A hidden-variables version of Gisin's theorem

TL;DR

This paper explores a hidden-variables approach to Gisin's theorem, demonstrating that local realism models can only describe pure separable states and proposing an efficient test for local realism using existing experimental setups.

Contribution

It introduces a hidden-variables version of Gisin's theorem, linking local realism to separable states and proposing a practical test for local realism deviations.

Findings

Local realism models only describe pure separable states.

Deviation of a specific function G from zero tests local realism.

Existing experiments can efficiently test the proposed local realism criterion.

Abstract

It is generally assumed that {\em local realism} represented by a noncontextual and local hidden-variables model in $d=4$ such as the one used by Bell always gives rise to CHSH inequality $|\langle B\rangle|\leq 2$. On the other hand, the contraposition of Gisin's theorem states that the inequality $|\langle B\rangle|\leq 2$ 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 $G({\bf a},{\bf b})= 4[\langle \psi|P({\bf a})\otimes P({\bf b})|\psi\rangle-\langle \psi|P({\bf a})\otimes{\bf 1}|\psi\rangle\langle \psi|{\bf 1}\otimes P({\bf b})|\psi\rangle]=0$ for any two projection operators $P({\bf a})$ and $P({\bf b})$. The test of local realism by the deviation of $G({\bf a},{\bf b})$ from $G({\bf a},{\bf b})=0$ is shown to be very efficient using the past experimental setup of Aspect and his collaborators in 1981.