Tight Bound on Randomness for Violating the CHSH Inequality

TL;DR

This paper establishes a precise limit on the amount of randomness needed for local hidden variable models to violate the CHSH inequality, considering realistic device setting assumptions.

Contribution

It provides the first tight bound on randomness required for CHSH violation under independent and biased device settings, using novel information theoretic and profile-based techniques.

Findings

Proved a tight bound on randomness for CHSH violation.

Developed a profile-based converse technique for binary sequences.

Achieved a clear achievability and converse proof structure.

Abstract

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.