Statistical Link Between Bell Nonlocality and Uncertainty Relations in Quantum Mechanics

Bell nonlocality refers to the nonlocal correlation strength among quantum systems that cannot be duplicated using local hidden variables, and uncertainty relations set the lower bound on the sum or product of variances for incompatible observables within a quantum system. In this work, the study of linking these two fundamental quantum features in terms of statistical characteristics within Bell tests is initiated. Two statistics determine the state-dependent quantum upper bounds of the Bell values: the expectation values of local observables, and the joint uncertainty that is the product of local uncertainty. Specifically, it is showed that the joint measurement uncertainty determines the upper bounds of covariant Bell inequalities in the quantum region, with more joint measurement uncertainty required for more covariant correlation strengths. Conversely, the covariant Bell value defines the lower bound of joint uncertainty. In addition, given the maximally entangled states, Tsirelson's bounds of Bell inequalities can be achieved when the joint uncertainty reaches the maximum.