Does CHSH inequality test the model of local hidden variables?

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.