Tight Bound on Randomness for Violating the CHSH Inequality

Yifeng Teng, Siwei Wang, Mingfei Zhao, Shenghao Yang

Published 2015-04-21Version 1

Free will has been studied to achieve loophole-free Bell's inequality test and to provide device-independent quantum key distribution security proofs. The required free will (or randomness) such that a local hidden variable model (LHVM) can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality has been studied, but a tight bound has not been proved for a practical case that i) the device settings of the two parties in the Bell test are independent; and ii) the device settings of each party can be correlated or biased across different runs. For a randomness measure related to min-entropy, we prove in this paper a tight bound of the required randomness for this case such that the CHSH inequality can be violated by certain LHVM. Our technique used in the proof is also of independent interest.