Optimal free-will on one side in reproducing the singlet correlation

TL;DR

This paper demonstrates that simulating singlet correlations with hidden variables depends on the degree of measurement independence, showing that 59% independence is optimal when only one party's measurements influence the hidden variables.

Contribution

It establishes the optimal measurement independence threshold of 59% for simulating singlet correlations with one-sided measurement dependence.

Findings

Optimal measurement independence is 59% for one-sided models.

Existing distributions can achieve this optimal lack of free will.

The result relates to the detection efficiency loophole in quantum experiments.

Abstract

Bell's theorem teaches us that there are quantum correlations that can not be simulated by just shared randomness (Local Hidden variable). There are some recent results which simulate singlet correlation by using either 1 cbit or a binary (no-signaling) correlation which violate Bell's inequality maximally. But there is one more possible way to simulate quantum correlation by relaxing the condition of independency of measurement on shared randomness. Recently, MJW Hall showed that the statistics of singlet state can be generated by sacrificing measurement independence where underlying distribution of hidden variables depend on measurement direction of both parties [Phys. Rev. Lett.105 250404 (2010)]. He also proved that for any model of singlet correlation, 86% measurement independence is optimal. In this paper, we show that 59% measurement independence is optimal for simulating singlet correlation when the underlying distribution of hidden variables depend only on measurements of one party. We also show that a distribution corresponding to this optimal lack of free will, already exists in literature which first appeared in the context of detection efficiency loophole.