Effects of measurement dependence on tilted CHSH Bell tests

Device-independent (DI) randomness generations based on tilted CHSH Bell tests, different from other Bell tests, use a little entanglement but can produce quite a lot of certified randomness. In all these protocols, the violations of Bell inequalities which imply the generation of certified randomness rely on an assumption, i.e., measurement independence. Since ensuring measurement independence in a practical Bell test is difficult, it is crucial to explore effects of relaxing this assumption (called as measurement dependence). Naturally, a question, how measurement dependence affects tilted CHSH Bell tests, arise. In this paper, we answer this question. Concretely, we introduce measurement dependence with flexible lower bounds which are different from fixed lower bound and then qualify the effect of measurement dependence on tilted CHSH Bell tests with different input distributions. The results show that the effect of measurement dependence on tilted CHSH Bell tests with flexible lower bound of measurement dependence is more powerful than that of only considering fixed lower bound. The relevance of this work highlights the study of DI randomness amplification and other DI quantum information tasks.