Tight Bound on Tilted CHSH Inequality with Measurement Dependence

Measurement dependence is the property that the distribution of the underlying variable is correlated with the measurement settings. If there is measurement dependence in a Bell test, the upper bound of Bell inequality will be affected. In the framework of device-independent quantum random number generation, the violation of Bell inequality indicates the existence of certified randomness. A tight upper bound on Bell inequality is critical to the amount of certified randomness. Recently, Sadhu and Das (2023) obtained a modified tilted CHSH inequality when measurement dependence exists. However, the upper bound of the modified tilted CHSH inequality they obtained is not tight. In this paper, we propose a method to quantify the measurement dependence applicable to the tilted CHSH inequality based on its structure. After that, we obtain a tight upper bound of modified tilted CHSH inequality and construct a local deterministic strategy to reach the upper bound.