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

Understand

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.