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Summary All approaches to quantum theory need to make sense of the objective probabilities which apparently correspond to the square of the amplitudes of components of the quantum state.
Key works The Born rule connecting probabilities and squared-amplitudes was first formulated in Born 1926. The interpretation of probabilities varies widely across different approaches to quantum mechanics: as ever, Bell 2004 is indispensable in setting out the main options. Wallace 2012 provides an authoritative treatment of probability in Everettian QM.
Introductions Dickson 2011
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  1. Luigi Accardi (1990). Quantum Probability and the Foundations of Quantum Theory. In Roger Cooke & Domenico Costantini (eds.), Boston Studies in the Philosophy of Science vol. 122: Statistics in Science. Springer Netherlands. 119-147.
    The point of view advocated, in the last ten years, by quantum probability about the foundations of quantum mechanics, is based on the investigation of the mathematical consequences of a deep and elementary idea developed by the founding fathers of quantum mechanics and accepted nowadays as a truism by most physicists, namely: one should be careful when applying the rules derived from the experience of macroscopic physics to experiments which are mutually incompatible in the sense of quantum mechanics.
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  2. Diedrik Aerts & Sven Aerts (1995). Applications of Quantum Statistics in Psychological Studies of Decision Processes. Foundations of Science 1 (1):85-97.
    We present a new approach to the old problem of how to incorporate the role of the observer in statistics. We show classical probability theory to be inadequate for this task and take refuge in the epsilon-model, which is the only model known to us caapble of handling situations between quantum and classical statistics. An example is worked out and some problems are discussed as to the new viewpoint that emanates from our approach.
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  3. James Aken (1986). Analysis of Quantum Probability Theory. II. Journal of Philosophical Logic 15 (3):333 - 367.
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  4. James Aken (1985). Analysis of Quantum Probability Theory. I. Journal of Philosophical Logic 14 (3):267 - 296.
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  5. Sergio Albeverio, Philippe Combe & M. Sirugue-Collin (eds.) (1982). Stochastic Processes in Quantum Theory and Statistical Physics: Proceedings of the International Workshop Held in Marseille, France, June 29-July 4, 1981. [REVIEW] Springer-Verlag.
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  6. Charis Anastopoulos (2006). Classical Versus Quantum Probability in Sequential Measurements. Foundations of Physics 36 (11):1601-1661.
    We demonstrate in this paper that the probabilities for sequential measurements have features very different from those of single-time measurements. First, they cannot be modelled by a classical stochastic process. Second, they are contextual, namely they depend strongly on the specific measurement scheme through which they are determined. We construct Positive-Operator-Valued measures (POVM) that provide such probabilities. For observables with continuous spectrum, the constructed POVMs depend strongly on the resolution of the measurement device, a conclusion that persists even if we (...)
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  7. D. M. Appleby (2005). Facts, Values and Quanta. Foundations of Physics 35 (4):627-668.
    Quantum mechanics is a fundamentally probabilistic theory (at least so far as the empirical predictions are concerned). It follows that, if one wants to properly understand quantum mechanics, it is essential to clearly understand the meaning of probability statements. The interpretation of probability has excited nearly as much philosophical controversy as the interpretation of quantum mechanics. 20th century physicists have mostly adopted a frequentist conception. In this paper it is argued that we ought, instead, to adopt a logical or Bayesian (...)
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  8. D. M. Appleby, Åsa Ericsson & Christopher A. Fuchs (2011). Properties of QBist State Spaces. Foundations of Physics 41 (3):564-579.
    Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement. But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted subset of all potentially available probabilities. A recent publication (Fuchs and Schack, arXiv:0906.2187v1, 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we study how this subset—quantum-state space—might be characterized. Our leading characteristic is that the inner (...)
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  9. István Aranyosi (2012). Should We Fear Quantum Torment? Ratio 25 (3):249-259.
    The prospect, in terms of subjective expectations, of immortality under the no-collapse interpretation of quantum mechanics is certain, as pointed out by several authors, both physicists and, more recently, philosophers. The argument, known as quantum suicide, or quantum immortality, has received some critical discussion, but there hasn't been any questioning of David Lewis's point that there is a terrifying corollary to the argument, namely, that we should expect to live forever in a crippled, more and more damaged state, that barely (...)
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  10. Guido Bacciagaluppi (2013). Insolubility Theorems and EPR Argument. European Journal for Philosophy of Science 3 (1):87-100.
    I present a very general and simple argument—based on the no-signalling theorem—showing that within the framework of the unitary Schrödinger equation it is impossible to reproduce the phenomenological description of quantum mechanical measurements (in particular the collapse of the state of the measured system) by assuming a suitable mixed initial state of the apparatus. The thrust of the argument is thus similar to that of the ‘insolubility theorems’ for the measurement problem of quantum mechanics (which, however, focus on the impossibility (...)
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  11. John C. Baez (1989). Is Life Improbable? Foundations of Physics 19 (1):91-95.
    E. P. Wigner's argument that the probability of the existence of self-reproducing units, e.g., organisms, is zero according to standard quantum theory is stated and analyzed. Theorems are presented which indicate that Wigner's mathematical result in fact should not be interpreted as asserting the improbability of self-reproducing units.
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  12. J. E. Baggott (2004). Beyond Measure: Modern Physics, Philosophy, and the Meaning of Quantum Theory. Oxford University Press.
    Quantum theory is one the most important and successful theories of modern physical science. It has been estimated that its principles form the basis for about 30 per cent of the world's manufacturing economy. This is all the more remarkable because quantum theory is a theory that nobody understands. The meaning of Quantum Theory introduces science students to the theory's fundamental conceptual and philosophical problems, and the basis of its non-understandability. It does this with the barest minimum of jargon and (...)
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  13. David Baker (2007). Measurement Outcomes and Probability in Everettian Quantum Mechanics. Studies in History and Philosophy of Science Part B 38 (1):153-169.
    The decision-theoretic account of probability in the Everett or many-worlds interpretation, advanced by David Deutsch and David Wallace, is shown to be circular. Talk of probability in Everett presumes the existence of a preferred basis to identify measurement outcomes for the probabilities to range over. But the existence of a preferred basis can only be established by the process of decoherence, which is itself probabilistic.
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  14. L. E. Ballentine (1990). Limitations of the Projection Postulate. Foundations of Physics 20 (11):1329-1343.
    The projection postulate, which prescribes “collapse of the state vector” upon measurement, is not an essential part of quantum mechanics. Rather it is only an optional discarding of certain branches of the state vector that are expected to be irrelevant for the purpose at hand. However, its use is hazardous, and there are examples of repeated measurements for which the conventional application of the projection postulate leads to incorrect results.
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  15. L. E. Ballentine (1973). Can the Statistical Postulate of Quantum Theory Be Derived?—A Critique of the Many-Universes Interpretation. Foundations of Physics 3 (2):229-240.
    The attempt to derive (rather than assume) the statistical postulate of quantum theory from the many-universes interpretation of Everett and De Witt is analyzed The many-universes interpretation is found to be neither necessary nor sufficient for the task.
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  16. A. O. Barut, M. Božić & Z. Marić (1988). Joint Probabilities of Noncommuting Operators and Incompleteness of Quantum Mechanics. Foundations of Physics 18 (10):999-1012.
    We use joint probabilities to analyze the EPR argument in the Bohm's example of spins.(1) The properties of distribution functions for two, three, or more noncommuting spin components are explicitly studied and their limitations are pointed out. Within the statistical ensemble interpretation of quantum theory (where only statements about repeated events can be made), the incompleteness of quantum theory does not follow, as the consistent use of joint probabilities shows. This does not exclude a completion of quantum mechanics, going beyond (...)
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  17. J. S. Bell (2004). Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy. Cambridge University Press.
    This book comprises all of John Bell's published and unpublished papers in the field of quantum mechanics, including two papers that appeared after the first edition was published. It also contains a preface written for the first edition, and an introduction by Alain Aspect that puts into context Bell's great contribution to the quantum philosophy debate. One of the leading expositors and interpreters of modern quantum theory, John Bell played a major role in the development of our current understanding of (...)
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  18. J. S. Bell (1992). Six Possible Worlds of Quantum Mechanics. Foundations of Physics 22 (10):1201-1215.
  19. Js Bell (1992). 6 Possible Worlds of Quantum-Mechanics (Reprinted From Possible Worlds in Humanities Arts and Sciences, Pg 359-373, 1989. [REVIEW] Foundations of Physics 22 (10):1201-1215.
  20. E. G. Beltrametti & S. Bugajski (2000). Remarks on Two-Slit Probabilities. Foundations of Physics 30 (9):1415-1429.
    The probability pattern emerging in two-slit experiments is a typical quantum feature whose essential ingredients are examined by translating them into the spin- $ \frac{1}{2} $ formalism. In view of the existence of extensions of quantum theory preserving some classical structure, we discuss how the two-slit probabilities behave under such extensions. We consider a generalization of the standard classical probability theory, to be called operational probability theory, that turns out to host the so called quantum probabilities.
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  21. Paul Benioff (1975). On Procedures for the Measurement of Questions in Quantum Mechanics. Foundations of Physics 5 (2):251-255.
    It is shown that there exist observablesA and Borel setsE such that the procedure “measureA and give as output the number 1 (0) if theA measurement outcome is (is not) inE” does not correspond to a measurement of the proposition observable ℰA(E) usually assigned to such procedures. This result is discussed in terms of limitations on choice powers of observers.
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  22. Paul Benioff (1973). On Definitions of Validity Applied to Quantum Theories. Foundations of Physics 3 (3):359-379.
    In this work, quantum theories are considered which consist in essence of a map from state preparation proceduresw to states and a map from decision proceduresQ to probability operator measures. Two definitions of validity, similar to that given elsewhere, are given and compared for these theories. One definition is given in terms of one carrying out of somew followed by someQ, denoted by(Q, w). The other is given in terms of infinite repetitions(Q, w) ofw followed byQ. Both definitions are discussed (...)
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  23. Rodney W. Benoist, Jean-Paul Marchand & Wolfgang Yourgrau (1978). Addendum to Statistical Inference and Quantum Mechanical Measurement. Foundations of Physics 8 (1-2):117-118.
  24. Rodney W. Benoist, Jean-Paul Marchand & Wolfgang Yourgrau (1977). Statistical Inference and Quantum Mechanical Measurement. Foundations of Physics 7 (11-12):827-833.
    We analyze the quantum mechanical measuring process from the standpoint of information theory. Statistical inference is used in order to define the most likely state of the measured system that is compatible with the readings of the measuring instrument and the a priori information about the correlations between the system and the instrument. This approach has the advantage that no reference to the time evolution of the combined system need be made. It must, however, be emphasized that the result is (...)
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  25. Carlton W. Berend (1942). A Note on Quantum Theory and Metaphysics. Journal of Philosophy 39 (22):608-611.
  26. J. Berkovitz (1995). What Econometrics Cannot Teach Quantum Mechanics. Studies in History and Philosophy of Science Part B 26 (2):163-200.
    Cartwright (1989) and Humphreys (1989) have suggested theories of probabilistic causation for singular events, which are based on modifications of traditional causal linear modelling. On the basis of her theory, Cartwright offered an allegedly local, and non-factorizable, common-cause model for the EPR experiment. In this paper I consider Cartwright's and Humphrey's theories. I argue that, provided plausible assumptions obtain, local models for EPR in the framework of these theories are committed to Bell inequalities, which are violated by experiment.
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  27. Joseph Berkovitz (2008). On Predictions in Retro-Causal Interpretations of Quantum Mechanics. Studies in History and Philosophy of Science Part B 39 (4):709-735.
  28. Joseph Berkovitz (2002). On Causal Loops in the Quantum Realm. In T. Placek & J. Butterfield (eds.), Non-Locality and Modality. Kluwer. 235--257.
  29. Hans Den Bervang, Dick Hoekzema & Hans Radder (1990). Accardi on Quantum Theory and the "Fifth Axiom" of Probability. Philosophy of Science 57 (1):149-.
    In this paper we investigate Accardi's claim that the "quantum paradoxes" have their roots in probability theory and that, in particular, they can be evaded by giving up Bayes' rule, concerning the relation between composite and conditional probabilities. We reach the conclusion that, although it may be possible to give up Bayes' rule and define conditional probabilities differently, this contributes nothing to solving the philosophical problems which surround quantum mechanics.
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  30. John C. Bigelow (1979). Quantum Probability in Logical Space. Philosophy of Science 46 (2):223-243.
    Probability measures can be constructed using the measure-theoretic techniques of Caratheodory and Hausdorff. Under these constructions one obtains first an outer measure over "events" or "propositions." Then, if one restricts this outer measure to the measurable propositions, one finally obtains a classical probability theory. What I argue is that outer measures can also be used to yield the structures of probability theories in quantum mechanics, provided we permit them to range over at least some unmeasurable propositions. I thereby show that (...)
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  31. M. Bitbol (1988). The Concept of Measurement and Time Symmetry in Quantum Mechanics. Philosophy of Science 55 (3):349-375.
    The formal time symmetry of the quantum measurement process is extensively discussed. Then, the origin of the alleged association between a fixed temporal direction and quantum measurements is investigated. It is shown that some features of such an association might arise from epistemological rather than purely physical assumptions. In particular, it is brought out that a sequence of statements bearing on quantum measurements may display intrinsic asymmetric properties, irrespective of the location of corresponding measurements in time t of the Schrodinger (...)
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  32. Michel Bitbol (2011). Traces of Objectivity: Causality and Probabilities in Quantum Physics. Diogenes 58 (4):30-57.
  33. Eftichios Bitsakis (1988). Quantum Statistical Determinism. Foundations of Physics 18 (3):331-355.
    This paper attempts to analyze the concept of quantum statistical determinism. This is done after we have clarified the epistemic difference between causality and determinism and discussed the content of classical forms of determinism—mechanical and dynamical. Quantum statistical determinism transcends the classical forms, for it expresses the multiple potentialities of quantum systems. The whole argument is consistent with a statistical interpretation of quantum mechanics.
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  34. Niels Bohr (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review 48 (696--702):696--702.
  35. Max Born (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37 (12):863-867.
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  36. Mirjana Božić & Zvonko Marić (1995). Compatible Statistical Interpretation of a Wave Packet. Foundations of Physics 25 (1):159-173.
    A compatible statistical interpretation of a wave packet is proposed. De Broglian probabilities which unite wave and particle features of quantons are evaluated for free wave packets and Jor a superposition of wave packets. The obtained expressions provide a very plausible and physically appealing explanation of coherence in apparently incoherent beams and of the characteristic modulation of the momentum distribution, found recently in neutron interferometry combined with spectral filtering. Certain conclusions about dualism and objectivity in quantum domain are also derived.
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  37. A. J. Bracken & R. J. B. Fawcett (1993). Compact Quantum Systems and the Pauli Data Problem. Foundations of Physics 23 (2):277-289.
    Compact quantum systems have underlying compact kinematical Lie algebras, in contrast to familiar noncompact quantum systems built on the Weyl-Heisenberg algebra. Pauli asked in the latter case: to what extent does knowledge of the probability distributions in coordinate and momentum space determine the state vector? The analogous question for compact quantum systems is raised, and some preliminary results are obtained.
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  38. Charles J. Brainerd, Zheng Wang & Valerie F. Reyna (2013). Superposition of Episodic Memories: Overdistribution and Quantum Models. Topics in Cognitive Science 5 (4):773-799.
    Memory exhibits episodic superposition, an analog of the quantum superposition of physical states: Before a cue for a presented or unpresented item is administered on a memory test, the item has the simultaneous potential to occupy all members of a mutually exclusive set of episodic states, though it occupies only one of those states after the cue is administered. This phenomenon can be modeled with a nonadditive probability model called overdistribution (OD), which implements fuzzy-trace theory's distinction between verbatim and gist (...)
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  39. Paul Busch & Gregg Jaeger (2010). Unsharp Quantum Reality. Foundations of Physics 40 (9-10):1341-1367.
    The positive operator (valued) measures (POMs) allow one to generalize the notion of observable beyond the traditional one based on projection valued measures (PVMs). Here, we argue that this generalized conception of observable enables a consistent notion of unsharp reality and with it an adequate concept of joint properties. A sharp or unsharp property manifests itself as an element of sharp or unsharp reality by its tendency to become actual or to actualize a specific measurement outcome. This actualization tendency—or potentiality—of (...)
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  40. Nancy Cartwright (1978). The Only Real Probabilities in Quantum Mechanics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1978:54 - 59.
    Position probabilities play a privileged role in the interpretation of quantum mechanics. The standard interpretation has it that |Ψ (r)| 2 represents the probability that the system is at (or will be found at) the location r. Use of these probabilities, however, creates tremendous conceptual difficulties. It forces us either to adopt a non-standard logic, or to be saddled with an intractable measurement problem. This paper proposes that we try to eliminate position probabilities, and instead to interpret quantum mechanics (...)
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  41. Carlton M. Caves, Christopher A. Fuchs, Kiran K. Manne & Joseph M. Renes (2004). Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements. Foundations of Physics 34 (2):193-209.
  42. C. T. K. Chari (1971). Towards Generalized Probabilities in Quantum Mechanics. Synthese 22 (3-4):438 - 447.
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  43. Leon Cohen (1988). Rules of Probability in Quantum Mechanics. Foundations of Physics 18 (10):983-998.
    We show that the quantum mechanical rules for manipulating probabilities follow naturally from standard probability theory. We do this by generalizing a result of Khinchin regarding characteristic functions. From standard probability theory we obtain the methods usually associated with quantum theory; that is, the operator method, eigenvalues, the Born rule, and the fact that only the eigenvalues of the operator have nonzero probability. We discuss the general question as to why quantum mechanics seemingly necessitates different methods than standard probability theory (...)
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  44. Leon Cohen (1966). Can Quantum Mechanics Be Formulated as a Classical Probability Theory? Philosophy of Science 33 (4):317-322.
    It is shown that quantum mechanics cannot be formulated as a stochastic theory involving a probability distribution function of position and momentum. This is done by showing that the most general distribution function which yields the proper quantum mechanical marginal distributions cannot consistently be used to predict the expectations of observables if phase space integration is used. Implications relating to the possibility of establishing a "hidden" variable theory of quantum mechanics are discussed.
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  45. R. Eugene Collins (1977). Quantum Theory: A Hilbert Space Formalism for Probability Theory. [REVIEW] Foundations of Physics 7 (7-8):475-494.
    It is shown that the Hilbert space formalism of quantum mechanics can be derived as a corrected form of probability theory. These constructions yield the Schrödinger equation for a particle in an electromagnetic field and exhibit a relationship of this equation to Markov processes. The operator formalism for expectation values is shown to be related to anL 2 representation of marginal distributions and a relationship of the commutation rules for canonically conjugate observables to a topological relationship of two manifolds is (...)
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  46. Nicola Cufaro-Petroni (1992). On the Structure of the Quantum-Mechanical Probability Models. Foundations of Physics 22 (11):1379-1401.
    In this paper the role of the mathematical probability models in the classical and quantum physics is shortly analyzed. In particular the formal structure of the quantum probability spaces (QPS) is contrasted with the usual Kolmogorovian models of probability by putting in evidence the connections between this structure and the fundamental principles of the quantum mechanics. The fact that there is no unique Kolmogorovian model reproducing a QPS is recognized as one of the main reasons of the paradoxical behaviors pointed (...)
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  47. Michael E. Cuffaro (2014). Review Of: Christopher G. Timpson, Quantum Information Theory and the Foundations of Quantum Mechanics. [REVIEW] Philosophy of Science 81 (4):681-684,.
  48. Michael Dickson (2011). Aspects of Probability in Quantum Theory. In Claus Beisbart & Stephan Hartmann (eds.), Probabilities in Physics. Oxford University Press. 171.
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  49. William Michael Dickson (1998). Quantum Chance and Non-Locality: Probability and Non-Locality in the Interpretations of Quantum Mechanics. Cambridge University Press.
    This book examines in detail two of the fundamental questions raised by quantum mechanics. First, is the world indeterministic? Second, are there connections between spatially separated objects? In the first part, the author examines several interpretations, focusing on how each proposes to solve the measurement problem and on how each treats probability. In the second part, the relationship between probability (specifically determinism and indeterminism) and non-locality is examined, and it is argued that there is a non-trivial relationship between probability and (...)
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  50. Dennis Dieks, Décio Krause & Christian de Ronde (2014). Preface Special Issue Foundations of Physics. Foundations of Physics 44 (12):1245-1245.
    The foundations of quantum mechanics are attracting new and significant interest in the scientific community due to the recent striking experimental and technical progress in the fields of quantum computation, quantum teleportation and quantum information processing. However, at a more fundamental level the understanding and manipulation of these novel phenomena require not only new laboratory techniques but also new understanding, development and interpretation of the formalism of quantum mechanics itself, a mathematical structure whose connection to what happens in physical reality (...)
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