  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.
