Does CHSH Inequality Test the Model of Local Hidden Variables?(General and Mathematical Physics)

概要

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It is pointed out that the local hidden-variable model of Bell and Clauser-Horne-Shimony-Holt (CHSH) gives ||≤2√<2> or ||≤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 ||≤2 does not provide a reliable test of the d=4 local non-contextual hidden-variable model. To achieve ||≤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 ||≤2, but no more a proper hidden-variable model in d=4. The conventional CHSH inequality ||≤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.
2012-06-25