Adv. Studies Theor. Phys., Vol. 4, 2010, no. 20, 945 949. CHSH and local hidden causality J.F. Geurdes C. vd. Lijnstraat 164 2593 NN Den Haag Netherlands han.geurdes@gmail.com Abstract Mathematics equivalent to Bell's derivation of the inequalities, also allows a local hidden variables explanation for the correlation between distant measurements. Keywords: Bell inequalities, classical probability, EPR paradox. 1 Introduction Bell inequalities [2] are a well studied subject. To many the experimental verification of the violation of inequalities e.g. [1], [5] is sufficient evidence for the completeness of quantum theory. Here, it will be demonstrated that Bell's form of local hidden correlation P (~a,~b) = ∫ λ∈Λ ρλAλ(~a)Bλ(~b)dλ (1) can be transformed to violate Bell's inequality. We have, ~a and ~b for unitary parameter vectors of e.g. Stern-Gerlach magnets in an ortho-positronium decay experiment. λ represents the extra hidden parameters in a set Λ. The probability density ρλ is a classical density. The measurement functions Aλ(~a) and Bλ(~b) project in {−1, 1}. Bell showed, using the expression below, that models with a classical probability density may not violate the inequality1. P (~a,~b)− P (~x, ~y) = ∫ λ∈Λ ρλAλ(~a)Bλ(~b)Aλ(~x)Bλ(~y) { Aλ(~x)Bλ(~y)− Aλ(~a)Bλ(~b) } (2) 1If there is no confusion the dλ will be suppressed. 2 J.F. Geurdes 1.1 Singlet state Bell inequality Bell expressed the singlet state of the electron and positron in the positronium as ∀ : ~a(|~a| = 1)∀ : λ(λ ∈ Λ) {Aλ(~a) +Bλ(~a) = 0}. The following steps are elementary. Let us take, ~x = ~b and ~y = ~c. With the singlet, we see that equation (2) can be written as P (~a,~b)− P (~b,~c) = ∫ λ∈Λ ρλ { Aλ(~b)Aλ(~c)− Aλ(~a)Aλ(~b) } (3) Or, noting 1− Aλ(~a)Aλ(~c) ≥ 0,∣∣∣P (~a,~b)− P (~b,~c)∣∣∣ ≤ ∫ λ∈Λ ρλ ∣∣∣Aλ(~c)Aλ(~b)∣∣∣ {1− Aλ(~a)Aλ(~c)} (4) Because, ∣∣∣Aλ(~c)Aλ(~b)∣∣∣ = 1 and ρλ classical, we have the Bell inequality∣∣∣P (~a,~b)− P (~b,~c)∣∣∣ ≤ 1 + P (~a,~c) (5) The quantum correlation is: Pqm(~x, ~y) = − (~x * ~y). If in two-dimensions, ~a =( −1√ 2 , 1√ 2 ) , ~b = ( 1√ 2 , 1√ 2 ) and ~c = (0, 1), then, inequality is violated because,∣∣∣0− −1√ 2 ∣∣∣ ≤ 1− 1√ 2 is false. Associated to this inequality in equation(5) a more general inequality, the CHSH inequality [3], exists. The principle is the same. 2 Sets and Integrals Keeping an eye on equation (2), hidden parameters sets can be defined Ω± = { λ ∈ Λ|Aλ(~a)Bλ(~b) = Aλ(~x)Bλ(~y) = ±1 } (6) and Ω0 = { λ ∈ Λ|Aλ(~a)Bλ(~b) = −Aλ(~x)Bλ(~y) = ±1 } (7) Given, ~a, ~b, ~x and ~y, either, Aλ(~a)Bλ(~b) = Aλ(~x)Bλ(~y) or Aλ(~a)Bλ(~b) = −Aλ(~x)Bλ(~y) for arbitrary, λ ∈ Λ. Moreover, Aλ(~a)Bλ(~b) = ±1 for arbitrary, λ ∈ Λ. Hence, Λ = Ω0 ∪ Ω+ ∪ Ω− and equation (2) is P (~a,~b)−P (~x, ~y) = ∫ λ∈Ω0 ρλAλ(~a)Bλ(~b)Aλ(~x)Bλ(~y) { Aλ(~x)Bλ(~y)− Aλ(~a)Bλ(~b) } (8) From Ω0 folowsAλ(~a)Bλ(~b)Aλ(~x)Bλ(~y) = −1 and { Aλ(~x)Bλ(~y)− Aλ(~a)Bλ(~b) } = 2Aλ(~x)Bλ(~y). Hence, P (~a,~b)− P (~x, ~y) = −2 ∫ λ∈Ω0 ρλAλ(~x)Bλ(~y) (9) CHSH & local hidden causality 3 Suppose, P (~a,~b) = 0, as 'starting position' in the experiment. This gives a reformulation of P (~x, ~y) where ~x and ~y are different form ~a and ~b. Hence, P (~x, ~y) = 2 ∫ λ∈Ω 0|P (~a,~b)=0 ρλAλ(~x)Bλ(~y) (10) Note that according to equation (1) and the Ω sets we may write for P (~a,~b) = 0 P (~a,~b) = 0 = ∫ λ∈Ω 0|P (~a,~b)=0 ρλAλ(~a)Bλ(~b) + ∫ λ∈Ω +|P (~a,~b)=0 ρλ − ∫ λ∈Ω−|P (~a,~b)=0 ρλ (11) Moreover, generally P (~x, ~y) 6= P (~a,~b) which follows from comparing equation (10) with (11). Because, in Ω0, we see for arbitary λ ∈ Ω0 that Aλ(~a)Bλ(~b) = −Aλ(~x)Bλ(~y) = ±1, it follows from equation (11) that we may rewrite P (~x, ~y) as 1 2 P (~x, ~y) = ∫ λ∈Ω +|P (~a,~b)=0 ρλ − ∫ λ∈Ω−|P (~a,~b)=0 ρλ (12) Equations (6) and (7) show that the Ω sets depend on ~a, ~b, ~x and ~y. Given P (~a,~b) = 0, this fixes the ~a and ~b. Hence, Ω±|P (~a,~b)=0 = Ω±|P (~a,~b)=0(~x, ~y), implicit in equation(12). Start the experiment with two parameters ~a and ~b that produces the condition P (~a,~b) = 0 and let ~x and ~y free2. ~x does not afect Bλ(~y) and vice versa, hence, no locality violation. 3 Violation CHSH We will show that there is a classical probability density that allows violation of the CHSH |D| ≤ 2, with, D = P (1A, 1B)− P (1A, 2B)− P (2A, 1B)− P (2A, 2B) (13) Here, 1A(B) and 2A(B) are unitary vectors randomly selected by A(B). 3.1 Probability density We postulate a density for (λ1, λ2) ∈ [−1√2 , 1√ 2 ]× [−1√ 2 , 1√ 2 ] = Λ with n = 1, 2 ρλn = { 1√ 2 , −1√ 2 ≤ λn ≤ 1√2 0, elsewhere (14) This density is Kolmogorovian. 2see the discussion section 4 J.F. Geurdes 3.2 Selection of parameters We establish the parameter vectors that the observers A and B will use. For A, 1A = (1, 0) and 2A = (0, 1). For B, 1B = ( 1√ 2 , −1√ 2 ) and 2B = ( −1√ 2 , −1√ 2 ). If we take the quantum correlation, it follows, Pqm(1A, 1B) = −1√ 2 , Pqm(1A, 2B) = 1√ 2 , Pqm(2A, 1B) = 1√ 2 and Pqm(2A, 2B) = 1√ 2 . Quantum mechanics violates |D| ≤ 2, because |D| = 2 √ 2 is found. Because, ρλ1ρλ2 = 1 2 for (λ1, λ2) ∈ [−1√ 2 , 1√ 2 ] × [−1√ 2 , 1√ 2 ] and Ω±|P (~a,~b)=0(~x, ~y) ⊂ [ −1√ 2 , 1√ 2 ] × [−1√ 2 , 1√ 2 ], we obtain from equation (12) P (~x, ~y) = ∫ λ∈Ω +|P (~a,~b)=0(~x,~y) dλ1dλ2 − ∫ λ∈Ω−|P (~a,~b)=0(~x,~y) dλ1dλ2 (15) If, subsequently, observer A selects 1A, then the hidden parameter λ1 is in [−1√ 2 , 1− 1√ 2 ] ⊂ [−1√ 2 , 1√ 2 ]. If, A selects 2A then λ1 is in [−1 + 1√2 , 1√ 2 ] ⊂ [−1√ 2 , 1√ 2 ]. Similarly, if B selects 1B, then then λ2 is in [0, 1√ 2 ] ⊂ [−1√ 2 , 1√ 2 ]. Finally, if B selects 2B, then λ2 is found in [ −1√ 2 , 0] ⊂ [−1√ 2 , 1√ 2 ]. The intervals responding to settings do not violate locality: A settings are associated to λ1 intervals, B settings to λ2 intervals. Suppose A selects 1A and B selects 1B. We turn to Ω±|P (~a,~b)=0(1A, 1B). If, Ω+|P (~a,~b)=0(1A, 1B) = ∅ and Ω−|P (~a,~b)=0(1A, 1B) = [−1√ 2 , 1− 1√ 2 ]×[0, 1√ 2 ], from equation (15) it follows that P (1A, 1B) = −1√ 2 . Hence, a selection of Ω±|P (~a,~b)=0(~x, ~y) is possible giving |D| > 2. 4 Conclusion and discussion The result of violating |D| ≤ 2 with proper Ω±|P (~a,~b)=0(~x, ~y) and locality obeying interval selection rules, is surprising. The mathematics was similar to the one used by Bell [2]. Moreover, no violations of locality were introduced. In a random selection experiment there is a non-zero probability that, combined with the deterministic interval selection, a proper selection of Ω±|P (~a,~b)=0(~x, ~y) is obtained. When Bell's reasoning is sound, no violation should be possible at all with the use of classical local hidden models given the employed parameters. Note that other violating instances can be treated similarly. If there can be no reasons given why locality and causality selections of Ω±|P (~a,~b)=0(~x, ~y) are impossible, then a local hidden variable explanation of experiments cannot be excluded. The transformation of (1) is based on a single fixing of ~a and ~b, independent of the ~x and ~y. If one assumes that the functional form of Aλ(*) and Bλ(*) changes in time (see also [4] for the role of time in Bell's theorem) then the fixing of P (~a,~b) = 0 can take place at times different than the measurement parameters selection and the sets in equations (6) and (7) will always be possible. CHSH & local hidden causality 5 References [1] A. Aspect, J. Dalibard, G. Roger, Experimental test of Bell's inequalities using timevarying analyzers, Phys. Rev. Lett. 49 (1982) 1804-1806. [2] J.S. Bell, On the Einstein Podolsky and Rosen paradox, Physics 1 (1964) 195-200. [3] J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23 (1969) 880-884. [4] K. Hess and W. Philipp, A possible loophole in the theorem of Bell, PNAS 98 (2001) 14224-14227. [5] G. Weihs, J. Jennewein, C. Simon, H. Weinfurter and A. Zeilinger, Violation of Bell's inequality under strict Einstein conditions, Phys. Rev. Lett. 81 (1998) 5039-5043. Received: Month 09,