Tight Bound on Randomness for Violating the CHSH Inequality

Free will (or randomness) has been studied to achieve loophole-free Bell's inequality test and to provide device-independent quantum key distribution security proofs. The required 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. Using some information theoretic techniques, we prove in this paper a tight bound on the required randomness for this case such that the CHSH inequality can be violated by certain LHVM. Our proof has a clear achievability and converse style. The achievability part is proved using type counting. To prove the converse part, we introduce a concept called profile for a set of binary sequences and study the properties of profiles. Our profile-based converse technique is also of independent interest.