 1 and P1,,   1/2, (ii) 1,,   such that 2,,  XX V, 1 0 and P2,,   1/2, (iii) 1,,  \2,, mod,P  0. Thus there exist: (i) 1,,   such that 1,,  XX V, 1 1 and P1,,   1/2, (ii) 2,,   such that 2,,  XX V, 1 0 and P2,,   1/2, (iii) 1,,  \2,, mod,P  0. (II) Let X H,  be a discrete random variable X H, :    such that   1,,  X H,   1  X V,  . 5.3.64 Therefore 1,,  XX H, 1 0,2,,  XX H, 1 1 and the probability mass function f X H, is f X H, x  1 2 if x  1 1 2 if x  0 5.3.65 Remark 5.3.31. Let BM4 be a Boolean algebra of physical events in Minkowski spacetime and let M4 ph  BM4 ,P be a measure algebra of physical events in Minkowski spacetime, i.e., M4 ph that is a Boolean 203 algebra BM4 with a probability measure P, see Chapter III subsection III.2, Definition 3.2.3. We remind that we denote such physical events by Ax,Bx, . . . etc., where x  t,x1 ,x2 ,x3   M4 or A,B, . . . etc., and we write for a short AOcx,BOcx, . . . iff the physical events Ax,Bx. . . were occurred. Remark 5.3.32. We assume that: (i) particle 1 is initially in the state |1    1 2 |V   |H , (ii) A|1  ,,; 1  ,,, t  M4 ph . Remark 5.3.33. Note that: (i) if the physical event A|1  ,,; 1  ,,, t was occurred then immediately after the measurement at the instant t particle 1 will be in the state V, 1  |V,  or in the state H, 1  |H, , (ii) immediately after the measurement at the instant t unconditional measure P collapses to conditional measure P X AOc|1  ,,; 1  ,,, t : P X AOc|1  ,,; 1  ,,, t   PX  AOc|1  ,,; 1  ,,, t PAOc|1  ,,; 1  ,,, t , 5.3.66 where X  M4 ph . Remark 5.3.34. Let A|1 , t be a physical event which consists of that at the instant particle 1 is in the state |1 . Note that: (i) A Oc|1  ,,; 1  ,,, t  AOc V, 1 , t  AOc H, 1 , t , (ii) A Oc V, 1 , t  AOc H, A , t  M4 ph , (iii) from (i), (ii) and (5.3.66) it follows that: P X AOc|1  ,,; 1  ,,, t  P X  A V, 1 , t P A V, 1 , t 5.3.67 or t 204 P X AOc|1  ,,; 1  ,,, t  P X  A H, 1 , t P A H, 1 , t , 5.3.68 where X  M4 ph . Remark 5.3.35. We assume now that: (i) a measure algebra M4 ph  BM4 ,P admits a representation  : M4 ph  ,B,PB  of the measure algebra M4 ph  BM4 ,P in the measure algebra B  ,B,PB , such that (ii) PBX  P 1X for any X  B and (iii) for physical events A V, 1 , t  M4 ph and A H, 1 , t  M4 ph (see Remark 5.3.33) the following conditions hold  AOc V, 1 , t  X V, A 1 1  1,, 5.3.69 and  AOc H, 1 , t  X H, 1 1 1  2,, , 5.3.70 where 1,,  XX V, 1 1 and 2,,  XX H, 1 1. Remark 5.3.36. We assume now that: (i) immediately after the measurement at the instant t particle 1 is in the state |V, 1 . In this case immediately after the measurement at the instant t unconditional measure P collapses to conditional measure P X A |V, 1 , t given by P X AOc |V, 1 , t  P X  AOc |V, 1 , t P A |V, 1 , t 5.3.71 where X  M4 ph , (ii)immediately after the measurement at the instant t particle 1 is in 205 the state |H 1 . In this case immediately after the measurement at the instant t unconditional measure P collapses to conditional measure P X A Oc |H, 1 , t given by P X AOc |H, 1 , t  P X  AOc |H, 1 , t P A |H, 1 , t , 5.3.72 where X  M4 ph . (1) From Eq. (5.3.71) and Eq. (5.3.69) - (5.3.70) we obtain P A |H, 2 , t A Oc |V, 1 , t  P A |H, 2 , t  A Oc |V, 1 , t P A |V, 1 , t  PB  A |H, 2 , t   A Oc |V, 1 , t PB  A |V, 1 , t  PB2,,  1,,  PB1,,   PB PB1,,   0. 5.3.73 Therefore A Oc |H, 2 , t and A Oc |V, 1 , t are mutually exclusive (disjoint) physical events, i.e., they both cannot occur simultaneously: AOc |H, 2 , t  A Oc |V, 1 , t  M4 ph . 5.3.74 (2) From Eq. (5.3.71) and Eq. (5.3.69) Eq.( 5.3.70) we obtain P A |V, 2 , t A Oc |V, 1 , t  P A |V, 2 , t  A Oc |V, 1 , t P A |V, 1 , t  PB  A |V, 2 , t   A Oc |V, 1 , t PB  A |V, 1 , t  PB1,,  1,  PB1,,   PB1,,  PB1,,   1. 5.3.75 206 Therefore physical events A Oc|V 2 , t and A Oc|V 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc |V, 2 , t  A Oc |V, 1 , t . 5.3.76 (3) From Eq. (5.3.72) and Eq. (5.3.69) - (5.3.70) we obtain P A |H, 2 , t A Oc |H, 1 , t  P A |H, 2 , t  A Oc |H, 1 , t P A |H, 1 , t  PB  A |H, 2 , t   A Oc |H, 1 , t PB  A |H, 1 , t  PB2,,  2,  PB1,,   PB2,  PB2,,   1. 5.3.77 Therefore physical events A Oc|H 2 , t and A Oc|H 1 , t always occur simultaneously even particle 1 and particle 2 are space-like separated: AOc |H, 2 , t  A Oc |H, 1 , t . 5.3.78 (4) From Eq. (5.3.40) and Eq. (5.3.20) Eq. (5.3.21) we obtain P A |V, 2 , t A Oc |H, 1 , t  P A |V, 2 , t  A Oc |H, 1 , t P A |H, 1 , t  PB  A |V, 2 , t   A Oc |H, 1 , t PB  A |H, 1 , t  PB1,,  2,,  PB2,,   PB PB2,,   0. 5.3.79 207 Therefore A Oc |H, 2 , t and A Oc |V, 1 , t are mutually exclusive (disjoint) physical events, i.e., they both cannot occur simultaneously: AOc |V, 2 , t  A Oc |H, 1 , t  M4 ph . 5.3.80 208 Chapter VI SCHRӦDINGER'S CAT MEASURED SPIN. SCHRӦDINGER'S CAT PARADOX RESOLUTION VI.1. Stern-Gerlach experiment revisited In 1922, by studying the deflection of a beam of silver atoms in a strongly inhomogeneous magnetic field (Fig. 6.1.1) Otto Stern and Walter Gerlach obtained an experimental result that contradicts the common sense prediction: the beam, instead of expanding, splits into two separate beams giving two spots of equal intensity N and N on a detector, at equal distances from the axis of the original beam. Historically, this is the experiment which helped establish spin quantization. Theoretically, it is the seminal experiment posing the problem of measurement in quantum mechanics. Fig. 6.1.1. Schematic configuration of the Stern-Gerlach experiment. Adapted from [19] z,y, t|t0   0z,y  1 0z2 0y. 6.1.1 209 We assume now that both density 0z and 0y is very narrow, in fact constrained such that 1 0z  1 0z,  0 iff |x| , 2 0y  2 0y,  0 iff |y| , 6.1.2 and 1 0z  1 0z,  2 00 2 1 4 e  z 2 4 0 2 cos 0 2 ei 0 2 sin 0 2 ei 0 2 iff |z| , 1 0z, 2 2  1; 2 0y  2 0y,  2 00 2 1 4 e  y 2 4 0 2 iff |y| , 2 0y, 2 2  1 0  1. 6.1.3 Silver atoms contained in the oven E (Fig. 6.1.1) are heated to a high temperature and escape through a narrow opening. A second aperture, T, selects those atoms whose velocity, v0 , is parallel to the y-axis. The atomic beam crosses the gap of the electromagnet A1 before condensing on the P1 detector. Before crossing the electromagnet, the magnetic moment of each silver atom is oriented randomly (isotropically). In the beam, we represent each atom by its wave function; one can assume that at the entrance to the electromagnet, A1 , and at the initial time t  0, each atom can be approximatively described by a quasi-Gaussian spinor in plain z,y given by Eqs. (6.1.1-6.1.3) corresponding to a pure state. As it will be proved later the variable y will be treated strictly quasiclassically, i.e. almost classically, with P y  vyv0 ,0 t    1, P y  vyv0 ,0 t    0 6.1.4 and 0  0   104m, where 0  corresponds to the size of the slot T along the z-axis and where the expression of the functions vy v0 ,0  210 and vy v0 ,0  will be given later. The approximation by a quasi-Gaussian initial spinor will allow explicit calculations. Because the slot is much wider along the x axis, the variable z will be also treated strictly quasiclassically with P z  z  vzu,0 t    1, P z  vzu,0 t    0, 6.1.5 where the expression of the functions vz u,0 ,u  BB0  t m will be given later. In order to obtain an explicit solution of the SternGerlach experiment, we take for the silver atom, we have m  1.8  1025kg,v0  500 m/s (corresponding to the temperature T  1000°K). In Eq. (6.1.3.) and in Fig. 6.1.2., 0 and 0 are the polar angles characterizing the initial orientation of the magnetic moment, 0 corresponds to the angle with the z-axis. The experiment is a statistical mixture of pure states where the 0 and the 0 are randomly chosen: 0 is drawn in a uniform way from 0, and that 0 is drawn in a uniform way from 0,2. Fig. 6.1.2. Orientation of the magnetic moment θ0 and φ0 are the polar angles characterizing the spin vector in the de Broglie-Bohm interpretation. Adapted from [19] 211 Assumption 6.1.1. We assume that a particle collapses in a magnetic field B at some instant t  by two particles, i.e. the spinor z,y, t collapses in a magnetic field B at some instant t  by two spinors z,y, t, t , and z,y, t, t , given by Eq. (6.1.9a) Eq .(6.1.9b). Note that such a collapse obviously occurs except spinors such that: 2 1  z  z   x , etc. Remark 6.1.1. Note that the standard assumption consists of that spinor collapses on detector P1 with respect to the Born rule. Thus the evolution of the spinor z,y, t, t    z,y, t, t   z,y, t, t   in a magnetic field B is then given by the nonlocal Pauli equation: i  dzdy  dt z,y, t, t   t  dzdy  dt z,y, t, t   t     2 2m  dt  dzdy z,y, t, t   z,y, t, t    B  dt  dzdyB z,y, t, t   z,y, t, t   6.1.6 where B  e 2me is the Bohr magneton and where   x,y,z corresponds to the three Pauli matrices. Remark 6.1.2. First the particle enters an electromagnetic field B directed along the z-axis, Bx  B0  x,By  0,Bz  B0  B0  z, with B0  5 Tesla, B0   Bz  10 3 Tesla/m over a length l  1 cm. Remark 6.1.3. In exiting the magnetic field, the both particles are free until they reach the detector P1 placed at distance D  20 cm. The particles stay within the magnetic field for a time t with t  lv0 . 6.1.7 Assumption 6.1.2. We assume now for simplification that 212 t   t. 6.1.8 Thus during this time t  0, t   0,t, the spinor z,y, t, t , is: z,y, t, t ,  z,y, t, t , z,y, t, t ,  z, t, t ,y, t, t , z, t, t ,y, t, t , , 6.1.9.a where z, t, t ,  cos 0 2 ei 0 2 20 2 1 2 exp  z  BB0  2m t2 2 40 2  exp i BB0  tz  B 2 B0 2 6m t3  BB0 t  0.50  iff z  BB0  2m t2  , z, t,  0 iff z  BB0  2m t2  , z, t, t ,  isin 0 2 ei 0 2 20 2 1 2 exp  z  BB0  2m t2 2 40 2  exp i BB0  tz  B 2 B0 2 6m t3  BB0 t  0.50  iff z  BB0  2m t2  , z, t, t ,  0 iff z  BB0  2m t2  ; y, t, t ,  20 2 1 2 exp  y  v0 t 2 40 2 iff |y  v0 t| , y, t, t ,  0 iff |y  v0 t|  . 6.1.9.b After the magnetic field, at time t  t t  0 in the free space, the both spinors become: z,y, t  t ,  z,y, t  t,  z, t  t,y, t  t, 6.1.10 213 and z,y, t  t ,  z,y, t  t,  z, t  t,y, t  t,. 6.1.11 Here z, t  t,  cos 0 2 20 2 1 2 exp  z  z  ut 2 40 2 ei muz  iff |z  z  ut |  , 0 iff |z  z  ut |   6.1.12 and z, t  t,  sin 0 2 20 2 1 4 exp  z  z  ut 2 40 2 ei muz  iff |z  z  ut |  , 0 iff |z  z  ut |  , 6.1.13 and y, t,  20 2 1 4 exp  y  v0t  t 2 40 2 iff |y  v0t  0 t| , y, t,  0 iff |y  v0t  t| . 6.1.14 where z  BB0  t2 2m , u  BB0  t m . 6.1.15 From Eq. (6.1.10) Eq. (6.1.12) and Eq. (6.1.14) we obtain z,y, t  t,  cos 0 2 20 2 1 2 exp  z  z  ut 2 40 2 ei muz   exp  y  v0t  t 2 40 2 iff |z  z  ut |   and |y  v0t  t|  0 otherwise 6.1.16 From Eq. (6.1.16) by the postulate for the probability density with respect to observable z we obtain the expression 214 c |z, t  20 2 1 2 exp  z 0   z  ut 2 20 2 iff z 0   z  ut   0 otherwise 0   cos2 0 2 6.1.17 and with respect to observable y we obtain the expression c |y, t  20 2 1 2 exp  y 0   v0t  t 2 40 2 iff y 0   v0t  t   0 otherwise 0   cos2 0 2 6.1.18 and therefore the corresponding particle moves by the strictly quasiclassical law P zt  z  vzu,0 t    1, P zt  z  vzu,0 t    0, P yt  vyv0 ,0 t  t    1, P yt  vyv0 ,0 t  t    0, z   0  z , vz 0   0  u,vy 0   0  v0 . 6.1.19 From Eq. (6.1.11), Eq. (6.1.13) and Eq. (6.1.14) we obtain 215 z,y, t  t,  sin 0 2 20 2 1 4 exp  z  z  ut 2 40 2 ei muz   exp  y  v0t  t 2 40 2 iff |z  z  ut |   and |y  v0t  0 t|  0 otherwise 6.1.20 From Eq. (6.1.20) by the postulate for the probability density with respect to observable z we obtain the expression c |z, t  20 2 1 2 exp  z 0   z  ut 2 20 2 iff z0   z  ut   0 otherwise 0   sin2 0 2 6.1.21 and with respect to observable y we obtain the expression c |y, t  20 2 1 2 exp  y 0   v0t  t 2 40 2 iff y 0   v0t  0 t   0 otherwise 0   sin2 0 2 6.1.22 and therefore the corresponding particle moves by the strictly quasiclassical law P zt  z  vzu,0 t    1, P zt  z  vzu,0 t    0, P yt  vyv0 ,0 t  t    1, P yt  vyv0 ,0 t  t    0, z   0  z , vz u,0   0  u,vy v0 ,0   0  v0 . 6.1.23 216 All interpretations are based on the Eq. (6.1.18) - (6.1.21). One deduce from Eq. (6.1.18) - (6.1.21) the probability density of a pure state in the free space after the electromagnet: 0z,y, t  t  20 2 1 2 0z, t  0 t  exp  y 0   v0t  t 2 40 2 ; 0z, t  t  20 2 1 2 cos2 0 2 20 2 1 2 exp   z 0   z  ut2 20 2  sin2 0 2 20 2 1 2 exp   z0   z  ut2 40 2 . 6.1.24 The decoherence time tdec, where the two spots N and N are separated, is then given by the equation: tdec  30  z u0   0    30  zu . 6.1.25 This decoherence time is usually the time required to diagonalize the marginal density matrix 0 S t, of spin variables associated with a pure state 0 S t,  |z,y, t  t,|2dzdy z,y, t  t,z,y, t  t,dzdy z,y, t  t,z,y, t  t,dzdy |z,y, t  t,|2dzdy 6.1.26 For t  tdec , the product z,y, t  t,z,y, t  t, is null and the density matrix 0 S t, is diagonal. We then obtain atoms with a spin oriented only along the z -axis (positively or negatively). Let us consider the spinor z,y, t  t, given by Eq. (6.1.10) - (6.1.15). Remark 6.1.4. Experimentally, we do not measure the spin directly, but the z position of the particle impact on the detector P1 (Fig.6.1.3.). 217 Fig. 6.1.3. Silver atom impacts on the detector P1. Adapted from [19] Remark 6.1.5. Note that if we measure the z -position of the particle at the instant t, we also measure the y -position of the particle at the same instant t. Remark 6.1.6. Let PtD  ,D,yt be the probability of obtaining the result y t  at the instant t, lying in the range D  ,D on measuring observable y in respect to spinor z,y, t  t,. From Eq. (6.1.19) we obtain PtD  ,D  ,yt   1 iff yt  D and yt  vyv0 ,0 t  t  . 6.1.27 From Eq. (6.1.27) it follows that: 218 PtD  ,D  ,yt   1 if t  tD  D vy v0 ,0   D v0 cos 2 0 2 . 6.1.28 Remark 6.1.7. Let Pt  z   ,  z   ,zt  be the probability of obtaining the result zt  at the instant t, lying in the range   z   ,  z   ,  z   N on measuring observable z in respect to spinor z,y, t  t,. From Eq.(6.1.19) we obtain Pt  z   ,  z   , zt    1 iff zt  z  and |zt  z  vzu,0 t |  . 6.1.29 From Eq. (6.1.29) it follows that: Pt  z   ,  z   , zt    1 if t  tz     z  vz u,0    z  ucos2 0 2 . 6.1.30 Remark 6.1.8. Note that from Remark 6.1.6 it follows that t  z   tD and therefore from Eq. (6.1.28) and Eq. (6.1.30) one obtains  z  ucos2 0 2  D v0 cos 2 0 2   z  u  D v0 6.1.31 as it should be, because the equality  z  u  D v0 is required by the condition of the Stern-Gerlach experiment. Remark 6.1.9. Let PtD  ,D,yt be the probability of obtaining the result y t  at the instant t, lying in the range D  ,D on measuring observable y in respect to spinor z,y, t  t,. From Eq. (6.1.23) we obtain 219 PtD  ,D  ,yt   1 iff yt  D and yt  vyv0 ,0 t  t  . 6.1.32 From Eq. (6.1.32) it follows that: PtD  ,D  ,yt   1 if t  tD  D vy v0 ,0   D v0 sin 2 0 2 . 6.1.33 Remark 6.1.10. Let Pt  z   ,  z   ,zt  be the probability of obtaining the result zt  at the instant t, lying in the range   z   ,  z   ,  z   N on measuring observable z in respect to spinor z,y, t  t,. From Eq.(6.1.32) we obtain Pt  z   ,  z   , zt    1 iff zt  z  and |zt  z  vzu,0 t |  . 6.1.34 From Eq. (6.1.29) it follows that: Pt  z   ,  z   , zt    1 if t  tz    |  z  | vz u,0   |  z  | u sin2 0 2 . 6.1.35 Remark 6.1.11. Note that from Remark 6.1.5 it follows that tz   tD and therefore from Eq. (6.1.33) and Eq. (6.1.35) one obtains |  z  | usin2 0 2  D v0 sin 2 0 2   z  u  D v0 6.1.36 as it should be, because the equality |  z  | u  D v0 is required by the condition of the Stern-Gerlach experiment. 220 VI.2. Schrödinger's cat which measures spin. Schrödinger's cat paradox resolution Another known in literature special sort of the Schrödinger cat paradox can be simply illustrated with the famous Stern-Gerlach experiment (Fig. 6.1.1). Silver atoms boiled off from a furnace are sent through a non-uniform magnetic field, and impinge on a photographic plate. Instead of a continuous distribution of spots, one sees two spots, corresponding to spin up and spin down relative to the magnetic field axis. Each atom goes up OR down, but one cannot predict which in any given run the results of the experiment are probabilistic. There is a 50% chance of an atom going up, and a 50% chance that it will go down. Fig. 6.1.4. Stern-Gerlach experiment. Adapted from [20] Remark 6.2.1. We remind that from the point of view of the Schrödinger equation of the quantum theory, this result has no any rigorous explanation. Remark 6.2.2. In the quantum theory, the state of the particle is described by its wave function, and the Schrödinger equation says that at a post-measurement final time t f , the wave function is related to that at a pre-measurement initial time t i , by known deterministic relation 221 t f   Ut f , t i t i , Ut f , t i   exp iHt f  t i  with the transition unitary operator U completely specified by the Hamiltonian H . To explain what is observed, the Schrödinger equation must be supplemented by the reduction postulate and the Born rule. This state that the wave function only gives a description of probabilities when a measurement is made, with the probabilities for an 'up' outcome and a 'down' outcome given by the squares of the coefficients of the corresponding components in the initial wave function t i , with the sum of the 'up' and 'down' probabilities equal to one. The reduction postulate and the Born rule are an add-on to the Schrödinger equation. According to the Copenhagen interpretation of quantum mechanics, the Schrödinger equation is applied when a microscopic system, the silver atom, is time-evolving in isolation. But when the atom interacts with a macroscopic measuring apparatus, as in the Stern-Gerlach setup, you have to use the reduction postulate and the Born rule. Fig. 6.1.5. The Stern-Gerlach apparatus with a Schrödinger cat as the outcome registration. Adapted from [20] 222 Remark 6.2.3. This situation leads to puzzles that have been debated for over eighty years. If quantum mechanics describes the whole universe, then why can't one use the Schrödinger equation to describe the system consisting of the silver atom plus the measuring apparatus? But we never see a superposition state of the atom plus apparatus. This is the Schrödinger's famous cat paradox. Arrange the experiment so that an up outcome triggers a mechanism that kills the cat, while a down outcome keeps the cat alive. Of course we don't do this, but if we were to do it, we would always see a live cat OR a dead one, never a superposition of the two (Fig. 6.1.5). So we have the problem of definite outcomes: where does the either or dichotomy arises? Let us consider again the Schrödinger's cat which measures spin by using the Stern-Gerlach apparatus, see (Fig. 6.1.4). When a measurement is made, with the up outcome Schrödinger's cat is dead. When a measurement is made, with the down outcome Schrödinger's cat is alive. As it is known many years that conventional QM with canonical explanation of the Stern-Gerlach experiment cannot give predicable and definite outcomes for Schrödinger's cat which measures spin. Theorem 6.2.1. Any spinor z,y, t, t   z,y, t, t   6.2.1 given by Eq. (6.1.9a) - (6.1.9b) with 0 such that cos 0 2  0 always kills the Schrödinger's cat at the instant t : t  D v0 cos 2 0 2 . 6.2.2 Proof. Immediately from Eq. (6.1.31) and Eq. (6.1.36). 223 Chapter VII THE BELL INEQUALITIES REVISITED One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. In [1], [2] Bell's inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell's inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained (see VII.2) under EPR-B nonlocality postulate. VII.1. Bell theorem without the hypothesis of locality and without the introduction of hidden variables VII.1.1. Clauser-Horne-Shimony-Holt (CHSH) inequality In a typical Bell experiment, two systems which may have previously interacted for instance they may have been produced by a common source are now spatially separated and are each measured by one of two distant observers, Alice and Bob (see Fig.7.2.1). Alice may choose one out of several possible measurements to perform on her system and we let x denote her measurement choice. For instance, x may refer to the position of a knob on her measurement apparatus. Similarly, we let y denote 224 Bob's measurement choice. Once the measurements are performed, they yield outcomes a and b on the two systems. Remark 7.1.1. The actual values assigned to the measurement choices x,y and outcomes a,b are purely conventional; they are mere macroscopic labels distinguishing the different possibilities. Remark 7.1.2. From one run of the experiment to the other, the outcomes a and b that are obtained may vary, even when the same choices of measurements x and y are made. Assumption 7.1.1. These outcomes a and b are thus in general governed by a Kolmogorovian probability distribution pab|xy, which can of course depend on the particular experiment being performed. By repeating the experiment a sufficient number of times and collecting the observed data, one can get a fair estimate of such Kolmogorovian probabilities [3]-[4]. Assumption 7.1.2. When such an experiment is actually performedsay , by generating pairs of spin 1/2 particles and measuring the spin of each particle in different directions it will in general be found that pab|xy  pa|xpb|y, 7.1.1 implying that the outcomes on both sides are not statistically independent of each other. Even though the two systems may be separated by a large distance – and may even be space-like separated the existence of such correlations is nothing mysterious. In particular, it does not necessarily imply some kind of direct influence of one system on the other, for these correlations some dependence relation between the two systems which was established when they interacted in the past may simply reveal. This is at least what one would expect in a local theory. 225 Let us formulate the idea of a local theory more precisely. Assumption 7.1.3. The assumption of locality implies that we should be able to identify a set of past factors, described by some variables λ, having a joint causal influence on both outcomes, and which fully account for the dependence between a and b. Once all such factors have been taken into account, the residual indeterminacies about the outcomes must now be decoupled, that is, the Kolmogorovian probabilities for a and b should factorize: pab|xy,  pa|x,pb|y,. 7.1.2 Remark 7.1.3. This factorability condition simply expresses that we have found an explanation according to which the probability for a only depends on the past variables and on the local measurement x , but not on the distant measurement and outcome, and analogously for the probability to obtain b . The variable will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because may involve physical quantities that are not fully controllable. The different values of across the runs should thus be characterized by a probability distribution qλ. Combined with the above factorability condition, we can thus write pab|xy    dqpa|x,pb|y,, 7.1.3 where we also implicitly assumed that the measurements x and y can be freely chosen in a way that is independent on , i.e., that qλ|x,y  qλ . This decomposition now represents a precise condition for locality in the context of Bell experiments. Remark 7.1.4. Note that no assumptions of determinism or of a classical behaviour are being involved in the condition (7.1.3): we λ λ λ λ λ 226 assumed that a (and similarly b ) is only probabilistically determined by the measurement x and the variable , with no restrictions on the physical laws governing this causal relation. Locality is the crucial assumption behind (7.1.3). In relativistic terms, it is the requirement that events in one region of space-time should not influence events in space-like separated regions. Let us consider for simplicity an experiment where there are only two measurement choices per observer x,y  0,1 and where the possible outcomes take also two values labelled a,b  1,1. Let axby  be the expectation value of the product ab for given measurement choices x,y : axby   a,b abpab|xy. 7.1.4 Consider the following expression S  a0b0   a0b1   a1b0   a1b1 , 7.1.5 which is a function of the probabilities pab|xy. If these probabilities satisfy the locality decomposition (7.1.3), we necessarily have that S  a0b0   a0b1   a1b0   a1b1   2, 7.1.6 which is known as the Clauser-Horne-Shimony-Holt (CHSH) inequality [5]. To derive this inequality, we can use (7.1.3) in the Definition (7.1.4) of axby , which allows us to express this expectation value as an average axby     dqdqax by  7.1.7 of a product of local expectations λ 227 ax   a apa|x, 7.1.8 and by   b bpb|y, 7.1.9 taking values in 1,1. Inserting these expressions (7.1.7) - (7.1.9) in Eq. (7.1.5), we can write S    dqS, 7.1.10 where S  a0 b0   a0 b1   a1 b0   a1 b1 . 7.1.11 Since a0 , b0   1,1, this last expression is smaller than Sλ  Sλ  Sλ   |b0   b1  |  |b0   b1  |. 7.1.12 Without loss of generality, we can assume that b0   b1   0 which yields Sλ  2b0   2 and thus S  2. Consider now the quantum predictions for an experiment in which the two systems measured by Alice and Bob are two qubits in the singlet state   1 2 |01  |10, where we have used the shortcut notation |ab  |a |b, and where |0 and |1 are conventionally the eigenstates of σz for the eigenvalues 1 and 1 respectively. Let the measurement choices x and y be associated with vectors x and y corresponding to measurements of x  σ on the first qubit and of y  σ on the second qubit, where σ  σ1 ,σ2 ,σ3 denotes the Pauli 228 vector. According to the quantum theory we (then) have the expectations axby    x y . Let the two settings x  0,1 correspond to measurements in the orthogonal directions  e 1 and  e 2 respectively and the settings y  0,1 to measurements in the directions  1 2 e1   e2 and 1 2 e1   e2 . We then have a0b0   a0b1   a1b0   1 2 and  a1b1    1 2 whence S  2 2 7.1.13 in contradiction with CHSH inequality (7.1.6). VII.1.2. Clauser Horne Inequality Suppose that some observable of the two particles is registered as a count in a detector. If the composite state consists of two photons, the detector registers a hit if the polarization is along some direction. The inequality will be determined by counting. There will be a total of N events, with N1a counts in detector 1 when it is set to select a and N2b counts in detector 2 when it is set to select b. The number of coincidences of the two detectors with settings a and b respectively is N12a,b. The probabilities are p1a  N1a N ,p2  b  N2b N ,p12  a  N12a,b N . 7.1.14 Remember that in the Bell formulation, the hidden variable determined absolutely the value of the polarization for a particular measurement. Then 229 p1a   dwp1a,,p2b   dwp2b,, p12a,b,  p1a,p2b,, p12a,b   dwp1a,p2b,. 7.1.15 Remind that for any four real numbers x,x ,y,y  0,1 the inequality holds xy  xy  x y  x y  x   y. 7.1.16 We denote now x  p1a,,y  p2b,,x   p1a,,y  p2b,, 7.1.17 and substitute into inequality (7.1.16) we get p1a,p2b,  p1a,p2b,  p1a,p2b,  p1a,p2b,   p1a,  p2b,. 7.1.18 Next multiplying by w and integrating over all  we get p12a,b  p12a,b  p12a,b  p12a,b  p1a  p2b. 7.1.19 Consider now the quantum predictions for an experiment with 2 photons. An atomic s-state with zero total angular momentum and even parity decays in two steps. Photon 1 is emitted in the E1 transition from the S-state to a P -state with m  1,0. Photon 2 is emitted in the second E1 transition to the ground state. The initial state of the atom also has zero angular momentum and even parity. Therefore the photons which are emitted back to back have the same helicity, so that their total angular momentum is zero. 230 The two photons have different energies, 1 and 2 . The helicity of each of the photons is determined by the intermediate state. If the intermediate state is m  1 then the helicity of both photons is odd and if m  1 then the helicities are even. The energy of the intermediate state is degenerate. There is no magnetic field that might split the energies of the m  1,0 levels. The final pure photon state is therefore the linear combination |  1 2 |1 |1  |1 |1. 7.1.20 It will be more interesting if the measurements of the photon polarizations are in the linear basis. Then we can look for correlations of the measurement of linear polarization by detector 1 along a and by 2 along b. So let us write | in the linear polarization basis. The linear and circular polarization bases are related according to |x,y,k   1 2 |1  i|1 7.1.21 and |x,y,k   1 2 |1  i|1. 7.1.22 Thus we can rewrite |  1 2 |x 1  |x 2  |y1  |y2 . 7.1.23 Evidently if detector 1 measures horizontal polarization then so will detector 2, etc. In general we want to measure the correlation p121 ,2 , that is the probability that we get a count on detector 1 with polarization axis 1 coincident with a count in detector 2 with polarization axis 2 . The observable is the operator 231 1 ,2   |1 1 |2 21 |12 |2 . 7.1.24 Assume now that 1 ,2   1  2 , 7.1.25 since there is zero angular momentum in the final state, there is rotation symmetry so the observable can only depend on the difference of the polarization angles. The expectation value of  is 1  2   |1  2 |  1 2 x |1x |2  y|1y|2 |1 1 |2 21 |12 |2|x 1 |x 2  |y1 |y2 . 7.1.26 Note that x|  cos, y|  sin. Finally one obtains 1  2   14 1  cos21  2 . 7.1.27 The Clauser Horne inequality is N12a,b  N12b,a  N12a,b  N12a,b N1a  N2b  1. 7.1.28 Assume now that a,b,a,b are all separated by the angle  then N12  N12  N12  N123 N1a  N2b  3N12  N123 N1a  N2b  1. 7.1.29 Next relate coincidences to expectation values. Note that N12  N||. As regards the singles counts N1a and N2b , we know that the number of counts must be independent of the direction of a  or b and that for any direction N1  1 2 N , since half the photons will be polarized along and direction. Therefore the 232 inequality (7.1.29) becomes   3 4 1  cos2  1 4 1  cos6 1 2  1 2  1 2  3 4 cos2  1 4 cos6  1. 7.1.30 The inequality (7.1.30) is maximally violated if    8 :   8  1 2  3 2 4  1 2 2  1.2 7.1.31 which is not less than 1. VII.1.3. Violation of Bell's inequality under strict Einstein locality conditions The assumption of locality in the derivation of Bell's theorem requires that the measurement processes of the two observers are space-like separated (Fig. 7.1.1). This means that it is necessary to freely choose a direction for analysis, to set the analyzer and finally to register the particle such that it is impossible for any information about these processes to travel via any (possibly unknown) channel to the other observer before he, in turn, finishes his measurement. Selection of an analyzer direction has to be completely unpredictable which necessitates a physical random number generator. A numerical pseudo-random number generator can not be used, since its state at any time is predetermined. Furthermore, to achieve complete independence of both observers, one should avoid any common context as would be conventional registration of coincidences as in all previous experiments. Rather the individual events should be registered on both sides completely independently and compared only after the measurements are finished. This requires independent and highly accurate time bases on both sides. In our experiment for the first time any mutual influence between the two observations is excluded within the realm of Einstein locality. 233 To achieve this condition the observers Alice and Bob were spatially separated by 400 m across the Innsbruck university science campus. In [5] polarization entangled photon pairs which were sent to the observers through optical fibers were used. About 250 m of each 500 m long cable was laid out and the rest was left coiled at the source. This, we remark, has no influence on the timing argument because the optical elements of the source and the locally coiled fibers can be seen as jointly forming the effective source of the experiment (Fig. 7.1.1). Remark 7.1.5. The difference in fiber length was less than 1m which means that the photons were registered simultaneously within interval 5ns. Fig. 7.1.1. Spacetime diagram of Bell experiment [5] Selecting a random analyzer direction, setting the analyzer and finally detecting a photon constitute the measurement process. This 234 process on Alice's side must fully lie inside the shaded region which is, during Bob's own measurement, invisible to him as a matter of principle. For setup this means that the decision about the setting has to be made after point "X" if the corresponding photons are detected at spacetime points "Y" and "Z" respectively. In this experiment the measurement process (indicated by a short black bar) including the choice of a random number only took less than a tenth of the maximum allowed time. The vertical parts of the kinked photon world lines emerging from the source represent the fiber coils at the source location. The source of polarization entangled photon pairs is degenerate type-II parametric down-conversion where a BBO-crystal was pumped with 400 mW of 351nm light from an Argon-ion-laser. A telescope was used to narrow the UV-pump beam, in order to enhance the coupling of the 702 nm photons into the two single mode glass fibers. On the way to the fibers, the photons passed a half-wave plate and the compensator crystals necessary to compensate for in-crystal birefringence and to adjust the internal phase of the entangled state |  1/ 2|H1 |V2  e i |V1 |H2 , which was chosen   . Remark 7.1.6. The horizontal and the vertical polarization jointly define a basis denoted by  z, which can take on the values |H or |V . The modulation systems (high-voltage amplifier and electro-optic modulator) had a frequency range from DC to 30 MHz. In operating the systems at high frequencies a reduced polarization contrast of 97% (Bob) and 98% (Alice) was observed. This, however, is no real depolarization but merely reflects the fact that we are averaging over the polarization rotation induced by an electrical signal from the high-voltage amplifier, which is not of perfectly rectangular shape. 235 Fig. 7.1.2. One of the two observer stations [5] A random number generator is driving the electro-optic modulator. Silicon avalanche photodiodes are used as detectors. A time tag is stored for each detected photon together with the corresponding random number 0 or 1 and the code for the detector + or corresponding to the two outputs of the Wollaston prism polarizer. All alignments and adjustments were pure local operations that did not rely on a common source or on communication between the observers. The actual orientation for local polarization analysis was determined independently by a physical random number generator. This generator has a light-emitting diode (coherence time tc  10 fs) illuminating a beam splitter whose outputs are monitored by photomultipliers. The subsequent electronic circuit sets its output to 0(1) upon receiving a pulse from photomultiplier 0(1). Remark 7.1.7. Events where both photomultipliers register a photon within △t ≤ 2 ns are ignored [5]. 236 Fig. 7.1.3. (i) t1 t2 = 0, (ii) t1 t2 = δ > 0, (iii) t1 t2 = -δ < 0 A down-converter (one way to produce an entangled pair) throws two entangled photons 1 and 2 in opposite directions. Polarization of the photons 1 and 2 is measured by polarizers I and II respectively. Remark 7.1.8. Assume that photon 1 collapses in polarizer I at instant t1 and photon 2 collapses in polarizer II at instant t2 respectively. Note that in general case t1  t2 even if photons 1 and 2 were registered simultaneously (within 5ns interval, see Remark 7.1.5). Notice that obviously there exist only three possibilities: (i) t1  t2  0, (ii) t1  t2  min    0, (iii) t1  t2  min    0. We have chosen here min  const  . The resulting binary random number generator has a maximum toggle frequency of 500 MHz. By changing the source intensity the mean interval was adjusted to about 10 ns in order to have a high primary random bit rate. Certainly this kind of random-number generator is not necessarily evenly distributed. For a test of Bell's inequality it is, however, not necessary to have perfectly even distribution, because all correlation functions are normalized to the total number of events for a certain combination of the analyzers' settings. Still, we kept the distribution even to within 2% in order to obtain an approximately equal number of samples for each setting. The distribution was adjusted by equalizing the number of counts of the two photomultipliers through changing their internal photoelectron amplification. Due to the limited speed of the 237 subsequent modulation system it was sufficient to sample this random number generator periodically at a rate of 10 MHz. There are many variants of Bell's inequalities. In G. Weihs, T. Jennewein experiment [5] a version first derived by Clauser et al. [4] (CHSH) was used since it applies directly to Zeilinger's experimental configuration. The number of coincidences between Alice's detector i and Bob's detector j is denoted by C ij, with i, j  , where  and  are the directions of the two polarization analyzers and  and  denote the two outputs of a two-channel polarizer respectively. If we assume that the detected pairs are a fair sample of all pairs emitted, then the normalized expectation value E, of the correlation between Alice's and Bob's local results is E,  C,  C,  C,  C,/N, 7.1.32 where N is the sum of all coincidence rates. Remark 7.1.9. We define now: (i) Cij ,  Cij,, t1 , t2 , where t1  t2  0; (ii) Cij ,  Cij,, t1 , t2 , where t1  t2  ; (iii) Cij ,  Cij,, t1 , t2 , where t1  t2  . Remark 7.1.10. Note that C ij,  C ij ,  C ij ,  C ij ,. 7.1.33 In a rather general form the CHSH inequality reads S,,,  |E,  E,|  |E,  E,|  2. 7.1.34 Quantum theory predicts a sinusoidal dependence for the coincidence rate C QM,  sin2   7.1.35 238 on the difference angle of the analyzer directions in Alice's and Bob's experiments. The same behavior can also be seen in the correlation function EQM,  cos2  . 7.1.36 Thus, for various combinations of analyzer directions ,,, these functions violate CHSH inequality. Maximum violation is obtained using the following set of angles Smax QM  SQM0, 45, 22.5, 67.5  2 2  2.82  2. 7.1.37 If, however, the perfect correlations (    0 or 90  ) have a reduced visibility V ≤ 1 then the quantum theoretical predictions for E and S are reduced as well by the same factor independent of the angle. Thus, because the visibility of the perfect correlations in this experiment was about 97% and was expected S to be not higher than 2.74 if alignment of all angles is perfect and all detectors are equally efficient. Various measurements with the described setup were performed [5]. The data presented in Fig. 7.1.4 are the result of a scan of the DC bias voltage in Alice's modulation system over a 200 V range in 5 V steps. At each point a synchronization pulse triggered a measurement period of 5 s on each side. From the timetag series we extracted coincidences after all measurements had been finished. Fig. 7.1.4 shows four of the 16 resulting coincidence rates as functions of the bias voltage. Each curve corresponds to a certain detector and a certain modulator state on each side. A nonlinear χ2 fit showed perfect agreement with the sine curve predicted by quantum theory. Visibility was 97% as one could have expected from the previously measured polarization contrast. No oscillations in the singles count rates were found. We want to stress again that the accidental coincidences have not been subtracted from the plotted data. 239 Fig. 7.1.4. Four out of sixteen coincidence rates between various detection channels as functions of bias voltage (analyzer rotation angle) on Alice's modulator. A+1/B−0, for example, are the coincidences between Alice's "+" detector with switch having been in position "1" and Bob's "−" detector with switch position "0". The difference in height is explained by different efficiencies of the detectors In order to give quantitative results for the violation of Bell's inequality with better statistics, experimental runs were performed with the settings 0, 45 for Alice's and 22.5,67.5 for Bob's polarization analyzer. A typical observed value of the function S in 240 such a measurement was S 2.73  0.02 for 14700 coincidence events collected in 10s. This corresponds to a violation of the CHSH inequality of 30 standard deviations assuming only statistical errors. If we allow for asymmetries between the detectors and minor errors of the modulator voltages this result agrees very well with the quantum theoretical prediction. VII.1.4. CHSH theorem without the hypothesis of locality One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., of the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of local realism. In [1, 2], CHSH inequality was derived in its traditional form, without resorting to the hypothesis of locality, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality is not relevant to violations of CHSH inequalities. Let A,A,B,B be random variables with values in the set 1,1, i.e., A  1,A  1,B  1,B  1. 7.1.38 Assume that there exist joint probability distribution functions WA,A,B,B, of A,A,B,B defining probabilities for each possible set of outcomes such that: (i) PA,A,B,B  0,PA,B,B  0,PA,B,B  0,etc., 7.1.39 241 (ii)  A,A ,B,B  PA,A ,B,B   1,  A,A ,B,B  PA,B,B   1,  A,A ,B,B  PA ,B,B   1,etc., 7.1.40 (iii) PA,A ,B,B   PA,A ,B,B   PA ,B,B   PA,A ,B,B , PA,A ,B,B   PA,A ,B,B   PA ,B,B   PA,A ,B,B , etc. 7.1.41 From (7.1.41) one obtains 0  PA,B,B   PA,A ,B,B   PA,A ,B,B   PA ,B   PA ,B  PA ,B   PB  PA ,B. 7.1.42 Similarly one obtains 0  PA,B,B   PA,B  PA,B,B   PA  PA,B  PA,B   PA,B,B  7.1.43 and therefore PA,B,B  PA  PA,B  PA,B . 7.1.44 From (7.1.42) and (7.1.44) we obtain 0  PA,B,B   PA,B,B   PA ,B   PB  PA ,B  PA  PA,B  PA,B  7.1.45 and therefore 0  PA ,B   PB  PA ,B  PA  PA,B  PA,B    PA  PB  PA,B  PA ,B  PA,B   PA ,B . 7.1.46 242 From (7.1.46) one obtains A,A,B,B   PA,B  PA,B  PA,B   PA,B   PA  PB  0. 7.1.47 Note that PB,B  PB  PB,B 7.1.48 and PB,B  PB  PB,B  1  PB  PB,B. 7.1.49 From (7.1.49) and (7.1.48) we obtain PB,B  1  PB  PB  PB,B . 7.1.50 Note that 0  PA,B,B  PB,B   PA,B,B . 7.1.51 Inserting (7.1.43) and (7.1.50) into (7.1.51) we obtain 0  1  PA  PB  PB   PA,B  PA ,B  PA,B   PB,B   PA,B,B   1  PA  PB  PB   PA,B  PA ,B  PA,B,B . 7.1.52 Note that PA,B,B   PA,A ,B,B   PA,A ,B,B   PA ,B  PA ,B    PA ,B  PB   PA ,B . 7.1.53 From (7.1.53) we obtain 243 0  1  PA  PB  PB  PA,B  PA,B  PA,B  PA,B. 7.1.54 From (7.1.54) and (7.1.47) we obtain 1  A,A,B,B  0. 7.1.55 Note that the following representation of the quantities AB, AB, AB , AB  holds AB  PAB   PAB   PAB   PAB , etc., 7.1.56 where PAB   PA  1,B  1,PAB   PA  1,B  1,etc. 7.1.57 From (7.1.56) and (7.1.57) we obtain AB  A B  AB   A B                     . 7.1.58 From (7.1.55) we obtain 2            0 7.1.59 and 0            2. 7.1.60 From (7.1.59) and (7.1.60) we obtain 2                      2. 7.1.61 244 From (7.1.58) and (7.1.61) finally we obtain |AB  AB  AB   AB |  2. 7.1.62 VII.1.5. CHSH theorem without the introduction of hidden variables A hidden-variable theory is the traditional, but not unique, basis for constructing various types of Bell's theorem. The starting point may also be a recognition of the existence of a positive-definite probability distribution function. This assumption alone is used to formulate and prove Bell's paradoxes of different types [6, 7]. Let A,A,B,B be random variables with values in the set 1,1, i.e., A  1,A  1,B  1,B  1. 7.1.63 Assume that there exists joint probability distribution function PA,A,B,B of A,A,B,B defining probabilities for each possible set of outcomes such that: PA,A,B,B   0, 7.1.64 and  A,A ,B,B  PA,A ,B,B   1, 7.1.65 and PA,A,B,B  PA,A,B,B   PA,B,B . 7.1.66 245 Let us consider Bell inequality of the form   1 2 |AB  A B  AB   A B |  1. 7.1.67 It is well known [1], [6] that under assumptions (7.1.64)-(7.1.66) Bell inequality is directly provable without any reference to Kolmogorov probability space ,,P. We abbreviate now for short [1, 6]: P1  PA,A ,B,B      PA  1,A   1,B  1,B  1, P2  PA,A ,B,B      PA  1,A   1,B  1,B  1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P16  PA,A ,B,B      PA  1,A   1,B  1,B  1. 7.1.68 For the quantities AB, AB, AB  and A B  using Eq. (7.1.68) one obtains the representatives AB  PAB   PAB   PAB   PAB , etc., 7.1.69 where PAB   PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B    , etc. 7.1.70 Substituting Eq. (7.1.69) into the expression AB  AB  AB   AB  one obtains 246   1 2 AB  A B  AB   A B   PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B      PA,A ,B,B    . 7.1.71 From (7.1.64) - (7.1.65) it obviously follows that 1    1, and therefore Bell inequality (7.1.67) holds. VII.2. Physical nature of the violation of the Bell inequalities VII.2.1. Physical interpretation of the Bell test experiment under EPR-B nonlocality postulate Actually, most experiments have been performed using polarization of photons. The quantum state of the pair of entangled photons is not the singlet state. The polarization of a photon is measured in a pair of perpendicular directions. Relative to a given orientation, polarization is either vertical (denoted by V or by ) or horizontal (denoted by H or by ). The photon pairs are generated in the quantum state |EPR   1 2 |Vs  |Vi  |Hs  |Hi , 7.2.1 where |V and |H denote the state of a single vertically or horizontally polarized photon, respectively (relative to a fixed and common reference direction for both particles) and subscripts s and i indicate signal or idler photon respectively. 247 The source S produces pairs of "photons" sent in opposite directions. Each photon encounters a two-channel polariser whose orientation (a or b) can be set by the experimenter. Emerging signals from each channel are detected and coincidences of four types (  , ,  and  ) are counted by the coincidence monitor. This state cannot be factored into a simple product of signal and idler states: |EPR  |As  |Bi for any choice of |As and |Bi . This means the state of one particle cannot be specified without making reference to the other particle. Such particles are said to be "entangled" and |EPR  is an entangled state. If we measure the polarizations of signal and idler photons in the H,V basis there are two possible outcomes: both vertical or both horizontal. Each occurs half of the time. We could instead measure the polarizations with polarizers rotated by an angle  . We use the rotated polarization basis |V   cos|V  sin|H, |H   sin|V  cos|H. 7.2.2 Here |V  describes a state with polarization rotated by  from the vertical, while |H  is  from the horizontal. In this basis the state is |EPR   1 2 |V s|V i  |H s|H i . 7.2.3 Remark 7.2.1. We will denote the events corresponding to coincidences (at the instant t ) of four types  , , ,  on the coincidence monitor by symbols ,; t,,; t,,; t,,; t or by symbols a,b;t,a,b;t,a,b;t,a,b;t respectively or simply a,b,a,b,a,b,a,b . 248 Fig. 7.2.1. Scheme of a "two-channel" Bell test Remark 7.2.2. Clearly, if we measure in this rotated basis we get the same results: half the time both are |V  and half of the time both are |H . Knowing this, we can measure the signal polarization and infer with certainty the idler polarization. This is the situation EPR described, but we have used polarizations instead of position and momentum. Remark 7.2.3. Note that there is an uncertainty relationship between polarizations in different bases. Knowledge of a photon polarization after the measurement such a polarization in the V0  ,H0  basis implies complete uncertainty of its polarization in the V45  ,H45  basis, for example. 249 Fig. 7.2.2. Schematic of experimental setup [8] Symbols: LD Laser Diode, CL Collimating Lens, BF Blue Filter, BA Beam Aperture, LP Laser Polarizer, QP Quartz Plate, MI Mirror, CR Downconversion Crystals, RA Rail, PA Polarizer A, PB Polarizer B, ID Iris Diaphragm, RF Red Filter, FL Focusing Lens, CA Cage Assembly, DA Detector A, DB Detector B, ST Beam Stop. Figure 7.2.2 shows a schematic of an experimental setup to produce polarization entangled photons [8]. A 5 mW free-running InGaN diode laser produces a beam of violet (405 nm) photons which passes through a blue filter, a linear polarizer, and a birefringent plate before reaching a pair of beta barium borate (BBO) crystals. In the crystals, a small fraction of the laser photons spontaneously decays into pairs of photons by the process of spontaneous parametric downconversion (SPD). In a given decay the downconverted photons emerge at the same time and on opposite sides of the laser beam. The detectors, two single-photon counting modules (SPCMs), are preceded by linear polarizers and red filters to block any scattered laser light. Even so, it is necessary to use coincidence detection to separate the downconverted photons from the background of other photons reaching the detectors. Because the photons of a downconverted pair are produced at the same time 250 they cause coincident, i.e., nearly simultaneous, firings of the SPCMs. Coincidences are detected by a fast logic circuit and recorded by a personal computer. The detection components (SPCMs, irises, lenses and filters) are mounted on rails which pivot about a vertical axis passing through the crystals. This allows the detection of SPD photons at different angles with minimal realignment. The rails were positioned at A  B  2.5 and the focusing lenses adjusted for maximum singles rates. With the irises fully open and polarizers both set to vertical, more than 300 counts per second were observed [8]. Fig. 7.2.3. Two-crystal down conversion source The crystals are 0.1 mm thick and in contact face-to-face, while the pump beam is approximately 1 mm in diameter. Thus the cones of downconverted light from the two crystals overlap almost completely. These BBO crystals are cut for Type I phase matching, which means that the signal and idler photons emerge with the same polarization, which is orthogonal to that of the pump photon. Each crystal can only support downconversion of one pump polarization. The other polarization passes through the crystal unchanged. We 251 use two crystals, one rotated 90 from the other, so that either pump polarization can downconvert according to the rules |Vp  |Hs|Hi, |Hp  expi|Vs|Vi. 7.2.4 where  is a phase due to dispersion and birefringence in the crystals. The geometry is shown schematically in Figure 7.2.3. To create an entangled state, we first linearly polarize the laser beam at an angle  l from the vertical and then shift the phase of one polarization component by  l with the birefringent quartz plate. The laser photons (pump photons) are then in the state |pump   cos l|Vp  expi lsin l|Hp 7.2.5 when they reach the crystals. The downconverted photons emerge in the state |DC  cos l|Hs|Hi  expisin l|Vs|Vi 7.2.6 where    l   is the total phase difference of the two polarization components [8]. Remark 7.2.4. This state is an entangled state and is already quite adjustable. Further modifications can be made with ordinary optical components. For example, if  l  /4 ,    then a half-wave plate in the signal beam could be used to switch the signal polarization |Hs  |Vs to produce |DC    |Vs|Hi  |Hs|Vi/ 2 . By placing polarizers rotated to angles  and  in the signal and idler paths, respectively, we measure the polarization of the downconverted photons. For a pair produced in the state |DC , the probability of coincidence detection is 252 PVV,  |V |sV |i|DC| 2 . 7.2.7 The VV subscripts on P indicate the measurement outcome VV , both photons vertical in the bases of their respective polarizers. More generally, for any pair of polarizer angles ,, there are four possible outcomes, VV,VH,HV and HH indicated by VV,VH,HV and HH, respectively. Using the basis of equation (7.2.2), we find PVV,  |sinsincos l  expicoscossin l | 2 7.2.8 or PVV,  sin2sin2cos2 l  cos2cos2sin2 l  14 sin2sin2sin2 l cos. 7.2.9 A special case occurs when |DC  |EPR , i.e., when  l  /4 and   0. In this case PVV,  12 cos 2  , 7.2.10 which depends only on the relative angle    . The last term in Eq. (7.2.9) is a cross term which accounts for the interference between the H,H and V,V parts of the state. The  in this term is, through its dependence on  , a complicated function of pump photon wavelength, signal photon wavelength and angle as well as crystal characteristics. Because the laser has a finite line width and we collect photons over a finite solid angle and wavelength range, we collect a range of  . To account for this, we replace cos by its average cos  cosm to get PVV,  sin2sin2cos2 l  cos2cos2sin2 l  14 sin2sin2sin2 l cosm 7.2.11 In the experiment, a fixed interval T of data acquisition (typically in the range 0.5 seconds to 15 seconds) was chosen and the number of coincidences N, during that interval was recorded [8]. 253 Assuming a constant flux of photon pairs, the number collected will be N,  A sin2sin2cos2 l  cos2cos2sin2 l  14 sin2sin2sin2 l cosm  C 7.2.12 where A is the total number of entangled pairs produced and C is an offset to account for imperfections in the polarizers and alignment of the crystals. This is necessary to account for the fact that some coincidences are observed even when the polarizers are set to   0,  90  . Remark 7.2.5. We emphasize that (7.2.9) and (7.2.10) hold iff the measurements on photon ν 1 and photon ν 2 occur simultaneously, i.e., photon ν 1 and photon ν 2 collapse in polarizers I and II respectively at an instant t  t1  t2 , t  0,T, (see Remark 7.1.8) and we will denote such events by ν1 t1 ,ν2 t2  or simply ν1 ,ν2. Remark 7.2.6. We will denote such entangled pairs of photons also by ν1 t1 ,ν2 t2  or simply ν1 ,ν2 and we will denote the total number of entangled pairs ν1 t1 ,ν2 t2  produced during interval T by AT . The number of coincidences during interval T corresponding exactly to entangled pairs ν1 t1 ,ν2 t2  we will denote by NT ,. We rewrite now Eq. (7.2.11) in the following form: PVV  ,| ν 1 t1 ,ν 2 t2   PVV  , t1 ;, t2 | ν 1 t1 ,ν 2 t2   sin2 sin2cos2 l  cos2cos2sin2 l  14 sin2 sin2sin2 l cosm 7.2.13 254 Remark 7.2.7. Note that PVV  , t1 ;, t2 | ν1 t1 ,ν2 t2  is the conditional probability of the event ,; t (see Remark 7.2.1) under condition that the event ν1 t1 ,ν2 t2  was occured at instant t  t1  t2 . We rewrite now Eq. (7.2.12) in the following form: NT ,  AT  sin2sin2cos2 l  cos2cos2sin2 l  14 sin2sin2sin2 l cosm  C1 7.2.14 Remark 7.2.8. Let us now consider the complete set of probabilities pa, t1 ;b, t2  of joint detections of ν 1 and ν 2 in the channels  or  of polarisers I or II, in orientations a and b, see Fig. 7.2.4. The canonical Quantum Mechanical predictions for the the joint detection probabilities are pa, t1 ;b, t2   pa, t1 ;b, t2   12 cos 2a,b, pa, t1 ;b, t2   pa, t1 ;b, t2   12 sin 2a,b, t1  t2 . 7.2.15 We emphasize that (7.2.15) holds iff the measurements on photon ν 1 and photon ν 2 occur simultaneously, i.e., iff photon ν 1 and photon ν 2 collapse in polarizers I and II respectively at instant t  t1  t2 , see Remark 7.1.8. Remark 7.2.9. Suppose now that the measurement on photon ν 1 occurs first, at instant t1 , and gives the result , with the polarizer I in the orientation a and therefore the measurement on photon ν 2 occurs at instant t2 , where t1  t2    0, (see Remark 7.1.8) and we will denote such events (i.e., if photon ν 1 collapses in polarizer I at instant t1 and photon ν 2 collapses in polarizer II at instant t2 , where t1 , t2  0,T respectively) by ν 1 t1 ,ν2 t2 , or simply ν1 ,ν2. 255 Remark 7.2.10. We will denote such entangled pairs of photons also by ν1 t1 ,ν2 t2 , t1 , t2  0,T, or simply ν1 ,ν2. Fig. 7.2.4. Schematic of the experiment testing EPR non-locality. (i) 1 2 0t t  , (ii) 1 2 0t t    , (iii) 1 2 0t t     Entangled photons from the source are sent to two fast switches, that direct them to polarizing detectors. The switches change settings very rapidly, effectively changing the detector settings for the experiment while the photons are in flight. ν1 ,ν2  describing the pair is obtained by projection of the initial state vector |ν1 ,ν2   1 2 |x,x  |y,y, where |x  and |y are linear polarizations states, onto the eigenspace associated to the result  : this two dimensional eigenspace has a basis |a,x, |a,y. Using the corresponding projector, one finds after a little algebra ν1 ,ν2   |a,a. 7.2.16 This means that (i) immediately after the first measurement, photon ν 1 takes the polarization |a: this is obvious because it has been measured with a polarizer oriented along a , and the result  has been found, (ii) the distant photon ν 2 , which has not yet interacted with any polarizer, at instant t1 has also been projected exactly into 256 the state |a with a well defined polarization, parallel to the one found for photon 1 . Remark 7.2.11. Note that the standard Heisenberg's uncertainty principle predicts that if the polarization of the photon ν 2 along the direction a becomes certainty, i.e., known exactly, all information about the polarization of the photon ν 2 along the direction b becomes uncertainty, i.e., it will be completely lost. In order to overcome this problem we apply Heisenberg's noise disturbance uncertainty relation, see Appendix A. This relation is generally formulated as follows: for any apparatus A to measure an observable A, the relation A,,AB,,A  1 2 ||A,B|| 7.2.17 holds for any input state  and any observable B, where A,,A stands for the noise of the A measurement in state  using apparatus A and B,,A stands for the disturbance of B in state  caused by apparatus A. From (7.2.17) one obtains a ,12 ,PIb,12 ,PI  12 |12 |a ,b |12 |, 7.2.18 where the measurement operators a and b measure the polarization in the a -direction and b-direction respectively and where |12   |ν1 ,ν2 . Remark 7.2.12. Note that after the measurement on the photon ν 1 along the direction a the polarization of the photon ν 1 along the direction b obtains finite uncertainty b  b,12 ,PI. Thus for the joint detection probabilities pa, t1 ;b, t2  instead classical Eq. (7.2.15) by using the weak postulate of nonlocality, 257 (see subsection I.4.1) we obtain p  a, t1 ;b, t2   p a, t1 ;b, t2   12 cos 2a a ,b b , p  a, t1 ;b, t2   p a, t1 ;b, t2   12 sin 2a a ,b b . 7.2.19 Where a  a ,12 ,PI, t1  t2    0. Remark 7.2.13. Suppose now that the measurement on photon ν 2 occurs first, at instant t2 , and gives the result , with the polarizer I in the orientation a and therefore the measurement on photon ν 1 occurs at instant t1 , where t1  t2    0, (see Remark 7.1.8) and we will denote such events (i.e., if photon ν 1 collapses in polarizer I at instant t1 and photon ν 2 collapses in polarizer II at instant t2 respectively, where t1  t2    0, t1 , t2  0,T ) by ν1 t1 ,ν2 t2  or simply ν1 ,ν2. Remark 7.2.14. We will denote such entangled pairs of photons by ν1 t1 ,ν2 t2 , t1 , t2  0,T or simply ν1 ,ν2. Remark 7.2.15. Note that after the measurement on the photon ν 2 along the direction b the polarization of the photon ν 1 along the direction a obtains finite uncertainty a  a ,12 ,PII. Thus for the joint detection probabilities p a, t1 ;b, t2  instead classical Eq.(7.2.15) by using the weak postulate of nonlocality, (see subsection I.4.1) we obtain p  a, t1 ;b, t2   p a, t1 ;b, t2   12 cos 2a a ,b b, p  a, t1 ;b, t2   p a, t1 ;b, t2   12 sin 2a a ,b b , 7.2.20 258 where b  b,12 ,PII, t1  t2    0. Remark 7.2.16. Note that similarly as above, instead Eq. (7.2.11) and Eq. (7.2.12) we obtain: (I) PVV  ,  PVV  , t1 ;, t2   |V |sV |i|DC| 2 , 7.2.21 PVV  ,  PVV  , t1 ;, t2   sin2  sin2cos2 l  cos2  cos2 sin2 l  14 sin2  sin2sin2 l cosm 7.2.22 where t1  t2    0 and NT ,  AT sin2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C2  AT sin2 sin2cos2 l  cos2cos2sin2 l   1 4 sin2 sin2sin2 l cosm  AT C,, l,m ,,  C2 , 7.2.23 where AT  is the total number of entangled pairs ν1 t1 ,ν2 t2  produced during interval T and NT , is the number of coincidences corresponding to entangled pairs ν1 t1 ,ν2 t2  produced during interval T. Remark 7.2.17. Note that PVV  , t1 ;, t2 | ν1 t1 ,ν2 t2  is the conditional probability of the event ,; t2 (see Remark 7.2.1) under condition that the event ν1 t1 ,ν2 t2  was occurred at instant t  t2 . (II) PVV  ,  PVV  , t1 ;, t2   |V |sV |i|DC| 2 , 7.2.24 259 PVV  ,  PVV  , t1 ;, t2   sin2  sin2cos2 l  cos2  cos2 sin2 l  14 sin2  sin2sin2 l cosm 7.2.25 where t1  t2    0 and NT ,  AT sin2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C3  AT sin2 sin2cos2 l  cos2cos2sin2 l   1 4 sin2 sin2sin2 l cosm  AT C,, l,m ,,   C3 , 7.2.26 where AT  is the total number of entangled pairs ν1 t1 ,ν2 t2  produced during interval T and NT , is the number of coincidences corresponding to entangled pairs ν1 t1 ,ν2 t2  produced during interval T. Remark 7.2.18. Note that PVV  , t1 ;, t2 | ν1 t1 ,ν2 t2  is the conditional probability of the event ,; t1 (see Remark 7.2.1) under condition that the event ν1 t1 ,ν2 t2  was occurred at instant t  t1 . From Eq. (7.2.14), (7.2.23) and (7.2.26) we obtain (I) PVV   PVV  ,   PVV  ,  ν 1 ,ν 2   NT , AT   AT   AT   AT  AT   AT   AT  sin 2 sin2cos2 l  cos2cos2sin2 l  1 4 sin2 sin2sin2 l cosm  C, 7.2.27 260 where PVV  , is the joint probability of both events , and ν1 ,ν2 being true. (II) PVV  ,   PVV  ,  ν 1 ,ν 2   NT , AT   AT   AT   AT  AT   AT   AT  sin 2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C  AT sin2 sin2cos2 l  cos2cos2sin2 l   1 4 sin2 sin2sin2 l cosm  C,, l,m ,,, 7.2.28 where PVV  , is the joint probability of both events , and ν1 ,ν2 being true. (III) PVV  ,   PVV  ,  ν 1 ,ν 2   NT , AT   AT   AT   AT  AT   AT   AT  sin 2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C  Asin2 sin2cos2 l  cos2cos2sin2 l   1 4 sin2 sin2sin2 l cosm  C,, l,m ,, , 7.2.29 where PVV  , is the joint probability of both events , and ν1 ,ν2 being true. From Eq. (7.2.27), (7.2.28) and (7.2.29) we obtain PVV  PVV,  PVV  ,  PVV  ,  PVV  ,, 7.2.30 where PVV, is the unconditional probability of the event ,. 261 Remark 7.2.19. Note that PVH  ,  |V |sH |i|DC | 2 ,PHV  ,  |V |sH |i|DC | 2 , PHH  ,  |H |sH |i|DC | 2 , PVV  ,  |V |sV |i|DC | 2 , PVV  ,  |V |sV |i|DC | 2 7.2.31 From Eqs. (7.2.31) similarly as above we obtain (I) PVH  PVH,  PVH  ,  PVH  ,  PVH  ,, 7.2.32 where (i) PVH  PVV, is the unconditional probability of the event ,, (ii) PVH  , is the joint probability of both events , and ν1 ,ν2 being true, (iii) PVH  , is the joint probability of both events , and ν1 ,ν2 being true, (iv) PVH  , is the joint probability of both events , and ν1 ,ν2 being true, (II) PHV  PHV,  PHV  ,  PHV  ,  PHV  ,, 7.2.33 where (i) PHV  PHV, is the unconditional probability of the event ,, (ii) PHV  , is the joint probability of both events , and ν1 ,ν2 being true, (iii) PHV  , is the joint probability of both events , and ν1 ,ν2 being true, 262 (iv) PHV  , is the joint probability of both events , and ν1 ,ν2 being true, (III) PHH  PHH,  PHH  ,  PHH  ,  PHH  ,, 7.2.34 where (i) PHV  PHV, is the unconditional probability of the event ,, (ii) PHH  ,  is the joint probability of both events , and ν1 ,ν2 being true, (iii) PHH  ,  is the joint probability of both events , and ν1 ,ν2 being true, (iv) PHH  ,  is the joint probability of both events , and ν1 ,ν2 being true. From Eq. (7.2.14), (7.2.23) and (7.2.26) we obtain where AT  AT   AT   AT  is the total number of entangled pairs produced and NT tot,  NT ,  NT ,  NT ,  AT  sin2 sin2cos2 l  cos2cos2sin2 l  14 sin2 sin2sin2 l cosm  C1  AT sin2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C2  AT sin2   sin2cos2 l  cos2cos2sin2 l   1 4 sin2   sin2sin2 l cosm  C3  AT   AT   AT     sin2 sin2cos2 l  cos2cos2sin2 l  14 sin2 sin2sin2 l cosm  AT C2,, l,m ,,  AT C3,, l,m ,,   C  AT sin 2 sin2cos2 l  cos2cos2sin2 l  14 sin2 sin2sin2 l cosm  AT C2,, l,m ,,  AT C3,, l,m ,,   C1  C2  C3  AT sin 2 sin2cos2 l  cos2cos2sin2 l  14 sin2 sin2sin2 l cosm  C, 7.2.35 263 C  AT C2,, l,m ,,  AT C3,, l,m ,,  C1  C2  C3 . 7.2.36 Remark 7.2.20. To create the state |EPR  or something close to it, it is necessarily to adjust the parameters which determine the laser polarization. First one adjusts  l to equalize the coincidence counts N0, 0 and N90,90 . Next one set  l by rotating the quartz plate about a vertical axis to maximize N45,45 . When performing these optimizations, one typically collects a few hundred photons per point which requires an acquisition window of a few seconds. Remark 7.2.21. A rough idea of the purity of the entangled state can be found by measuring N0,0, N90,90,N45,45 and N0, 90 . Using the model of Eq. (7.2.19), one obtains C  N0,90, 7.2.37 A  N0, 0  N90, 90  2C, 7.2.38 tan2 l  N90, 90  C N0, 0  C , 7.2.39 cosm  1 sin2 l 4 N45, 45  C A  1 . 7.2.40 In a typical acquisition, after optimizing  l and  l we find, with T  10 seconds, N0,0  293 , N90,90  307 , N0,90  22 ,and N45,45  286 . These give C  22,A  556, l  46, and m  26 . More extensive data are shown in Fig. 7.2.5 along with a fit to Eq. (7.2.19). The best fit parameters, C  31,A  539, l  46 and m  26 are in good agreement with the rough estimates made with just four points. 264 Fig. 7.2.5a Experimental polarization correlations.   0 (open circles) and   45 (filled circles) Fig. 7.2.5b. Experimental polarization correlations.   90 (open circles) and   135 (filled circles) 265 Fig. 7.2.5a shows   0 (open circles) and   45 (filled circles). Fig.7.2.5b shows   90 (open circles) and   135 (filled circles). Error bars indicate plus/minus one standard deviation statistical uncertainty. Curves are a fit to Eq. (7.2.19). Remark 7.2.22. Remind that in his comment on Bohr's lecture, Einstein noted that quantum mechanics allows a measurement of one particle to influence the state of another. To illustrate this for polarizations, we consider again the state |EPR  of Eq. (7.2.3). If the signal photon is measured with a polarizer set to  , the result will be H or V , each occurring half the time. In the usual Copenhagen interpretation the state has collapsed, at the moment of measurement, from |EPR  to either |V s|V i or |H s|H i . But the mere choice of  does not determine the state of the idler photon; it is the (random) outcome of the measurement on the signal photon that decides whether the idler ends up as |V i or |H i . Despite the randomness, the choice of  clearly has an effect on the state of the idler photon: it gives it a definite polarization in the |V i, |H i basis, which it did not have before the measurement. Remark 7.2.23. After the signal photon is measured the idler is equally likely to be V or H . A measurement of its polarization, at any angle  , finds V with the probability PV  12 |Vb |V | 2  1 2 |Vb |H | 2   1 2 cos2b    sin2b    12 . 7.2.41 This gives no information about the choice of α. It is also the probability we would find if the signal photon had not been measured. 266 VII.2.2. EPR-B nonlocality is the physical nature of the violation of the Bell inequalities Remind that classical CHSH constrains the degree of polarization correlation under measurements at different polarizer angles. The proof involves two measures of correlation, see subsection VII.2.3. The first measure is E,  PVV,  PHH,  PVH,  PHV,. 7.2.42 This incorporates all possible measurement outcomes and varies from 1 when the polarizations always agree to 1 when they always disagree. Fig. 7.2.6. Polarizer angles for maximal SQM The second measure is S  Ea,b  Ea,b  Ea,b  Ea,b, 7.2.43 where a,a,b,b are four different polarizer angles. S does not have a clear physical meaning. Its importance comes from the fact that Clauser, Horne, Shimony and Holt proved 267 |SHVT|  2 7.2.44 (see subsection VII.2.3) for any classical Hidden Variable Theory (HVT) and arbitrary a,a,b,b . Remark 7.2.24. Quantum mechanics under (i) canonical Copenhagen interpretation, and (ii) canonical SRT (Special Relativity Theory) locality for certain settings, can violate this inequality. If we choose the polarizer angles, a  45, a  0, b  22.5 and b  22.5 , as shown in Fig. 7.2.6, then, using Eq. (7.2.15) SQM  2 2 . 7.2.45 This result is specific to the state |EPR  . Other states give lower S values. It is interesting to note that for these angles our simple HVT (see subsection VII.2.3) gives SHVT  2. 7.2.46 The CHSH (Clauser, Horne, Shimony and Holt) inequality shows that no theory which is both local and realistic (or `complete' in the EPR sense) will ever agree with quantum mechanics. Remark 7.2.25. Note that derivation of the CHSH inequality essentially depends on SR locality condition, see Remark 7.2.27. In subsection VII.2.4 we introduced generalized HVT (GHVT) based on EPR-B nonlocality condition. For such GHVT we derive the revised CHSH inequality |SGHVT|  4. 7.2.47 To find the probabilities P that make up E , we need four values of N, specificially 268 PVV,  NT tot,/NT tot,PVH,  NT tot,/NT tot, PHV,  NT tot,/NT tot,PHH,  NT tot,/NT tot, NT tot  NT tot,  NT tot,  NT tot,  NT tot,, 7.2.48 where N tot is the total number of pairs detected during interval of time 0,T and , are the polarizer settings   90,   90 . This requires counting coincidences for equal intervals with the polarizer set four different ways. The quantity E, requires four N measurements E,  NT tot,  NT tot,  NT tot,  NT tot, NT tot,  NT tot,  NT tot,  NT tot, 7.2.49 and S  Ea,b  Ea,b  Ea,b  Ea,b requires sixteen. Remark 7.2.26. By consideration above (see Eq. (7.2.35)) based on EPR-B nonlocality condition we find that the quantities: NT tot,,NT tot,,NT tot,,NT tot, have the representations NT tot,  NT ,  NT ,  NT ,, NT tot,  NT ,  NT ,  NT ,, NT tot,  NT ,  NT ,  NT ,, NT tot,  NT ,  NT ,  NT ,. 7.2.50 These representations have the rigorous physical meaning and they are essentially important for derivation of the revised CHSH inequality (7.2.47), see subsection VII.2.4. Remark 7.2.27. Remind that any classical Bell's type inequality was derived under the condition of SR locality and the canonical Copenhagen interpretation of QM. We consider now again the state |EPR  of Eq. (7.2.3). If the signal photon 1 is measured at instant t1 with a polarizer I set to  , the 269 result will be H or V , each occurring half the time. In the usual Copenhagen interpretation the state |EPR  has immediately collapsed, at the moment of measurement, from |EPR  to either |V s|V i or |H s|H i . We assume now for definiteness that the state |EPR  collapsed to |V s|V i and thus the state of the idler photon 2 is |V i. In agreement with a strong SR locality the idler photon 2 is not disturbed and its polarization can be measured with a polarizer II at instant t2 such that t2  t1 . Thus under the canonical physical interpretation of the Bell test experiment given in physical literature [8] it is not important which event was occurred: 1 t1 ,2 t2 , 1 t1 ,2 t2  or 1 t1 ,2 t2 , before the corresponding coincidence is revealed with detectors A and B. Remark 7.2.28. We emphasize that the violation of the classical CHSH inequality (7.2.44) confirms EPR-B nonlocality condition. VII.2.3. Canonical Local Realistic Hidden Variable Theory Einstein believed that a theory could be found to replace quantum mechanics, one which was complete and contained only local interactions. Here we describe such a theory, a local realistic hidden variable theory (HVT) [8]. In a such HVT, each photon has a polarization angle  , but this polarization does not behave like polarization in quantum mechanics. When a photon meets a polarizer set to an angle  , it will always register as V if  is closer to  than to   /2 , i.e., PV HVT,  1 |  | /4 1 |  | 3/4 0 otherwise. 7.2.51 In each pair, the signal and idler photon have the same polarization 270 s  i   . As successive pairs are produced  changes in an unpredictable manner that uniformly covers the whole range of possible polarizations. The quantity  is the hidden variable, a piece of information that is absent from quantum mechanics. HVTs do not have the spooky features of quantum mechanics. The theory is local: measurement outcomes are determined by features of objects present at the site of measurement. Any measurement on the signal (idler) photon is determined by s and  ( i and ). The theory is also realistic: All measurable quantities have definite values, independent of our knowledge of them. Furthermore, the theory specifies all of these values (for a given  ), so it is complete in Einstein's sense of the word. Finally, there is no requirement that  be random; it could be that  is changing in a deterministic way that remains to be discovered. Fig. 7.2.7. Predicted polarization correlations for a quantum mechanical entangled state (solid curve) and a hidden-variable theory (dashed line) To compare this theory to quantum mechanics, we need a prediction for the coincidence probability PVV HVT, . A coincidence occurs when  is in a range such that both  and  are close to  . The probability of this is 271 PVV HVT,  1  0  PV HVT,PV HVT,d  1 2  |  | . 7.2.52 This function and the corresponding quantum mechanical probability from Eq. (7.2.10) are plotted in Fig. 7.2.7. The predictions are fairly similar. Where they disagree quantum mechanics predicts stronger correlations (or stronger anti-correlations) than the HVT. Remind that for any canonical HVT, the distribution of the hidden variable  is described by a function  , where   0 and d  1. The assumptions of locality and realism are embodied in the following: It is assumed that for the signal photon the outcome of a measurement is determined completely by  and the measurement angle  . These outcomes are specified by the function A, , which can take on the values 1 for detection as V and 1 for detection as H . Similarly, a function B, describes the outcomes for the idler photon as 1 for V and 1 for H . A HVT would specify the functions ,A and B. The probability of a particular outcome, averaged over an ensemble of photon pairs, is given by an integral. In particular PVV,   1A,2 1B, 2 d,PVH,   1A,2 1B, 2 d, PHV,   1A,2 1B, 2 d,PHH,   1A,2 1B, 2 d. 7.3.53 Let E, be E,  PVV,  PHH,  PVH,  PHV,. 7.2.54 It is easy to see that E,, given in Eq. (7.2.54), is E,  A,B,d. 7.2.55 272 We define now the quantity s , which describes the polarization correlation in a single pair of particles: s,a,a,b,b  A,aB,b  A,aB,b  A,aB,b  A,aB,b  A,aB,b  B,b  A,aB,b  B,b, 7.2.56 where a,a,b,b are four angles. Note that s can only take on the values 2. The average of s over an ensemble of pairs is s,a,a,b,b   s,a,a,b,bd  Ea,b  Ea,b  Ea,b  Ea,b  Sa,a,b,b. 7.2.57 Because s can only take on the values 2, its average S must satisfy 2  S  2 , which is the Clauser, Horne, Shimony and Holt inequality |S|  2. 7.2.58 VII.2.4. Local Hidden Variable Theory revisited. Generalized Local Hidden Variable Theory Validity of CHSH-inequality for correlations taking into account EPR-B nonlocality For any GHVT, the distribution of the hidden variable  is described by a function , t1 , t2 , where , t1 , t2  0 7.2.59 and , t1 , t2d  1. 7.2.60 The assumptions of EPR-B nonlocality and realism are embodied in 273 the following: It is assumed that for the signal photon the outcome of a measurement is determined completely by  and the measurement angle  . These outcomes are specified by the function A,, t1 , which can take on the values 1 for detection as V and 1 for detection as H . Similarly, a function B,, t2 describes the outcomes for the idler photon as 1 for V and 1 for H . A GHVT would specify the functions ,A and B. The probability of a particular outcome, averaged over an ensemble of photon pairs, is given by an integral. In particular PVV, t1 ;, t2   1  A,, t1  A,, t2 2 1  B,, t1  B,, t2 2 , t1 , t2d, PVH, t1 ;, t2   1  A,, t1  A,, t2 2 1  B,, t1  B,, t2 2 , t1 , t2d, PHV, t1 ;, t2   1  A,, t1  A,, t2 2 1  B,, t1  B,, t2 2 , t1 , t2d, PHH, t1 ;, t2   1  A,, t1  A,, t2 2 1  B,, t1  B,, t2 2 , t1 , t2d. 7.2.61 Let E, t1 ;, t2 be E, t1 ;, t2  PVV, t1 ;, t2  PHH, t1 ;, t2  PVH, t1 ;, t2  PHV, t1 ;, t2. 7.2.62 It is easy to see that E, t1 ;, t2 , given in Eq. (7.2.62), is 274 E, t1 ;, t2  A,, t1  A,, t2B,, t1  B,, t2, t1 , t2d  A,, t1B,, t1, t1 , t2d  A,, t1B,, t2, t1 , t2d  A,, t2B,, t1, t1 , t2d  A,, t2B,, t2, t1 , t2d  A,, t1B,, t2, t1 , t2d  A,, t2B,, t1, t1 , t2d  A,, t1B,, t1, t1 , t2d  A,, t2B,, t2, t1 , t2d 7.2.63 We assume now for simplicity that: (i) |t1  t2 |    0, i.e. t1  t2   t1  t2   t1  t2  and (ii) A,, t1B,, t1, t1 , t2d  0,A,, t2B,, t2, t1 , t2d  0. 7.2.64 From Eq. (7.2.63) and Eq. (7.2.64) we obtain E, t1 ;, t2  A,, t1B,, t2, t1 , t2d  A,, t2B,, t1, t1 , t2d 7.2.65 We assume now for definiteness that: t1  t2 and rewrite Eq. (7.2.65) in the following form E, t1 ;, t2  E1, t1 ;, t2  E2, t1 ;, t2 E1, t1 ;, t2  A,, t1B,, t2, t1 , t2d, E2, t1 ;, t2  A,, t2B,, t1, t1 , t2d. 7.2.66 The second measure Sa,a,b,b; t1 , t2  now is 275 Sa,a,b,b; t1 , t2   S1a,a,b,b; t1 , t2   S2a,a,b,b; t1 , t2   Ea, t1 ;b, t2  Ea, t1 ;b, t2  Ea, t1 ;b, t2  Ea, t1 ;b, t2  E1, t1 ;, t2  E1a, t1 ;b, t2  E1a, t1 ;b, t2  E1a, t1 ;b, t2  E2a, t1 ;b, t2  E2a, t1 ;b, t2  E2a, t1 ;b, t2  E2a, t1 ;b, t2, 7.2.67 where S1a,a,b,b; t1 , t2   E1, t1 ;, t2  E1a, t1 ;b, t2  E1a, t1 ;b, t2  E1a, t1 ;b, t2, S2a,a,b,b; t1 , t2   E2a, t1 ;b, t2  E2a, t1 ;b, t2  E2a, t1 ;b, t2  E2a, t1 ;b, t2. 7.2.68 Note that S1a,a,b,b; t1 , t2    s1,a,a,b,b; t1 , t2 , t1 , t2d, S2a,a,b,b; t1 , t2    s2,a,a,b,b; t1 , t2 , t1 , t2d, 7.2.69 where s1,a,a,b,b; t1 , t2   A,a, t1B,b, t2  A,a, t1B,b, t2  A,a, t1B,b, t2  A,a, t1B,b, t2  A,a, t1B,b, t2  B,b , t2  A,a, t1B,b, t2  B,b, t2 7.2.70 and s2,a,a,b,b; t1 , t2   A,a, t2B,b, t1  A,a, t2B,b, t1  A,a, t2B,b, t1  A,a, t2B,b, t1  A,a, t2B,b, t1  B,b, t1  A,a, t2B,b, t1  B,b, t1. 7.2.71 Remark 7.2.29. Note that s1,a,a,b,b;t1 , t2  and s2,a,a,b,b;t1 , t2  can only take on the values 2 and therefore its averages 276 S1a,a,b,b; t1 , t2  and S2a,a,b,b; t1 , t2  must satisfy 2  S1  2,2  S2  2. 7.2.72 From Eq. (7.2.67) we obtain SGHVTa,a,b,b; t1 , t2   |S1a,a,b,b; t1 , t2   S2a,a,b,b; t1 , t2 |  |S1a,a,b,b; t1 , t2 |  |S2a,a,b,b; t1 , t2 | 7.2.73 Thus from Eq. (7.2.72) and (7.2.73) finally we obtain SGHVT  4. 7.2.74 VII.3. Bell inequalities revisited VII.3.1. Clauser-Horne-Shimony-Holt (CHSH) inequality revisited. Validity of revised CHSH inequality In a typical Bell experiment, two systems which may have previously interacted for instance they may have been produced by a common source are now spatially separated and are each measured by one of two distant observers, Alice and Bob (see Fig.7.2.1). Alice may choose one out of several possible measurements to perform on her system and we let x t1 denote her measurement choice at instant t1 . For instance, x t1 may refer to the position of a knob on her measurement apparatus at instant t1 . Similarly, we let yt2 denote Bob's measurement choice. Once the measurements are performed, they yield outcomes a t1 and b t2 on the two systems. Remark 7.3.1. The actual values assigned to the measurement choices x t1 ,yt2 and outcomes at1 ,bt2 are purely conventional; they are mere macroscopic labels distinguishing the different possibilities. 277 Remark 7.3.2. From one run of the experiment to the other, the outcomes a t1 and b t2 that are obtained may vary, even when the same choices of measurements x t1 and yt2 are made. Assumption 7.3.1. These outcomes a t1 and b t2 are thus in general governed by a Kolmogorovian probability distribution pa, t1 ;b, t2 |x t1yt2, which can of course depend on the particular experiment being performed. By repeating the experiment a sufficient number of times and collecting the observed data, one can get a fair estimate of such Kolmogorovian probabilities. Assumption 7.3.2. The assumption of locality implies that we should be able to identify a set of past factors, described by some variables λ, having a joint causal influence on both outcomes, and which fully account for the dependence between a t1 and bt2 . Once all such factors have been taken into account, the residual indeterminacies about the outcomes must now be decoupled, that is, the Kolmogorovian joint probabilities for a t2 and b t2 should factorize: pa, t1 ;b, t2 |xy,  pa, t1 |x,pb, t2 |y,. 7.3.1 Remark 7.3.3. This factorability condition simply expresses that we have found an explanation according to which the probability for a t1 only depends on the past variables and on the local measurement x t1 , but not on the distant measurement and outcome, and analogously for the probability to obtain b t2 . The variable will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because may involve physical quantities that are not fully controllable. The different values of across the runs should thus be characterized by a probability distribution qλ, t1 , t2. Combined with the above factorability condition, we can thus write λ λ λ λ 278 pa, t1 ;b, t2 |xy    dq, t1 , t2 pa, t1 |x,pb, t2 |y,, 7.3.2 where we also implicitly assumed that the measurements x t1 and yt2 can be freely chosen in a way that is independent of , i.e., that qλ, t1 , t2 |x t1 ,yt2  qλ, t1 , t2 . This decomposition now represents a precise condition for locality in the context of Bell experiments. Let us consider for simplicity an experiment where there are only two measurement choices per observer x t1 ,yt2  0,1 and where the possible outcomes take also two values labelled at1 ,bt2  1,1. Let at1x t1 bt2y t2 be the expectation value of the product at1bt2 for given measurement choices x t1 ,yt2 : at1x t1 bt2y t2  a,b abpab, t1 , t2 |x t1 yt2. 7.3.3 Assumption 7.3.3. We assume now that t1 , t2   2 ,    ,0    1, pab, t1 , t2 |x t1 yt2  pab, t1  t2 |x t1 yt2  |t1  t2 |    |t1  t2 |  0, q, t1 , t2   q, t1  t2   |t1  t2 |    |t1  t2 |  0 . 7.3.4 Thus at1x t1 bt2y t2  a,b abpab, t1  t2 |x t1yt2. 7.3.5 Remark 7.3.4. We denote at1x t1 bt2y t2  axby  7.3.6 iff |t1  t2 |  0. We denote λ 279 at1x t1 bt2y t2  axby  7.3.7 iff |t1  t2 |   and t1  t2 . We denote at1x t1 bt2y t2  axby  7.3.8 iff |t1  t2 |   and t1  t2 . We denote axby   axby   axby   axby . 7.3.9 Consider the following expression S  a0b0   a0b1   a1b0   a1b1 , 7.3.10 which is a function of the probabilities pab|xy. If these probabilities satisfy the locality decomposition (7.3.2) and Eq. (7.3.4), we necessarily have that S  S  S  S  a0b0   a0b1   a1b0   a1b1   6, 7.3.11 where S  a0b0   a0b1   a1b0   a1b1 , S  a0b0   a0b1   a1b0   a1b1 , S  a0b0   a0b1   a1b0   a1b1 . 7.3.12 To derive this inequality, we can use (7.3.2) and Eq. (7.3.4) in the definitions (7.3.6) - (7.3.8) of a0b0 , axby  and axby  which allows us to express these expectation values as averages: (i) 280 axby     dq, t1  t2 ax  by   , 7.3.13 where t1  t2  0, and where we denote at1x t1   ax  , bt2y t2   by   , (ii) axby     dq, t1  t2 ax  by   , 7.3.14 where t1  t2 and t1  t2  , and where we denote at1x t1   ax  , bt2y t2   by   , (iii) axby     dq, t1  t2 ax  by   , 7.3.15 where t1  t2 , t1  t2  , and where we denote at1x t1   ax  , bt2y t2   by   , of a product of corresponding local expectations: ax     a apa, t1 |x,, by     b bpb, t2 |y,, 7.3.16 and ax     a apa, t1 |x,, by     b bpb, t2 |y,, 7.3.17 and ax     a apa, t1 |x,, by     b bpb, t2 |y,, 7.3.18 281 taking values in 1,1. Inserting this expressions (7.3.16)-(7.3.18) in Eqs. (7.3.12), one obtains S    dq, 0S  , S    dq,S  , S    dq,S, 7.3.19 where S   a0  b0    a0  b1    a1  b0    a1  b1  , S   a0   b0    a0   b1    a1   b0    a1   b1   , S   a0  b0    a0  b1    a1  b0    a1  b1  , 7.3.20 Since a0 , b0   1,1, these last expressions are smaller than  Sλ  ,  S  and  S  correspondingly, where Sλ    Sλ   |b0    b1   |  |b0    b1   |, S    S   |b0    b1   |  |b0    b1   |, S    S   |b0    b1   |  |b0    b1   |. 7.3.21 Without loss of generality, we can assume that b0    b1    0, b0    b1    0, b0    b1    0, 7.3.22 which yields Sλ   2b0    2,S  2b0    2,S  2b0    2 7.3.23 and thus S  2,S  2,S  2. 7.3.24 From (7.3.24) we obtain 282 S  S  S  S  6. 7.3.25 The inequality (7.3.25) finalized the proof. VII.3.2. Clauser-Horne inequality revisited. Validity of revised Clauser-Horne inequality The inequality will be determined by counting during the large time t. There will be a total of N t events, with N1a, t counts in detector 1 by the time t when it is set to select a and N2b, t counts in detector 2 by the time t when it is set to select b . The number of coincidences of the two detectors with settings a and b respectively is N12a,b, t. The probabilities are p1a, t  N1a, t N t ,p2b, t  N2b, t N t ,p12a,b, t  N12a,b, t N t , p12a,b, t  N12a,b, t N t ,p12a,b, t  N12a,b, t N t ,p12a,b, t  N12a,b, t N t . 7.3.26 Remember that in the Bell formulation, the hidden variable determined absolutely the value of the polarization for a particular measurement. Then p1a, t1   dwp1a, t1 ,,p2b, t2   dwp2b, t2 ,, p12a,b, t1 , t2 ,  p1a, t1 ,p2b, t2 ,, p12a,b, t1 , t2   dwp1a, t1 ,p2b, t2 ,, 7.3.27 where t1 , t2  0, t. Remind that for any four real numbers x,x ,y,y  0,1 the inequality holds xy  xy  x y  x y  x   y. 7.3.28 283 We denote now x  p1a, t1 ,,y  p2b, t2 ,,x   p1a, t1 ,,y  p2b, t2 ,, 7.3.29 and substitute into inequality (7.3.28), we get p1a, t1 ,p2b, t2 ,  p1a, t1 ,p2b, t2 ,  p1a, t1 ,p2b, t2 ,  p1a, t1 ,p2b, t2 ,  p1a, t1 ,  p2b, t2 ,. 7.3.30 Next, multiply by w and integrate over all  we get p12a,b, t1 , t2  p12a,b, t1 , t2  p12a,b, t1 , t2  p12a,b, t1 , t2   p1a, t1  p2b, t2. 7.3.31 Remark 7.3.5. We assume now that t1 , t2   2 ,    ,0    1, p12a,b, t1 , t2  p12a,b, t1  t2  |t1  t2 |    |t1  t2 |  0, p12a,b, t1 , t2  p12a,b, t1  t2  |t1  t2 |    |t1  t2 |  0, p12a,b, t1 , t2  p12a,b, t1  t2  |t1  t2 |    |t1  t2 |  0, p12a,b, t1 , t2  p12a,b, t1  t2  |t1  t2 |    |t1  t2 |  0. . 7.3.32 Where t1 , t2  0, t. Remark 7.3.6. We denote p12a,b, t1  t2  p12  a,b 7.3.33 iff t1  t2  0. We denote p12a,b, t1  t2  p12  a,b 7.3.34 iff t1  t2   and t1  t2 . We denote 284 p12a,b, t1  t2  p12  a,b 7.3.35 iff t1  t2   and t1  t2 . Remark 7.3.7. We denote p12a,b, t1  t2  p12  a,b 7.3.36 iff t1  t2  0. We denote p12a,b, t1  t2  p12  a,b 7.3.37 iff t1  t2   and t1  t2 . We denote p12a,b, t  t2  p12  a,b 7.3.38 iff t1  t2   and t1  t2 . Remark 7.3.8. We denote p12a,b, t1  t2  p12  a,b 7.3.39 iff t1  t2  0. We denote p12a,b, t1  t2  p12  a,b 7.3.40 iff t1  t2   and t1  t2 . We denote p12a,b, t1  t2  p12  a,b 7.3.41 iff t1  t2   and t1  t2 . Remark 7.3.9. We denote p12a,b, t1  t2  p12  a,b 7.3.42 iff t1  t2  0. We denote p12a,b, t1 , t2  p12  a,b 7.3.43 iff t1  t2   and t1  t2 . We denote p12a,b, t1  t2  p12  a,b 7.3.44 iff t1  t2   and t1  t2 . Remark 7.3.10. We denote 285 p12a,b  p12  a,b  p12  a,b  p12  a,b, p12a,b  p12  a,b  p12  a,b  p12  a,b, p12a,b  p12  a,b  p12  a,b  p12  a,b, p12a,b  p12  a,b  p12  a,b  p12  a,b. 7.3.45 From (7.3.31) - (7.3.32) we obtain p12a,b, t1  t2  p12a,b, t1  t2  p12a,b, t1  t2  p12a,b, t1  t2   p1a, t1  p2b, t2, 7.3.46 where |t1  t2 |    |t1  t2 |  0. From (7.3.46) we obtain: (i) p12  a,b  p12  a,b  p12  a,b  p12  a,b   p1a, t1  p2b, t2, 7.3.47 where t1  t2  0; (ii) p12  a,b  p12  a,b  p12  a,b  p12  a,b   p1a, t1  p2b, t2, 7.3.48 where t1  t2   and t1  t2 ; (iii) p12  a,b  p12  a,b  p12  a,b  p12  a,b   p1a, t1  p2b, t2, 7.3.49 where t1  t2   and t1 < t2. From (7.3.47) we obtain (i) 286 p12  a,b  p12  a,b  p12  a,b  p12  a,b   t1 lim p1a, t1  t2 lim p2b, t2  p1a  p2b, 7.3.50 where we denote p1a  t1 lim p1a, t1,p2b  t2 lim p2b, t2; (ii) p12  a,b  p12  a,b  p12  a,b  p12  a,b   t1 lim p1a, t1  t2 lim p2b, t2  p1a  p2b, 7.3.51 where we denote p1a  t1 lim p1a, t1,p2b  t2 lim p2b, t2; (iii) p12  a,b  p12  a,b  p12  a,b  p12  a,b   t1 lim p1a, t1  t2 lim p2b, t2  p1a  p2b, 7.3.52 where we denote p1a  t1 lim p1a, t1,p2b  t2 lim p2b, t2; From (7.3.50) - (7.3.52) we obtain (i) p12  a,b  p12  a,b  p12  a,b  p12  a,b  p1a  p2b, 7.3.53 (ii) p12  a,b  p12  a,b  p12  a,b  p12  a,b  p1a  p2b, 7.3.54 (iii) p12  a,b  p12  a,b  p12  a,b  p12  a,b  p1a  p2b. 7.3.55 From (7.3.53) - (7.3.55) we obtain 287 p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b  p12  a,b   3p1a  3p2b, 7.3.56 From (7.3.53) by Eqs. (7.3.45) we obtain p12a,b  p12a,b  p12a,b  p12a,b  3p1a  3p2b. 7.3.57 The revised Clauser Horne inequality is N12a,b  N12b,a  N12a,b  N12a,b N1a  N2b  3. 7.3.58 Assume now that a,b,a,b are separated by the angle  then N12  N12  N12  N123 N1a  N2b  3N12  N123 N1a  N2b  3. 7.3.59 Next relate coincidences to expectation values. Note that N12  N||. As regards the singles counts N1a and N2b , we know that the number of counts must be independent of the direction of a  or b and that for any direction N1  1 2 N , since half the photons will be polarized along the direction. Therefore, the inequality (7.3.59) becomes   3 4 1  cos2  1 4 1  cos6 1 2  1 2  1 2  3 4 cos2  1 4 cos6  3. 7.3.60 The inequality (7.3.60) is not violated even if    8 :  max     8  1 2  3 2 4  1 2 2  1.2, 7.3.61 which of course is less than 3. 288 VII.3.3. Revised CHSH inequality without the hypothesis of locality Let At,At ,Bt,Bt , t      ,0    1, be stochastic processes defined on a common probability space ,,P with values in the set 1,1, i.e., At  1,At   1,Bt  1,Bt   1, t  0,T. 7.3.62 Assume that there exist joint probability distribution functions WAt1 ,At2  ,Bt3 ,Bt4  , of At1 ,At2  ,Bt3 ,Bt4  (where |t i  t j |  0  |t i  t j |  , i, j  1,2,3,4 ) defining probabilities for each possible set of outcomes such that: (i) PAt1 ,At2  ,Bt3 ,Bt4    0,PAt1 ,Bt3 ,Bt4    0,PAt2  ,Bt3 ,Bt4    0,etc., 7.3.63 (ii)  A t1 ,A t2  ,B t3 ,B t4  PA t1 ,A t2  ,Bt3 ,Bt4    1,  A t1 ,B t3 ,B t4  PA t1 ,Bt3 ,Bt4    1,  A t2  ,B t3 ,B t4  PA t2  ,Bt3 ,Bt4    1, etc., 7.3.64 (iii) PA t1 ,A t2  ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4    PA t2  ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4  , PA t1 ,A t2  ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4    PA t2  ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4  , etc. 7.3.65 From (7.3.65) one obtains 289 0  PA t1 ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4     PA t2  ,Bt4    PA t2  ,Bt3   PA t2  ,Bt4    PBt3   PA t2  ,Bt3 . 7.3.66 Similarly one obtains 0  PA t1 ,Bt3 ,Bt4    PA t1 ,Bt3   PA t1 ,Bt3 ,Bt4    PA t1   PA t1 ,Bt3   PA t1 ,Bt4    PA t1 ,Bt3 ,Bt4   7.3.67 and therefore PAt1 ,Bt3 ,Bt4    PAt1   PAt1 ,Bt3   PAt1 ,Bt4  . 7.3.68 From (7.3.66) and (7.3.68) we obtain 0  PA t1 ,Bt3 ,Bt4    PA t1 ,Bt3 ,Bt4    PA t2  ,Bt4    PBt3   PA t2  ,Bt3   PA t1   PA t1 ,Bt3   PA t1 ,Bt4   7.3.69 and therefore 0  PA t2  ,Bt4    PBt3   PA t2  ,Bt3   PA t1   PA t1 ,Bt3   PA t1 ,Bt4     PA t1   PBt3   PA t1 ,Bt3   PA t2  ,Bt3   PA t1 ,Bt4    PA t2  ,Bt4  . 7.3.70 From (7.3.70) one obtains A t1 ,A t2  ,Bt3 ,Bt4    PA t1 ,Bt3   PA t2  ,Bt3   PA t1 ,Bt4    PA t2  ,Bt4    PA t1   PBt3   0. 7.3.71 Note that PBt3 ,Bt4    PBt3   PBt3 ,Bt4   7.3.72 290 and PBt3 ,Bt4    PBt4    PBt3 ,Bt4    1  PBt4    PBt3 ,Bt4  . 7.3.73 From (7.3.73) and (7.3.72) we obtain PBt3 ,Bt4    1  PBt3   PBt4    PBt3 ,Bt4  . 7.3.73a Note that 0  PAt1 ,Bt3 ,Bt4    PBt3 ,Bt4    PAt1 ,Bt3 ,Bt4  . 7.3.74 Inserting (7.3.67) and (7.3.73a) into (7.3.74) we obtain 0  1  PA t1   PBt3   PBt4    PA t1 ,Bt3   PA t2  ,Bt3   PA t1 ,Bt4    PBt3 ,Bt4    PA t1 ,Bt3 ,Bt4    1  PA t1   PBt3   PBt4    PA t1 ,Bt3   PA t2  ,Bt3   PA t1 ,Bt3 ,Bt4  . 7.3.75 Note that PA t1 ,Bt3 ,Bt4    PA t1 ,A t2  ,Bt3 ,Bt4    PA,A ,B,B    PA t2  ,Bt3   PA t2  ,Bt4    PA t2  ,Bt3   PBt4    PA t2  ,Bt4  . 7.3.76 From (7.3.76) we obtain 0  1  PA t1   PBt3   PBt4    PA t1 ,Bt3   PA t2  ,Bt3   PA t1 ,Bt4    PA t2  ,Bt4  . 7.3.77 From (7.3.77) and (7.3.71) we obtain 1  At1 ,At2  ,Bt3 ,Bt4    0. 7.3.78 291 (I)Let us define the following quantities: AB , AB, AB , AB  AB  PAB     PAB     PAB     PAB   , A B  P A B     P A B     P A B     P A B   , AB   P AB      P AB      P AB      P AB    , A B   P A B      P A B      P A B      P A B    , 7.3.79 where PAB     PA t1  1,Bt3  1  t1  t3  0, PAB     PA t1  1,Bt3  1  t1  t3  0, PAB     PA t1  1,Bt3  1  t1  t3  0, PAB     PA t1  1,Bt3  1  t1  t3  0, P A B     PA t2   1,Bt3  1  t2  t3  0, etc. 7.3.80 From Eqs. (7.3.71), (7.3.79) and (7.3.80) we obtain AB  A B  AB   A B                     . 7.3.81 From (7.3.78) we obtain 2            0 7.3.82 and 0            2 7.3.83 From (7.3.82) and (7.3.83) we obtain 2                      2. 7.3.84 292 (II)Let us define the following quantities: AB , AB, AB , AB  AB  PAB     PAB     PAB     PAB   , A B  P A B     P A B     P A B     P A B   , AB   P AB      P AB      P AB      P AB    , A B   P A B      P A B      P A B      P A B    , 7.3.85 where PAB     PA t1  1,Bt3  1  t1  t3  , PAB     PA t1  1,Bt3  1  t1  t3  , etc. 7.3.86 From Eqs. (7.3.71), (7.3.85) and (7.3.86) we obtain AB  A B  AB   A B                     . 7.3.87 From (7.3.78) we obtain 2            0 7.3.88 and 0            2 7.3.89 From (7.3.88) and (7.3.89) we obtain 2                      2. 7.3.90 (III) Let us define the following quantities: AB , AB, AB , AB  : 293 AB  PAB     PAB     PAB     PAB   , A B  P A B     P A B     P A B     P A B   , AB   P AB      P AB      P AB      P AB    , A B   P A B      P A B      P A B      P A B    , 7.3.91 where PAB     PA t1  1,Bt3  1  t1  t3  , PAB     PA t1  1,Bt3  1  t1  t3  , etc. 7.3.92 From Eqs. (7.3.71), (7.3.91) and (7.3.92) we obtain AB  A B  AB   A B                     . 7.3.93 From (7.3.78) we obtain 2            0 7.3.94 and 0            2. 7.3.95 From (7.3.94) and (7.3.95) we obtain 2                      2. 7.3.96 (IV) Let us define the following quantities:    ,   ,    and     294                   ,                   ,                   ,                 . 7.3.97 From Eq. (7.3.97) and the inequalities (7.3.96), (7.3.90) and (7.3.84) we obtain 6                      6. 7.3.98 (V) Let us define the following quantities: AB, AB AB  AB  AB  AB, A B  A B  A B  A B, AB   AB   AB   AB , A B   A B   A B   A B . 7.3.99 From Eqs. (7.3.93), (7.3.87), (7.3.84), (7.3.97) and (7.3.99) we obtain AB  A B  AB   A B                     . 7.3.100 From inequality (7.3.98) and Eq. (7.3.100) finally we obtain |AB  AB  AB   AB |  6. 7.3.101 VII.3.4. Revised CHSH inequality without the introduction of hidden variables Let At,At ,Bt,Bt , t      ,0    1, be stochastic processes defined on a common probability space ,,P with values in the 295 set 1,1, i.e., At  1,At   1,Bt  1,Bt   1, t  0,T. 7.3.102 Assume that there exist joint probability distribution functions PAt1 ,At2  ,Bt3 ,Bt4  , of At1 ,At2  ,Bt3 ,Bt4  (where |t i  t j |  0  |t i  t j |    0, i, j  1,2,3,4 ) defining probabilities for each possible set of outcomes such that: (i) PAt1 ,At2  ,Bt3 ,Bt4    0,PAt1 ,Bt3 ,Bt4    0,PAt2  ,Bt3 ,Bt4    0,etc., 7.3.103 and  A t1 ,A t2  ,B t3 ,B t4  PA t1 ,A t2  ,Bt3 ,Bt4    1, 7.3.104 and PAt1 ,At2  ,Bt3 ,Bt4    PAt1 ,At2  ,Bt3 ,Bt4    PAt2  ,Bt3 ,Bt4  . 7.3.105 We abbreviate now for short: 296 and PA t1B t3     PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2 ,B t3 ,B t4      t1  t3 , etc., 7.3.107 PA t1B t3     PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2 ,B t3 ,B t4      t1  t3  , etc., 7.3.108 P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3 , P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3 ; P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  , P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  ; P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  , P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P A,A ,B,B        P A t1 ,A t2  ,B t3 ,B t4        PA t1  1,A t2   1,Bt3  1,Bt4   1  t1  t3  ; 7.3.106 297 PA t1B t3     PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2  ,B t3 ,B t4       PA t1 ,A t2 ,B t3 ,B t4      t1  t3  , etc. 7.3.109 For the quantities (i) AB , AB, AB , AB ; (ii) AB , AB, AB , AB ; and (iii) AB , AB, AB , AB  , using Eqs. (7.3.107) - (7.3.109) one obtains the representatives AB  PAB     PAB     PAB     PAB   , etc., AB  PAB     PAB     PAB     PAB   , etc., AB  PAB     PAB     PAB     PAB   . 7.3.110 Substituting Eqs. (7.3.110) into expressions   1 2 AB  A B  AB   A B  ,   1 2 AB  A B  AB   A B  ,   1 2 AB  A B  AB   A B  , 7.3.111 one obtains   1 2 AB  A B  AB   A B    P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B      ; 7.3.112 298   1 2 AB  A B  AB   A B    P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B      ; 7.3.113   1 2 AB  A B  AB   A B    P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B        P A,A ,B,B      . 7.3.114 From (7.3.103)-(7.3.105) it obviously follows that  2    2,2    2,2    2. 7.3.115 Thus we obtain   2, | |  2, | |  2. 7.3.116 We define now the quantityes        7.3.117 and 299 AB  AB  AB  AB, A B  A B  A B  A B, AB   AB   AB   AB , A B   A B   A B   A B . 7.3.118 From (7.3.116) and (7.3.117) we obtain ||  |     |  | |  | |  | |  6. 7.3.119 From (7.3.117) and (7.3.118) we obtain         AB  AB  AB   AB . 7.3.120 From (7.3.119) and (7.3.120) finally we obtain |AB  AB  AB   AB |  6. 7.3.121 VII.4. Leggett inequality revisited VII.4.1. Classical Leggett inequality Leggett have introduced the class of non-local models and formulated an incompatibility theorem [8]. Such models were extended so as to make them applicable to real experimental situations and also to allow simultaneous tests of all local hiddenvariable models. Finally, an experiment was performed that violates the new inequality and hence excludes a broad class of non-local hidden-variable theories [9]. These theories are based on the following assumptions: (1) all measurement outcomes are determined by pre-existing properties of particles independent of the measurement (realism); (2) physical 300 states are statistical mixtures of subensembles with definite polarization, where (3) polarization is defined such that expectation values taken for each subensemble obey Malus' law (that is, the well-known cosine dependence of the intensity of a polarized beam after an ideal polarizer). These assumptions are in a way appealing, because they provide a natural explanation of quantum mechanically separable states (polarization states indeed obey Malus' law). In addition, they do not explicitly demand locality; that is, measurement outcomes may very well depend on parameters in space-like separated regions. As a consequence, such theories can explain important features of quantum mechanically entangled (non-separable) states of two particles: first, they do not allow information to be transmitted faster than the speed of light; second, they reproduce perfect correlations for all measurements in the same bases, which is a fundamental feature of the Bell singlet state; and third, they provide a model for all thus far performed experiments in which the Clauser, Horne, Shimony and Holt (CHSH) inequality was violated. A general framework of such models is the following: assumption (1) requires that an individual binary measurement outcome A for a polarization measurement along direction a (that is, whether a single photon is transmitted or absorbed by a polarizer set at a specific angle) is predetermined by some set of hidden-variables  , and a three-dimensional vector u , as well as by some set of other possibly non-local parameters  (for example, measurement settings in space-like separated regions) that is, A  A,u,a, . According to assumption (3), particles with the same u but with different  build up subensembles of 'definite polarization' described by a probability distribution u . The expectation value Au , obtained by averaging over  , fulfills Malus' law, that is, 301 Au   duA,u,a,  u  a. Finally, with assumption (2), the measured expectation value for a general physical state is given by averaging over the distribution Fu of subensembles, that is, A   duFuAu . Let us consider a specific source, which emits pairs of photons with well-defined polarizations u and v to laboratories of Alice and Bob, respectively. The local polarization measurement outcomes A and B are fully determined by the polarization vector, by an additional set of hidden variables  specific to the source and by any set of parameters  outside the source. For reasons of clarity, we choose an explicit non-local dependence of the outcomes on the settings a and b of the measurement devices. Note, however, that this is just an example of a possible non-local dependence, and that one can choose any other set out of  . Each emitted pair is fully defined by the subensemble distribution u,v . In agreement with assumption (3) we impose the following conditions on the predictions for local averages of such measurements (all polarizations and measurement directions are represented as vectors on the Poincaré sphere [10]): Au   du,vAa,b,  u  a, 7.4.1 Bv   du,vBb,a,  v  b. 7.4.2 It is important to note that the validity of Malus' law imposes the nonsignalling condition on the investigated non-local models, as the local expectation values do only depend on local parameters. The correlation function of measurement results for a source emitting well-polarized photons is defined as the average of the products of the individual measurement outcomes: ABu,v   du,vAa,b,Bb,a, 7.4.3 302 For a general source producing mixtures of polarized photons the observable correlations are averaged over a distribution of the polarizations Fu,v , and the general correlation function E is given by: E  AB   dudvFu,vABu,v 7.4.4 It is a very important trait of this model that there exist subensembles of definite polarizations (independent of measurements) and that the predictions for the subensembles agree with Malus' law. It is clear that other classes of non-local theories, possibly even fully compliant with all quantum mechanical predictions, might exist that do not have this property when reproducing entangled states. There the non-local correlations are a consequence of the non-local quantum potential, which exerts suitable torque on the particles leading to experimental results compliant with quantum mechanics. In that theory, neither of the two particles in a maximally entangled state carries any angular momentum at all when emerging from the source [11]. In contrast, in the Leggett model, it is the total ensemble emitted by the source that carries no angular momentum, which is a consequence of averaging over the individual particles' well defined angular momenta (polarization). The theories described here are incompatible with quantum theory. Remind the basic idea of the incompatibility theorem [8] uses the following identity, which holds for any numbers A  1 and B  1 : 1  |A  B| AB  1  |A  B|. 7.4.5 One can apply this identity to the dichotomic measurement results A  Aa,b,  1 and B  Bb,a,  1 . The identity holds even if the values of A and B mutually depend on each other. For example, the 303 value of a specific outcome A can depend on the value of an actually obtained result B. In contrast, in the derivation of the CHSH inequality it is necessary to assume that A and B do not depend on each other. Therefore, any kind of non-local dependencies used in the present class of theories are allowed. Taking the average over the subensembles with definite polarizations we obtain: 1   du,v|A  B|  du,vAB  1   du,v|A  B| 7.4.6 Denoting these averages by bars, one arrives at the shorter expression: 1  |A  B|  AB  1  |A  B|. 7.4.7 As the average of the modulus is greater than or equal to the modulus of the averages, one gets the set of inequalities: 1  |A  B| AB  1  |A  B|. 7.4.8 By inserting Malus' law, equations (7.4.1) and (7.4.2), in equation (7.4.8), and by using expression (7.4.4), one arrives at a set of inequalities for experimentally accessible correlation functions (for a detailed derivation see Appendix B). In particular, if we let Alice choose her observable from the set of two settings a1 and a2 , and Bob from the set of three settings b  1 , b  2 and b  3  a2 , the following generalized Leggett-type inequality is obtained: SNLHV  |E11  E230||E22  E230| 4  4 |sin  2 |, 7.4.9 where Ekl is a uniform average of all correlation functions, defined in the plane of ak and b  l , with the same relative angle  ; the 304 subscript NLHV stands for 'non-local hiddenvariables'. For the inequality to be applied, vectors a1 and b  1 necessarily have to lie in a plane orthogonal to the one defined by a2 and b  2 . This contrasts with the standard experimental configuration used to test the CHSH inequality, which is maximally violated for settings in one plane. Fig. 7.4.1. Testing non-local hidden-variable theories [9] (a) Diagram of a standard two-photon experiment to test for hidden variable theories. When pumping a nonlinear crystal (NL) with a strong pump field, photon pairs are created via spontaneous parametric down-conversion (SPDC) and their polarization is detected with single-photon counters (PC). Local measurements at A and B are performed along directions a and b on the Poincaré sphere, respectively. Depending on the measurement directions, the obtained correlations can be used to test Bell inequalities (b) or Leggett-type inequalities (c). (b) Correlations in one plane. Shown are measurements along directions in the linear plane of the Poincaré sphere (H (V) denotes horizontal (vertical) polarization). The original experiments by Wu and Shaknov [12] and Kocher and Commins [13], designed to test quantum predictions for correlated photon pairs, measured perfect correlations (solid lines). Measurements along the dashed line allow a Bell test, as was first performed by Freedman and Clauser [14]. (c) Correlations in orthogonal planes. All current experimental 305 tests to violate Bell's inequality (CHSH) are performed within the shaded plane. Out-of-plane measurements are required for a direct test of the class of non-local hidden-variable theories, as was first suggested by Leggett [15]. The situation resembles in a way the status of the Einstein, Podolsky and Rosen (EPR) paradox before the advent of Bell's theorem and its first experimental tests. The experiments of Wu and Shaknov [12] and of Kocher and Commins [13] were designed to demonstrate the validity of a quantum description of photon-pair correlations. As this task only required the testing of correlations along the same polarization direction, their results could not provide experimental data for the newly derived Bell inequalities (Fig. 7.4.1a, b). Curiously, as was shown by Clauser, Horne, Shimony and Holt, only a small modification of the measurement directions, such that nonperfect correlations of an entangled state are probed, was sufficient to test Bell's inequalities. The seminal experiment by Freedman and Clauser [14] was the first direct and successful test. Today, all Bell tests that is, tests of local realism are performed by testing correlations of measurements along directions that lie in the same plane of the Poincaré sphere. Similar to the previous case, violation of the Leggett-type inequality requires only small modifications to that arrangement: To test the inequality, correlations of measurements along two orthogonal planes have to be probed (Fig. 7.4.1c). Therefore, the existing data of all Bell tests cannot be used to test the class of nonlocal theories considered in [14]. Quantum theory violates inequality (7.4.9). Consider the quantum predictions for the polarization singlet state of two photons, |AB  1 2 |HA |VB  |VA |HB  , where, for example, |HA denotes a horizontally polarized photon propagating to Alice. The quantum correlation function for the measurements ak and b  l performed on 306 photons depends only on the relative angles between these vectors, and therefore Ekl  ak  bl  cos . Thus the left hand side of inequality (7.4.9), for quantum predictions, reads |2cos  1| . The maximal violation of inequality (7.4.9) is for max  18.8 . For this difference angle, the bound given by inequality (7.4.9) equals 3.792 and the quantum value is 3.893. Although this excludes the nonlocal models, it might still be possible that the obtained correlations could be explained by a local realistic model. In order to avoid that, we have to exclude both local realistic and non-local realistic hiddenvariable theories. Note however that such local realistic theories need not be constrained by assumptions (1) - (3). The violation of the CHSH inequality invalidates all local realistic models. If one takes SCHSH  |E11  E12  E21  E22 |  2 7.4.10 the quantum value of the left hand side for the settings used to maximally violate inequality (7.4.9) is 2.2156. The correlation function determined in an actual experiment is typically reduced by a visibility factor V to Eexp  Vcos owing to noise and imperfections. Thus to observe violations of inequality (7.4.9) (and inequality (7.4.10)) in the experiment, one must have a sufficiently high experimental visibility of the observed interference. For the optimal difference angle max  18.8 , the minimum required visibility is given by the ratio of the bound (3.792) and the quantum value (3.893) of inequality (7.4.9), or  97.4%. We note that in standard Bell-type experiments, a minimum visibility of only  71% is sufficient to violate the CHSH inequality, inequality (7.4.10), at the optimal settings. For the settings used here, the critical visibility reads 2/2.2156  90.3%, which is much lower than 97.4%. 307 Fig. 7.4.2. Experimental set-up [9] A 2-mm-thick type-II barium-borate (BBO) crystal is pumped with a pulsed frequencydoubled Ti: sapphire laser (180 fs) at  = 395 nm wavelength and ~150 mW optical c.w. power. The crystal is aligned to produce the polarization-entangled singlet state |AB  1 2 |HA |VB  |VA |HB  . Spatial and temporal distinguishability of the produced photons (induced by birefringence in the BBO) are compensated by a combination of half-wave plates /2 and additional BBO crystals (BBO/2), while spectral distinguishability (due to the broad spectrum of the pulsed pump) is eliminated by narrow spectral filtering of 1 nm bandwidth in front of each detector. In addition, the reduced pump power diminishes higher-order SPDC emissions of multiple photon pairs. This allows us to achieve a twophoton visibility of about 99%, which is well beyond the required threshold of 97:4%. The arrows in the Poincaré spheres indicate the measurement settings of Alice's and Bob's polarizers for the maximal violation of inequality (7.4.9). Note that setting b 2 lies in the y  z plane and therefore a quarter-wave plate has to be 308 introduced on Bob's side. The coloured planes indicate the measurement directions for various difference angles  for both inequalities. In the experiment [9] (see Fig. 7.4.2), a pairs of polarization entangled photons was generated via SPDC. The photon source is aligned to produce pairs in the polarization singlet state. We observed maximal coincidence count rates (per 10 s), in the H/V basis, of around 3,500 with single count rates of 95,000 (Alice) and 105,000 (Bob), 3,300 coincidences in the 45 basis (75,000 singles at Alice and 90,000 at Bob), and 2,400 coincidences in the R/L basis (70,000 singles at Alice and 70,000 at Bob). The reduced count rates in the R/L basis are due to additional retarding elements in the beam path. The two-photon visibilities are approximately 99.0  1.2% in the H/V basis, 99.2  1.6% in the 45 basis and 98.9  1.7% in the R/L basis, which to our knowledge is the highest reported visibility for a pulsed SPDC scheme. So far, no experimental evidence against the rotational invariance of the singlet state exists. We therefore replace the rotation averaged correlation functions in inequality (7.4.9) with their values measured for one pair of settings (in the given plane). In terms of experimental count rates, the correlation function Ea,b for a given pair of general measurement settings is defined by Ea,b  N  N  N  N N  N  N  N 7.4.11 where NAB denotes the number of coincident detection events between Alice's and Bob's measurements within the integration time. We ascribe the number 1, if Alice (Bob) detects a photon polarized along a b  , and 1 for the orthogonal direction a b  . For example, N denotes the number of coincidences in which Alice obtains a and Bob b . Note that Eak,bl  Ekl , where  is the 309 difference angle between the vectors a and b on the Poincaré sphere. To test inequality (7.4.9), three correlation functions (Ea1 ,b1 , , Ea2 ,b2 , Ea2 ,b3) have to be extracted from the measured data. We choose observables a1 and b  1 as linear polarization measurements (in the x z plane on the Poincaré sphere; see Fig. 7.4.2) and a2 and b2 as elliptical polarization measurements in the y z plane. Two further correlation functions (Ea2 ,b1 and Ea1 ,b2) are extracted to test the CHSH inequality (7.4.10). The first set of correlations, in the x z plane, is obtained by using linear polarizers set to 1 and 1 (relative to the z axis) at Alice's and Bob's location, respectively. In particular, 1  45 , while 1 is chosen to lie between 45 and 160 (green arrows in Fig. 7.4.2). The second set of correlations (necessary for CHSH) is obtained in the same plane for 2  0/90 and 1 between 45 and 160 . The set of correlations for measurements in the y z plane is obtained by introducing a quarter-wave plate with the fast axis aligned along the (horizontal) 0 direction at Bob's site, which effectively rotates the polarization state by 90 around the z-axis on the Poincaré sphere (red arrows in Fig. 7.4.2). The polarizer angles are then set to 2  0/90 and 2 is scanned between 0 and 115 . With the same 2 and 1  45 , the expectation values specific only for the CHSH case are measured. The remaining measurement for inequality (7.4.9) is the check of perfect correlations, for which we choose 2  3  0 , that is, the intersection of the two orthogonal planes. Fig. 7.4.3 shows the experimental violation of inequalities (7.4.9) and (7.4.10) for various difference angles. Maximum violation of inequality (7.4.9) is achieved, for example, for the settings 1 ,2 ,1 ,2 ,3  45,0,55,10,0 . 310 Fig. 7.4.3. Experimental violation of the inequalities for non-local hidden-variable theories (NLHV) and for local realistic theories (CHSH) (a) Dashed line indicates the bound of inequality (7.4.9) for the investigated class of nonlocal hidden variable theories. The solid line is the quantum theoretical prediction reduced by the experimental visibility. The shown experimental data were taken for various difference angles  (on the Poincaré sphere) of local measurement settings. The bound is clearly violated for 4    36. Maximum violation is observed for max  20. (b) At the same time, no local realistic theory can model the correlations for the investigated settings as the same set of data also violates the CHSH inequality (7.4.10). The bound (dashed line) is overcome for all values  around max , and hence excludes any local realistic explanation of the observed correlations in a. Again, the solid line is the quantum prediction for the observed experimental visibility. Error bars indicate s.d. The following expectation values for a difference angle   20 (the errors are calculated assuming that the counts follow a poissonian distribution) were obtained [9]: Ea1 ,b1  0.9298  0.0105 , Ea2 ,b2  0.942  0.0112 , Ea2 ,b3  0.9902  0.0118 . This results in 311 SNLHV  3.8521  0.0227 , which violates inequality (7.4.9) by 3.2 standard deviations (see Fig. 7.4.3). At the same time, we can extract the additional correlation functions Ea2 ,b1  0.3436  0.0088 , Ea1 ,b2  0.0374  0.0091 required for the CHSH inequality. We obtain SCHSH  2.178  0.0199 , which is a violation by  9 standard deviations. The stronger violation of inequality (7.4.10) is due to the relaxed visibility requirements on the probed entangled state. VII.4.2. Derivation of the Canonical Leggett inequality With the assumption that photons with well defined polarization obey Malus' law: A  u  a, B  v  b, 7.4.12 the upper bound of Eq. (7.4.8) becomes: AB  1  |u  ak  v  bl|, 7.4.13 where ak and bl are unit vectors associated with the k -th measurement setting of Alice and the l -th of Bob, respectively. Taking the average over arbitrary polarizations we obtain Ekl HVT  1   dudvFu,v|u  ak  v  bl|, 7.4.14 where Ekl HVT is the correlation function which can be experimentally measured when Alice chooses to measure ak and Bob chooses bl . Let us denote by ukl and v kl the length of projections of vectors u and v onto the plane spanned by ak and bl . Since one can decompose vectors u and v into a vector orthogonal to the plane of the settings and a vector within the plane the scalar products read 312 u  ak  ukl cosa k  u,v  bl  v kl cosb l  v, 7.4.15 where all the  angles are relative to some axis within the plane of the settings; angles u and v describe the position of the projections of vectors u and v , respectively, whereas angles a k and b l describe the position of the setting vectors. With this notation the inequality (7.4.14) becomes: Ekl HVT  1   dudvFu,v|ukl cosa k  u  v kl cosb l  v|. 7.4.16 The magnitudes of the projections can always be decomposed into the sum and the difference of two real numbers ukl  n1  n2 and v kl  n1  n2 . We insert this decomposition into the last inequality, and hence the terms multiplied by n1 and n2 are cosa k  u  cosb l  v  2sin a k  b l  u  v 2 sin a k  b l  u  v 2 7.4.17 and cosa k  u  cosb l  v  2cos a k  b l  u  v 2 cos a k  b l  u  v 2 , 7.4.18 respectively. We make the following substitution for the measurement angles kl  a k  b l 2 ,kl  a k  b l , 7.4.19 and parameterize the position of the projections within their plane by 313 uv  u  v 2 , uv  u  v. 7.4.20 Using these new angles one obtains that Ekl HVTkl,kl  1  2  dudvFu,v|n2 cos kl  uv 2 coskl  uv  n1 sin kl  uv 2 sinkl  uv|, 7.4.21 where in the correlation function Eklkl,kl we explicitly state the angles it is dependent on. The expression within the modulus is a linear combination of two harmonic functions of kl  uv , and therefore is a harmonic function itself. Its amplitude reads n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2 , 7.4.22 and the phase is some fixed real number  Ekl HVTkl,kl  1  2  dudvFu,v n22 cos2 kl  uv 2  n1 2 sin2 kl  uv 2  |coskl  uv  |. 7.4.23 In the next step we average both sides of this inequality over the measurement angle kl  a k  b l 2 . This means an integration over kl  0,2 and a multiplication by 1 2 . The integral of the kl dependent part of the right-hand side of (7.4.23) reads: 1 2  0 2 dkl|coskl  uv  | 2 . 7.4.24 By denoting the average of the correlation function over the angle kl as: 314 Ekl HVTkl  1 2  0 2 dklEklkl,kl, 7.4.25 one can write (7.4.23) as Ekl HVTkl  1  4  dudvFu,v n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2 7.4.26 This inequality is valid for any choice of observables in the plane defined by ak and bl . One can introduce two new observable vectors in this plane and write the inequality for the averaged correlation function Ek l HVT k l   of these new observables. The sum of these two inequalities is Ekl HVTkl  Ek l  HVT k l     2  4  dudvFu,v  n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2   n2 2 cos2  k l  uv 2   n1 2 sin2  k l  uv 2  7.4.27 One can use the triangle inequality ||x  y|| ||x||||y||, x 1  y12  x 2  y22  x 1 2  x 2 2  y1 2  y2 2 , 7.4.28 for the two-dimensional vectors x  x1 ,x2 and y  y1 ,y2 , with components defined by x 1  n2 cos kl  uv 2 , y1  n2 cos  k l   uv 2 , 7.4.29 and 315 x 2  n1 sin kl  uv 2 , y2  n1 sin  k l   uv 2 . 7.4.30 One can further estimate this bound by using the following relations cos kl  uv 2  cos  k l   uv 2  sin kl  k l  2 7.4.31 and sin kl  uv 2  sin  k l   uv 2  sin kl  k l  2 . 7.4.32 This estimate follows if one uses the formula for the sine of the difference angle to the right-hand side argument kl  k l  2  kl  uv 2   k l   uv 2 . Namely, sin kl  k l  2  sin kl  uv 2 cos  k l    uv 2  cos kl  uv 2 sin  k l   uv 2  sin kl  uv 2 cos  k l   uv 2  cos kl  uv 2 sin  k l    uv 2 . 7.4.33 After these estimates, the lower bound of Ekl  Ek l (following form the left-hand side inequality in (7.4.8)) is equal to minus the upper bound, and thus one can apply the upper bound to the modulus of the left hand side of (7.4.27). This is because the only formal difference between expressions in the estimates seeking the lower bound of the averaged Eq. (7.4.8) compared to those seeking the upper bound boils down to the interchange between n1 and n2 . After applying (7.4.31) and (7.4.32), this makes no difference anymore. One can shortly write 316 |Ekl HVTkl  Ek l HVT k l  | 2  4 sin kl  k l  2  dudvFu,v n22  n12 . 7.4.34 Going back to the magnitudes: |Ekl HVTkl  Ek l HVT k l  | 2  2 2 sin kl  k l  2  dudvFu,v ukl2  v kl2 . 7.4.35 This inequality is valid for any choice of the plane of observables. The bound involves only the projections of vectors u and v onto the plane of the settings. The integrations in the bound can be thought of as a mean value of expression ukl 2  v kl 2 averaged over the distribution of the vectors. For the plane orthogonal to the initial one the inequality is |Epq HVTpq  Ep q  HVT p q   |  2  2 2 sin pq  p q   2  dudvFu,v upq2  v pq2 , 7.4.36 where upq and v pq denote the projections of vectors u and v , respectively, onto the plane spanned by the settings ap and b  q (which is by construction orthogonal to the plane spanned by ak and bl ). We add the inequalities for orthogonal observation planes, (7.4.35) and (7.4.36), and choose k l    p q    0 and kl  pq   . This gives |Ekl HVT  E k l  HVT0||EpqHVT  Ep q  HVT0| 4  2 2 sin  2  dudvFu,v ukl2  v kl2  upq2  v pq2 7.4.37 We apply the triangle inequality (7.4.28) to the expression within the bracket. These time vectors x and y have the following components: 317 x  ukl,upq,y  v kl,vpq. 7.4.38 The integrand is bounded by: ukl 2  v kl 2  upq2  v pq2  ukl  upq2  v kl  v pq2 . 7.4.39 Let us consider the term involving vector u only. Since the lengths are positive ukl  upq2  ukl 2  upq2 . 7.4.40 Recall that ukl and upq are projections onto orthogonal planes. One can introduce normal vectors to these planes, nkl and npq , respectively, and write nkl  u2  ukl 2  1,npq  u2  upq2  1. 7.4.41 Note that the scalar products are two components of vector u in the Cartesian frame build out of vectors nkl , npq , and the one which is orthogonal to these two. Since vector u is normalized one has: nkl  u2  npq  u2  1, 7.4.42 which implies for the sum of equations (7.4.41) ukl 2  upq2  1. 7.4.43 The same applies to vector v and one can conclude that ukl 2  v kl 2  upq2  v pq2  2 . 7.4.44 318 Since the weight function Fu,v is normalized, the final Laggett type inequality is Ekl HVT  E k l HVT0||EpqHVT  Ep q  HVT0  4  4 sin  2 . 7.4.45 VII.4.3. Leggett inequality revisited. Validity of revised Leggett inequality A general framework of such models is the following: assumption (1) requires that an individual binary measurement outcome A t1 for a polarization measurement at instant t1 along direction a (that is, whether a single photon is transmitted or absorbed at instant t1 by a polarizer set at a specific angle) is predetermined by some set of hidden-variables , and a three-dimensional vector u , as well as by some set of other possibly non-local parameters  (for example, measurement settings in space-like separated regions) that is, At1,u,a,  A,u,a,, t1 . According to assumption (3), particles with the same u but with different  build up subensembles of 'definite polarization' described by a probability distribution u, t1 . The expectation value At1u  Au, t1, obtained by averaging over  , fulfils Malus' law, that is, At1u   du, t1A,u,a,, t1  u  a . Finally, with assumption (2), the measured expectation value for a general physical state is given by averaging over the distribution Fu, t1 of subensembles, that is, At1   duFu, t1At1u . Let us consider a specific source, which emits pairs of photons with well-defined polarizations u and v to laboratories of Alice and Bob, respectively. The local polarization measurement outcomes A t1 and Bt2 are fully determined by the polarization vector, by an additional set of hidden variables  specific to the source and by any set of parameters  outside the source. For reasons of clarity, we choose an explicit non-local dependence of the outcomes on the settings a 319 and b of the measurement devices. Note, however, that this is just an example of a possible non-local dependence, and that one can choose any other set out of  . Each emitted pair is fully defined by the subensemble distribution u,v, t1 , t2 . In agreement with assumption (3) we impose the following conditions on the predictions for local averages of such measurements (all polarizations and measurement directions are represented as vectors on the Poincaré sphere [10]): At1u   du,v, t1 , t2Aa,b,, t1  u  a, 7.4.46 Bt2v   du,v, t1 , t2Bb,a,, t2  v  b. 7.4.47 It is important to note that the validity of Malus' law imposes the nonsignalling condition on the investigated non-local models, as the local expectation values do only depend on local parameters. The correlation function of measurement results for a source emitting well-polarized photons is defined as the average of the products of the individual measurement outcomes: At1 Bt2u,v  ABu,v, t1 , t2   du,v, t1 , t2Aa,b,, t1Bb,a,, t2 7.4.48 For a general source producing mixtures of polarized photons the observable correlations are averaged over a distribution of the polarizations Fu,v, t1 , t2 , and the general correlation function Et1 , t2  is given by: Et1 , t2   At1Bt2F   dudvFu,v, t1 , t2ABu,v, t1 , t2 7.4.49 It is a very important trait of this model that there exist subensembles of definite polarizations (independent of measurements) and that the predictions for the subensembles agree with Malus' law. There the non-local correlations are a consequence 320 of the non-local quantum potential, which exerts suitable torque on the particles leading to experimental results compliant with quantum mechanics. In that theory, neither of the two particles in a maximally entangled state carries any angular momentum at all when emerging from the source [11]. In contrast, in the Leggett model, it is the total ensemble emitted by the source that carries no angular momentum, which is a consequence of averaging over the individual particles' well defined angular momenta (polarization). Assumption 7.4.1. We assume now that |t1  t2 |    0, and ABu,v, t1 , t2  ABu,v, t1  t2, Fu,v, t1 , t2  Fu,v, t1  t2, Et1 , t2   Et1  t2 . 7.4.50 Remark 7.4.1. We abbreviate now for short A t1 Bt2u,v  ABu,v  iff t1  t2    0, A t1 Bt2u,v  ABu,v  iff t1  t2    0, Et1 , t2   E iff t1  t2    0, Et1 , t2   E iff t1  t2    0. 7.4.51 We take a source which distributes pairs of well-polarized photons. Different pairs can have different polarizations. The size of a subensemble in which photons have polarizations u and v is described by the weight function Fu,v, t1 , t2  Fu,v, t1  t2 . All polarizations and measurement directions are represented as vectors on the Poincaré sphere. In every such subensemble individual measurement outcomes are determined by hidden variables . The hidden variables are allocated according to the distribution u,v, t1 , t2  u,v, t1  t2 . For any dichotomic measurement results, At1  1 and Bt2  1 , the following identity holds: 321 1  |At1  Bt2 | At1Bt2  1  |At1  Bt2 |. 7.4.52 If the signs of A t1 and Bt2 are the same |At1  Bt2 | 2 and |At1  Bt2 | 0 , and if At1  Bt2 then |At1  Bt2 | 0 and |At1  Bt2 | 2 . Any kind of nonlocal dependencies is allowed, i.e. At1  Aa,b ,u,v,, t1 . . . . and Bt2  Ba,b ,u,v,, t2 . . . . . Taking the average over the subensemble with definite polarizations gives 1   du,v, t1 , t2|A t1  Bt2 |   du,v, t1 , t2A t1 Bt2  1   du,v, t1 , t2|A t1  Bt2 |, 7.4.53 which in an abbreviated notation, where the averages are denoted by , is 1  |At1  Bt2 |  At1Bt2   1  |At1  Bt2 |. 7.4.54 As the average of the modulus is greater than or equal to the modulus of the averages, one gets the set of inequalities: 1  |At1   Bt2 |  At1Bt2   1  |At1   Bt2 |. 7.4.55 From now on only the upper bound will be considered, however all steps apply to the lower bound as well. With the assumption that photons with well defined polarization obey Malus' law At1   u  a, Bt2   v  b , 7.4.56 the upper bound of the inequality (7.4.55) becomes At1Bt2   1  |u  ak  v  bl|, 7.4.57 322 where ak and bl are unit vectors associated with the k -th measurement setting of Alice and the l -th of Bob, respectively. Taking the average over arbitrary polarizations we obtain Eklt1 , t2   1   dudvFu,v, t1 , t2|u  ak  v  bl|. 7.4.58 Remark 7.4.2. We assume now that t1  t2    0 and by using (7.4.50) – (7.4.51) we rewrite now the inequalities (7.4.57) and (7.4.58) as AB  1  |u  ak  v  bl| 7.4.59 and Ekl   1   dudvFu,v,|u  ak  v  bl|. 7.4.60 respectively. Remark 7.4.3. We assume now that t1  t2    0 and by using (7.4.50)-(7.4.51) we rewrite now the inequalities (7.4.57) and (7.4.58) as AB  1  |u  ak  v  bl| 7.4.61 and Ekl   1   dudvFu,v,|u  ak  v  bl|. 7.4.62 respectively. Remark 7.4.4. We define now the full avereges Ekl GHVT as Ekl GHVT  Ekl   Ekl  , 7.4.63 323 where Ekl GHVT is the correlation function which can be experimentally measured when Alice chooses to measure ak and Bob chooses bl . Let us denote by ukl and v kl the length of projections of vectors u and v onto the plane spanned by ak and bl . Thus from the inequalities (7.4.60) and (7.4.62) by using Eq. (7.4.63) we obtain Ekl GHVT  2   dudvFu,v|u  ak  v  bl|, Fu,v,  Fu,v,  Fu,v,. 7.4.64 Сompare the inequality (7.4.64) with the inequality (7.4.14). Let us denote by ukl and v kl the length of projections of vectors u and v onto the plane spanned by ak and bl . Since one can decompose vectors u and v into a vector orthogonal to the plane of the settings and a vector within the plane the scalar products read u  ak  ukl cosa k  u,v  bl  v kl cosb l  v, 7.4.65 where all the  angles are relative to some axis within the plane of the settings; angles u and v describe the position of the projections of vectors u and v , respectively, whereas angles a k and b l describe the position of the setting vectors. With this notation the inequality (7.4.64) becomes Ekl GHVT  2   dudvFu,v,|ukl cosa k  u  v kl cosb l  v|. 7.4.66 Сompare the inequality (7.4.66) with the inequality (7.4.16). The magnitudes of the projections can always be decomposed into the sum and the difference of two real numbers ukl  n1  n2 and v kl  n1  n2 . We insert this decomposition into the last inequality, and hence the terms multiplied by n1 and n2 are 324 cosa k  u  cosb l  v   2sin a k  b l  u  v 2 sin a k  b l  u  v 2 7.4.67 and cosa k  u  cosb l  v   2cos a k  b l  u  v 2 cos a k  b l  u  v 2 , 7.4.68 respectively. We make the following substitution for the measurement angles kl  a k  b l 2 ,kl  a k  b l , 7.4.69 and parameterize the position of the projections within their plane by uv  u  v 2 ,uv  u  v. 7.4.70 Using these new angles one obtains that Ekl GHVTkl,kl  2  2  dudvFu,v,|n2 cos kl  uv 2 coskl  uv  n1 sin kl  uv 2 sinkl  uv|, 7.4.71 where in the correlation function Eklkl,kl we explicitly state the angles it is dependent on. Сompare the inequality (7.4.71) with the inequality (7.4.21). The expression within the modulus is a linear combination of two harmonic functions of kl  uv, and therefore is a harmonic function itself. Its amplitude reads n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2 , 7.4.72 325 and the phase is some fixed real number  Ekl GHVTkl,kl  2  2  dudvFu,v, n22 cos2 kl  uv2  n1 2 sin2 kl  uv 2  |coskl  uv  |. 7.4.73 In the next step we average both sides of this inequality over the measurement angle kl  a k  b l 2 . This means an integration over kl  0,2 and a multiplication by 1 2 . The integral of the kl dependent part of the right-hand side of (7.4.73) reads: 1 2  0 2 dkl|coskl  uv  |  2 . 7.4.74 By denoting the average of the correlation function over the angle kl as: Ekl GHVTkl  1 2  0 2 dklEkl GHVTkl,kl, 7.4.75 one can write (7.4.73) as Ekl GHVTkl  2  4  dudvFu,v, n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2 7.4.76 This inequality is valid for any choice of observables in the plane defined by ak and bl . One can introduce two new observable vectors in this plane and write the inequality for the averaged correlation function Ek l GHVT k l   of these new observables. The sum of these two inequalities is 326 Ekl GHVTkl  Ek l  HVT k l     4  4  dudvFu,v,  n2 2 cos2 kl  uv 2  n1 2 sin2 kl  uv 2   n2 2 cos2  k l    uv 2  n1 2 sin2  k l    uv 2 . 7.4.77 One can use the triangle inequality ||x  y|| ||x||||y||, x 1  y12  x 2  y22  x 1 2  x 2 2  y1 2  y2 2 , 7.4.78 for the two-dimensional vectors x  x1 ,x2 and y  y1 ,y2 , with components defined by x 1  n2 cos kl  uv 2 , y1  n2 cos  k l   uv 2 , 7.4.79 and x 2  n1 sin kl  uv 2 , y2  n1 sin  k l   uv 2 . 7.4.80 One can further estimate this bound by using the following relations cos kl  uv 2  cos  k l   uv 2  sin kl  k l  2 7.4.81 and sin kl  uv 2  sin  k l   uv 2  sin kl  k l  2 . 7.4.82 This estimate follows if one uses the formula for the sine of the 327 difference angle to the right-hand side argument kl  k l  2  kl  uv 2   k l   uv 2 . Namely, sin kl  k l  2  sin kl  uv 2 cos  k l    uv 2  cos kl  uv 2 sin  k l   uv 2  sin kl  uv 2 cos  k l   uv 2  cos kl  uv 2 sin  k l    uv 2 . 7.4.83 After these estimates, the lower bound of Ekl GHVT  E k l GHVT (following form the left-hand side inequality in (7.4.55)) is equal to minus the upper bound, and thus one can apply the upper bound to the modulus of the left hand side of (7.4.27). This is because the only formal difference between expressions in the estimates seeking the lower bound of the averaged Eq. (7.4.55) compared to those seeking the upper bound boils down to the interchange between n1 and n2 . After applying (7.4.81) and (7.4.82), this makes no difference anymore. One can shortly write Ekl GHVTkl  Ek l GHVT k l    4  4 sin kl  k l  2 dudvFu,v, n22  n12 . 7.4.84 Going back to the magnitudes: |Ekl GHVTkl  Ek l GHVT k l  | 4  2 2 sin kl  k l  2  dudvFu,v, ukl2  v kl2 . 7.4.85 The inequality (7.4.85) is valid for any choice of the plane of observables. The bound involves only the projections of vectors u and v onto the plane of the settings. The integrations in the bound can be thought of as a mean value of expression ukl 2  v kl 2 averaged over the distribution of the vectors. For the plane orthogonal to the initial one the inequality is 328 |Epq GHVTpq  Ep q  GHVT p q   |  2  2 2 sin pq  p q   2  dudvFu,v, upq2  v pq2 , 7.4.86 where upq and v pq denote the projections of vectors u and v , respectively, onto the plane spanned by the settings ap and b  q (which is by construction orthogonal to the plane spanned by ak and bl ). We add the inequalities for orthogonal observatin planes, (7.4.85) and (7.4.86), and choose k l    p q    0 and kl  pq   . This gives Ekl GHVT  E k l  GHVT0||EpqGHVT  Ep q  GHVT0 |  8  2 2 sin  2  dudvFu,v, ukl2  v kl2  upq2  v pq2 7.4.87 We apply the triangle inequality (7.4.78) to the expression within the bracket. These time vectors x and y have the following components: x  ukl,upq, y  v kl,vpq. 7.4.88 The integrand is bounded by: ukl 2  v kl 2  upq2  v pq2  ukl  upq2  v kl  v pq2 . 7.4.89 Let us consider the term involving vector u only. Since the lengths are positive ukl  upq2  ukl 2  upq2 . 7.4.90 Recall that ukl and upq are projections onto orthogonal planes. One can introduce normal vectors to these planes, nkl and npq , respectively, and write 329 nkl  u2  ukl 2  1,npq  u2  upq2  1. 7.4.91 Note that the scalar products are two components of vector u in the Cartesian frame build out of vectors nkl , npq , and the one which is orthogonal to these two. Since vector u is normalized one has: nkl  u2  npq  u2  1, 7.4.92 which implies for the sum of equations (7.4.90) ukl 2  upq2  1. 7.4.93 The same applies to vector v and one can conclude that ukl 2  v kl 2  upq2  v pq2  2 . 7.4.94 Since the weight function Fu,v is normalized, the final Laggett type inequality is Ekl GHVT  E k l GHVT0||EpqGHVT  Ep q  GHVT0  8  4 sin  2 . 7.4.95 330 APPENDICES Appendix A HEISENBERG'S NOISE-DISTURBANCE UCERTAINTY PRINCIPLE In the wave description of a photon, polarization can be visualized as the way the wave is rotated. A photon possesses horizontal |H or vertical |V polarization, but until its polarization is measured, these two states are said to be in a superposition, described by |  |H  |V A.1 where ,   and ||2 is the probability of finding the photon in state |H and ||2 in state |V. The horizontal and the vertical polarization jointly define a basis denoted by  z , which can take on the values |H or |V. However, the polarization can be described in an additional basis as well, the  x basis, which is shifted 45 in positive direction, see Fig. A.1. Fig. A.1. A graphical representation of the bases  z and  x 331 Just like the particle superposition consists of |H and |V in  z basis, the two new states |H  and |V  will describe the particle superposition in the  x basis as |  |H   |V  A.2 where ||2 describes the probability of finding the photon in state |H  and ||2 is state |V . An analogy to the intrinsic property spin is the polarization of photons. Since the polarization is just another example of an intrinsic property a particle could exhibit, one can repeat the SternGerlach experiment with the use of photons. Like the spin of the electron, one can write a polarization state of one basis as a superposition constituting the eigenstates of the other basis. The eigenstates obtained in the  z basis are given by |H  1 2 |H   |V , |V  1 2 |H   |V . A. 3 while polarization states in the  x basis are described by |H   1 2 |H  |V, |V   1 2 |H  |V. A. 4 Every SG (Stern-Gerlach) apparatus is replaced by three half wave plates (HWP) and a polarizing beam splitter (PBS) in between, see Fig. A.2. This setup will have the same effect on a photon as the SG-apparatus had on the electrons since it measures the polarization of the photon which thereby collapses into one of the eigenstates. A mathematical description of the setup, in order to 332 explain the expectation value which will make us able to predict the paths of the photons throughout the measurements, follows below. Fig. A.2. A representation of a setup corresponding to a SGapparatus The HWP rotates the polarization of the transmitted light and thereby shifts between the two bases. The HWP is described mathematically by the operator  R  cos2 sin2 sin2 cos2 A. 5 and when set to basis  z :  0 , and when set to  x :   22.5. The PBS divides the incident beam into two output beams and can mathematically be described as the operator P  1 0 0 1 . A. 6 333 In this instance the matrix representing the measurement operator  z in the  z basis is derived by z   R0 P  R0  . A. 7 Similarly, the matrix representing the measurement operator x in the  x basis is derived by x   R22.5 P  R22.5  . A. 8 Though the expressions above look alike, note that does not take on the same values when set to measure  z and  x . The measurement operator  z measures the polarization in the z -direction. Hence the photon will always be thrown into one of the eigenstates |H or |V. When measured in x , it will always be thrown into |H  or |V . Thus, one can say that the states are eigenstates to the respective operator. The two operators  z and x are represented by z  1 0 0 1 ,x  0 1 1 0 . A. 9 Notice that z,x   2 0 1 1 0  2C. A. 10 The Robertson uncertainty relation is given by z x   1 2 |z,x |  C. A. 11 Or in general case 334 A,B,  1 2 ||A,B|| A. 12 for any observables A,B and any state , where the standard deviation X, of an observable X in state  is defined by 2X,  |X 2 |  |X|2 . This relation was proven mathematically from fundamental postulates of quantum mechanics. Nevertheless, this relation describes the limitation on preparing microscopic objects but has no direct relevance to the limitation of accuracy of measuring devices. It is a common understanding that the uncertainty principle implies or is implied by a limitation on measuring a system without disturbing it as a position measurement typically disturbs the momentum. However, the limitation has eluded a correct quantitative expression on the trade-off between noise and disturbance. Heisenberg noise-disturbance uncertainty relation By the  -ray thought experiment, Heisenberg [10, 16] argued that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than /2. This relation is generally formulated as follows: for any apparatus A to measure an observable A, the relation A,,AB,,A  1 2 ||A,B|| A. 13 holds for any input state  and any observable B, where A,,A stands for the noise of the A measurement in state  using apparatus A and B,,A stands for the disturbance of B in state  caused by apparatus A. We refer to the above relation as the Heisenberg noise-disturbance uncertainty relation. 335 Heisenberg uncertainty relation for joint measurements Very similarly to the above relation (A.13), the Heisenberg uncertainty relation for joint measurements is generally formulated as follows: for any apparatus A with two outputs for the joint measurement of A and B, the relation A,,AB,,A  1 2 ||A,B|| A. 14 holds for any input state, where X,,A stands for the noise of the X measurement in state  using apparatus A for X  A,B . This relation was proven under the joint unbiasedness condition requiring that the (experimental) mean values of the outcome x A of the A measurement and the outcome yB of the B measurement should coincide with the (theoretical) mean values of observables A and B, respectively, on any input state  . It is a common opinion that currently available measuring devices satisfy this relation [12]-[14]. 336 Appendix B CONDITIONAL PROBABILITY DENSITY FUNCTIONS If X is given to be a continuous random variable with a defined density function say f(x) and E is an event which has positive probability then we define the conditional density function as fx|E  fx PE  x  E 0  x  E B. 1 337 Appendix C FOURIER TRANSFORM AND HEISENBERG UNCERTAINTY PRINCIPLE Definition C.1. Given any 1,2 function f, where 1,2  1 2, we define two operators as follows: (1) fx   f p  fp    fxexp2ixpdx, (2) fp  1fx    fpexp2ixpdp. Theorem C.1. (Heisenberg's Uncertainty Principle). Let   1,2 with the condition   |x|2dx  1. Then   x 2 |x|2dx   p2  f p 2  1 162 , C. 1 and   x  x 0 2 |x|2dx   p  p0 2  f p 2  1 162 . C. 2 Thinking of x  x0 x2 as the standard deviation of the displacement written x and similarly for p  p0 p2 written p we may write xp  const. C.3 which is the usual quantum-mechanical way of Heisenberg's uncertainty principle. 338 REFERENCES Introduction [1] Bell J. S. 1964 On the Einstein Podolsky Rosen Paradox, Physics 1,3,195200 [2] Bell J. S. 1987 Speakable and Unspeakable in Quantum Mechanics (Cambridge: Cambridge Univ. Press) [3] Kolmogorov A. N. 1933 Grundbegriffe der Wahrscheinlichkeitsrechnung (Berlin: Springer Verlag); English translation: Kolmogorov A. N. 1956 Foundations of Theory of Probability (New York: Chelsea Publishing Company) [4] L. Accardi 1981 Topics in quantum probability. Phys. Rep. 77 169-192. [5] L. Accardi, M. Regoli, Locality and Bell's inequality. arXiv:quantph/0007005. [6] G. Adenier, A. Khrennikov, Anomalies in EPR-Bell experiments. Quantum theory: reconsideration of foundations-3,283-293, AIP Conf. Proc., 810, Amer. Inst. Phys., Melville, NY, 2006. [7] G. Adenier and A. Khrennikov, Is the fair sampling assumption supported by EPR experiments? Journal of Physics B: Atomic, Molecular and Optical Physics, 40, 131-141 (2007). [8] Avis D, Fischer P, Hilbert A, and Khrennikov A. 2009 Single, Complete, Probability Spaces Consistent With EPR-Bohm-Bell Experimental Data, Foundations of Probability and Physics-5 vol. 750 (Melville, NY: AIP) pp. 294-301. 339 [9] Khrennikov A., Bell Could Become the Copernicus of Probability, Open Systems & Information Dynamics, Vol. 23, No. 02, 1650008 (2016) https://doi.org/10.1142/S1230161216500086 [10] A. Aspect, Bell's Theorem: The Naive View of an Experimentalist, arXiv:quant-ph/0402001 [11] Belinskii A. V., "Bell's paradoxes without the introduction of hidden variables" Phys. Usp. 37 413-419 (1994) DOI: 10.1070/PU1994v037n04ABEH000024 [12] Clauser, J. F., M. A. Horne, A. Shimony, and R. A. Holt, 1969, Phys. Rev. Lett. 23, 880. [13] A. Einstein et al., Phys. Rev. 47, 777 (1935). [14] M. D. Reid, The Einstein-Podolsky-Rosen Paradox and Entanglement 1: Signatures of EPR correlations for continuous variables. http://arxiv.org/abs/quant-ph/0112038v1 [15] D. Bohm, Quantum Theory (New York, Prentice-Hall, 1951). Chapter I-III [1] Foukzon, J., Potapov, A., Men'kova, E. and Podosenov, S. (2017) Schrödinger's Cat Paradox Resolution Using GRW Collapse Model: Von Neumann Measurement Postulate Revisited. Journal of Applied Mathematics and Physics, 5, 494-521. doi: 10.4236/jamp.2017.52044. [2] Foukzon, J., Potapov, A., Men'kova, E. Schrödinger's Cat Paradox Resolution: A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. Paperback 84 pages, Publisher: LAP LAMBERT Academic Publishing (December 19, 2017). ISBN-10: 3659935433; ISBN-13: 340 978-3659935435 [3] S. L. Adler, Where is quantum theory headed? Journal of Physics: Conference Series 504 (2014) 012002. doi:10.1088/17426596/504/1/012002 [4] V. Man'ko, Classical propagators of quadratic quantum systems, Theoretical and Mathematical Physics, November 1999, Volume 121, Issue 2, pp. 1496-1505. DOI: http://dx.doi.org/10.4213/tmf809 http://link.springer.com/article/10.1007/BF02557220#page-1 [5] S. Mancini, V. I. Man'ko, P. Tombesi. Phys. Lett. A. 1996. V. 213. P. 1. [6] A. Einstein et al., Phys. Rev. 47, 777 (1935). [7] Bacciagaluppi, G. & Valentini, A. Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference 487 (Cambridge University Press, 2009). [8] Born, M. Natural Philosophy of Cause and Chance 109 (Oxford University Press, 1949). [9] Jones, S. J. & Wiseman, H. M. Nonlocality of a single photon: paths to an Einstein-Podolsky-Rosen-steering experiment. Phys. Rev. A 84, 012110 (2011). [10] M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, A. Furusawa. Experimental proof of nonlocal wave function collapse for a single particle using homodyne measurements. Nature Communications 6, Article number: 6665 doi:10.1038/ncomms7665 http://www.nature.com/ncomms/2015/150324/ncomms7665/full/nco mms7665.html [11] Schrödinger, E. Discussion of probability relations between separated systems. Proc. Cambridge Philos. Soc. 31, 553 (1935). 341 [12] Terhal, B. M. Bell inequalities and the separability criterion. Phys. Lett. A 271, 319-326 (2000). [13] Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195-200 (1964). [14] Mittelstaedt, P. "The Problem of Decoherence and the EPRParadox." Decoherence: Theoretical, Experimental, and Conceptual Problems, Edited by P. Blanchard, D. Giulini, E. Joos, C. Kiefer, I.O. Stamatescu, Lecture Notes in Physics, Published by Springer Berlin Heidelberg ISBN: 978-3-540-66899-2. DOI: 10.1007/3-54046657-6 [15] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, M. D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Phys. Rev. A 80, 032112 (2009). http://arxiv.org/abs/0907.1109v2 [16] M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, G. Leuchs. The EinsteinPodolsky-Rosen paradox: from concepts to applications, Rev. Mod. Phys. 81, 1727 (2009). http://arxiv.org/abs/0806.0270 [17] M. D. Reid, The Einstein-Podolsky-Rosen Paradox and Entanglement 1: Signatures of EPR correlations for continuous variables. http://arxiv.org/abs/quant-ph/0112038v1 [18] E. G. Cavalcanti, P. D. Drummond, H. A. Bachor, M. D. Reid, Spin entanglement, decoherence and Bohm's EPR paradox, Optics Express, Vol. 17, Issue 21, pp. 18693-18702 (2009) http://arxiv.org/abs/0711.3798 [19] Y. Yin, Y. Chen, D. Sank, P. J. J. O'Malley, T. C. White, R. 342 Barends, J. Kelly, E. Lucero, M. Mariantoni, A. Megrant, et al., Phys. Rev. Lett. 110, 107001 (2013). [20] R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, et al., Nature 508, 500 (2014). [21] T. Guerreiro, B.Sanguinetti, H. Zbinden, N. Gisin, A. Suarez, Single-photon space-like antibunching. arXiv: 1204.1712v1 [quantph] 8 Apr 2012 [22] D. H. Fremlin, Measure Theory, Volume 3: Measure Algebras, May 31, 2002. Chapter IV-VI [1] A. Einstein et al., Phys. Rev. 47, 777 (1935). [2] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, M. D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Phys. Rev. A 80, 032112 (2009). http://arxiv.org/abs/0907.1109v2 [3] M. D. Reid, The Einstein-Podolsky-Rosen Paradox and Entanglement 1: Signatures of EPR correlations for continuous variables. http://arxiv.org/abs/quant-ph/0112038v1 [4] E. G. Cavalcanti, P. D. Drummond, H. A. Bachor, M. D. Reid, Spin entanglement, decoherence and Bohm's EPR paradox, Optics Express, Vol. 17, Issue 21, pp. 18693-18702 (2009). http://arxiv.org/abs/0711.3798 [5] M. D. Reid, P. D. Drummond, E. G. Cavalcanti, W. P. Bowen, P. K. Lam, H. A. Bachor, U. L. Andersen, G. Leuchs The Einstein- 343 Podolsky-Rosen paradox: from concepts to applications, Rev. Mod. Phys. 81, 1727 (2009). http://arxiv.org/abs/0806.0270 [6] P. M. Walker, L. Tinkler, M. Durska, D. M. Whittaker, I. J. Luxmoore, B. Royall, D. N. Krizhanovskii, M. S. Skolnick, I. Farrer, and D. A. Ritchie, Applied Physics Letters 102, 012109 (2013). [7] M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, A. Furusawa, Experimental proof of nonlocal wave function collapse for a single particle using homodyne measurements. Nature Communications 6, Article number: 6665 doi:10.1038/ncomms7665 http://www.nature.com/ncomms/2015/150324/ncomms7665/full/nco mms7665.html [8] John C. Howell, Ryan S. Bennink, Sean J. Bentley, and R. W. Boyd Realization of the Einstein-Podolsky-Rosen Paradox Using Momentum and Position-Entangled Photons from Spontaneous Parametric Down Conversion. http://www.pas.rochester.edu/~howell/mysite2/EPR%20PRL.pdf [9] Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Realization of the Einstein-Podolsky-Rosen paradox for continuous variables. Phys. Rev. Lett. 68, 3663. [10] Foukzon, J., Potapov, A., Men'kova, E. and Podosenov, S. (2017) Schrödinger's Cat Paradox Resolution Using GRW Collapse Model: Von Neumann Measurement Postulate Revisited. Journal of Applied Mathematics and Physics, 5, 494-521. doi: 10.4236/jamp.2017.52044. [11] Foukzon, J., Potapov, A., Men'kova, E. Schrödinger's Cat Paradox Resolution: A New Quantum Mechanical Formalism Based on the Probability Representation of Continuous Observables. Paperback 84 pages, Publisher: LAP LAMBERT Academic Publishing (December 19, 2017). ISBN-10: 3659935433; ISBN-13: 344 978-3659935435. [12] Y. Sherkunov, David M. Whittaker, and Vladimir Fal'ko, Position-momentum-entangled photon pairs in nonlinear waveguides and transmission lines, Phys. Rev. A 93, 043842 http://arxiv.org/abs/1511.03514 [13] D., Bohm, Y., Aharonov, Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky. [14] P. M. Walker, L. Tinkler, D. V. Skryabin, A. Yulin, B. Royall, I. Farrer, D. A. Ritchie, M. S. Skolnick, and D. N. Krizhanovskii, Nat Commun 6, 8317 (2015). [15] J. Clauser, M. Horne, A. Shimony, R. Holt, (1969). "Proposed Experiment to Test Local Hidden-Variable Theories". Physical Review Letters. 23 (15): 880-884. [16] S. Mancini, V. I. Man'ko, P. Tombesi. Phys. Lett. A. 1996. V.213. P. 1. [17] D. Bohm, Quantum Theory (New York, Prentice-Hall, 1951). [18] D. Bohm, Y. Aharonov, Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky. PHYSI CAL REVIEW VOLUM E 108, NUMBER 4 NOVEMBER 15, 1957. [19] M. Gondran, "Measurement in the de Broglie-Bohm interpretation: Double-slit, Stern-Gerlach and EPR-B" arXiv:1309.4757 [quant-ph] [20] S. L. Adler, Where is quantum theory headed? Journal of Physics: Conference Series 504 (2014) 012002 doi:10.1088/17426596/504/1/012002 345 Chapter VII [1] Belinskii A. V., "Bell's theorem without the hypothesis of locality" Phys. Usp. 37. 219--222 (1994). DOI: 10.1070/PU1994v037n02ABEH000009 URL: https://ufn.ru/en/articles/1994/2/d/ [2] Fine A., Phys. Rev. Lett. 48, 291 (1982). [3] Kolmogorov A. N., 1933 Grundbegri e der Wahrscheinlichkeitsrechnung (Berlin: Springer Verlag); English translation: Kolmogorov A. N. 1956 Foundations of Theory of Probability (New York: Chelsea Publishing Company). [4] Clauser, J. F., M. A. Horne, A. Shimony, and R. A. Holt, 1969, Phys. Rev. Lett. 23, 880. [5] G. Weihs, T. Jennewein, Violation of Bell's inequality under strict Einstein locality conditions, Phys. Rev. Lett. 81 (1998) 5039-5043. DOI:10.1103/Phys Rev Lett.81.5039 arXiv:quant-ph/9810080 [6] Belinskii A. V., "Bell's paradoxes without the introduction of hidden variables" Phys. Usp. 37 413--419 (1994). DOI: 10.1070/PU1994v037n04ABEH000024 URL: https://ufn.ru/en/articles/1994/4/l/ [7] Belinskii A. V., Klyshko D. N., "Interference of light and Bell's theorem" Phys. Usp. 36 (8) 653--693 (1993). [8] D. Dehlinger and M. W. Mitchell, Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory, American Journal of Physics 70, 903 (2002). https://doi.org/10.1119/1.1498860 [9] S. Groblacher, T. Paterek, An experimental test of non-local realism, Nature 446, 871-875 (2007). DOI: 10.1038/nature05677 346 arXiv:0704.2529 [quant-ph] [10] Heisenberg, W.  Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43,172-198 (1927). The physical content of quantum kinematics and mechanics (Engl. Trans.), in Quantum Theory and Measurement (eds Wheeler, J. A. & Zurek, W. H.) (Princeton Univ. Press, Princeton, 1983). [11] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, Oxford, 1964), 2nd ed. [12] C. S. Wu and I. Shaknov, Phys. Rev. 77, 136 (1950). [13] C. A. Kocher and E. D. Commins, Phys. Rev. Lett. 18, 575 (1967). [14] S. J. Freedman and J. F. Clauser, Phys. Rev. Lett. 28, 938 (1972). [15] A. J. Leggett, Found. of Phys. 33, 1469 (2003). [16] Heisenberg, W. The Physical Principles of the Quantum Theory, (Univ. of Chicago Press, Chicago, 1930; Dover,New York, 1949, 1967). View publication stats