Quantum Discord, CHSH Inequality and Hidden Variables -- Critical reassessment of hidden-variables models

TL;DR

This paper critically reassesses hidden-variables models in quantum mechanics, revealing their limitations in describing quantum discord and clarifying the non-uniqueness of CHSH inequality predictions, especially in relation to non-contextual models and separable states.

Contribution

It demonstrates the inconsistency of Bell's hidden-variable model in 2D for quantum discord and clarifies the non-uniqueness of CHSH inequality predictions in 4D, emphasizing the absence of viable local non-contextual models.

Findings

Hidden-variable models in 2D cannot fully describe quantum discord.

CHSH inequality predictions are non-unique in 4D due to non-linearity.

No viable local non-contextual hidden-variable models exist in any dimension.

Abstract

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 $d=2$. The criterion of vanishing quantum discord is related to the notion of reduction and, surprisingly, the hidden-variable model in $d=2$, 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 $|B|\leq 2$ for the CHSH operator $B$ 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 $d=4$ used by Bell and CHSH, in agreement with Gleason's theorem which excludes $d=4$ non-contextual hidden-variables models. If one imposes the linearity condition, their model is converted to a factored product of two $d=2$ models which describes quantum mechanical separable states. The CHSH inequality thus does not test the hidden-variables model in $d=4$. 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.