Tight Bound on Randomness for Violating the Clauser–Horne–Shimony–Holt 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: 1) the device settings of the two parties in the Bell test are independent and 2) 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.