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arXiv:1204.3437 · Progress of Theoretical Physics

Does CHSH Inequality Test the Model of Local Hidden Variables?

K. Fujikawa

Published Jun 1, 2012• 6 citations
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Abstract

It is pointed out that the local hidden-variable model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives |≪B >| ≤2√2 or |≪B >| ≤2 for the quantum CHSH operator B = a·σ ⊗(b + b′) ·σ + a′ ·σ ⊗ (b - b′) ·σ depending on two different ways of evaluation, when it is applied to a d = 4 system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality |≪B >| ≤2 does not provide a reliable test of the d = 4 local non-contextual hidden-variable model. To achieve |≪B >| ≤2 uniquely, one needs to impose a linearity requirement on the hidden-variable model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual d = 2 hidden-variable models. This factored product implies pure separable quantum states and satisfies |≪B >| ≤2, but no more a proper hidden-variable model in d = 4. The conventional CHSH inequality |≪B >| ≤2 thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in d = 4, to be consistent with Gleason's theorem which excludes non-contextual models in d = 4. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.

DOI
10.1143/ptp.127.975

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It is pointed out that the local hidden-variable model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives |≪B >| ≤2√2 or |≪B >| ≤2 for the quantum CHSH operator B = a·σ ⊗(b + b′) ·σ + a′ ·σ ⊗ (b - b′) ·σ depending on two different ways of evaluation, when it is applied to a d = 4 system of two spin-1/2 particles. This is due to the failure of linearity, and it shows that the conventional CHSH inequality |≪B >| ≤2 does not provide a reliable test of the d = 4 local non-contextual hidden-variable model. To achieve |≪B >| ≤2 uniquely, one needs to impose a linearity requirement on the hidden-variable model, which in turn adds a von Neumann-type stricture. It is then shown that the local model is converted to a factored product of two non-contextual d = 2 hidden-variable models. This factored product implies pure separable quantum states and satisfies |≪B >| ≤2, but no more a proper hidden-variable model in d = 4. The conventional CHSH inequality |≪B >| ≤2 thus characterizes the pure separable quantum mechanical states but does not test the model of local hidden variables in d = 4, to be consistent with Gleason's theorem which excludes non-contextual models in d = 4. This observation is also consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to hidden variables.
Hidden-variables models are critically reassessed. It is first examined if the quantum discord is classically described by the hidden-variable model of Bell in the Hilbert space with d=2d=2d=2. The criterion of vanishing quantum discord is related to the notion of reduction and, surprisingly, the hidden-variable model in d=2d=2d=2, which has been believed to be consistent so far, is in fact inconsistent and excluded by the analysis of conditional measurement and reduction. The description of the full contents of quantum discord by the deterministic hidden-variables models is not possible. We also re-examine CHSH inequality. It is shown that the well-known prediction of CHSH inequality ∣B∣≤2|B|\leq 2∣B∣≤2 for the CHSH operator BBB introduced by Cirel'son is not unique. This non-uniqueness arises from the failure of linearity condition in the non-contextual hidden-variables model in d=4d=4d=4 used by Bell and CHSH, in agreement with Gleason's theorem which excludes d=4d=4d=4 non-contextual hidden-variables models. If one imposes the linearity condition, their model is converted to a factored product of two d=2d=2d=2 models which describes quantum mechanical separable states. The CHSH inequality thus does not test the hidden-variables model in d=4d=4d=4. This observation is consistent with an application of the CHSH inequality to quantum cryptography by Ekert, which is based on mixed separable states without referring to dispersion-free representations. As for hidden-variables models, there exist no viable local non-contextual models in any dimensions.
In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nthn\mathrm{th}nth particle pair would be independent of measurement choices and outcomes for the first $(n\ensuremath{-}1)$ pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the two-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the two-sided memory loophole allows a systematic violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality when the data are analyzed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in the existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearized CHSH inequality as well as other Bell-type inequalities.
The Bell-Clauser-Horne-Shimony-Holt (BCHSH) inequality, which is proven in the context of the local hidden-variable theory, has been used as a test to reveal the failure of the hidden-variable theory and to prove the validity of the quantum theory. We note that the violation of the BCHSH inequality is caused by the noncommutativity of quantum observables and find a systematic method of constructing generalizations of the BCHSH inequality. This method is used to invent a new quantity, which is defined in a two-qubit system and satisfies a new type of inequality. This provides a fair test of the quantum theory. Remaining problems are also discussed.
The validity of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality for hidden-variable models of the Bell type and the quantum prediction that said inequality can be violated are two indisputable assertions. However, one can prove wrong the assertion that no hidden-variable model is able to reproduce quantum predictions about the Bell-CHSH test. Here, we reproduce all quantum predictions of Bell-CHSH tests by means of the Bell-Kochen-Specker hidden-variable model for a qubit plus Einstein-Podolsky-Rosen “criterion of physical reality”.
A hidden variables (HVs) model is reported, which reproduces quantum predictions for Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) tests. The existence of such a model poses some limitations to quantum certifications that rely on Bell-CHSH inequality violations. The reported model does not prove wrong Bell's theorem. The latter assumes the factorability of the probability density pABp_{AB}pAB​, which rules the stochastic behavior of the HVs. The reported HVs model is based on an extended form of pABp_{AB}pAB​, which is suggested by Lebesgue's decomposition theorem for bounded functions. The considered pABp_{AB}pAB​ complies with locality and realism, and also with measurement independence, parameter independence and outcome independence.
It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in d=4d=4d=4 such as the one used by Bell always gives rise to CHSH inequality ∣⟨B⟩∣≤2|\langle B\rangle|\leq 2∣⟨B⟩∣≤2. On the other hand, the contraposition of Gisin's theorem states that the inequality ∣⟨B⟩∣≤2|\langle B\rangle|\leq 2∣⟨B⟩∣≤2 for arbitrary parameters implies (pure) separable quantum states. The fact that local realism can describe only pure separable quantum states is naturally established in hidden-variables models, and it is quantified by G(a,b)=4[⟨ψ∣P(a)⊗P(b)∣ψ⟩−⟨ψ∣P(a)⊗1∣ψ⟩⟨ψ∣1⊗P(b)∣ψ⟩]=0G({\bf a},{\bf b})= 4[\langle ψ|P({\bf a})\otimes P({\bf b})|ψ\rangle-\langle ψ|P({\bf a})\otimes{\bf 1}|ψ\rangle\langle ψ|{\bf 1}\otimes P({\bf b})|ψ\rangle]=0G(a,b)=4[⟨ψ∣P(a)⊗P(b)∣ψ⟩−⟨ψ∣P(a)⊗1∣ψ⟩⟨ψ∣1⊗P(b)∣ψ⟩]=0 for any two projection operators P(a)P({\bf a})P(a) and P(b)P({\bf b})P(b). The test of local realism by the deviation of G(a,b)G({\bf a},{\bf b})G(a,b) from G(a,b)=0G({\bf a},{\bf b})=0G(a,b)=0 is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.