Quantum complementarity yields Tsirelson bounds on Bell-type inequalities

Abstract

We show that quantum complementarity defines the boundaries of quantum correlations. We introduce a complementarity principle (CP), which states that for mutually complementary observables with measurement values of $\pm 1$, the sum of their squared expectation values cannot exceed 1. Using the CP, we derive the Tsirelson bounds for violations of the CHSH, Mermin, and Svetlichny inequalities. We show that the CP is closely related to the uncertainty principle, providing an uncertainty relation for complementary dichotomic operators. Our results offer new insights into the relation between the uncertainty principle and quantum correlations. Compared to the Exclusivity Principle (EP), which relies on graph-theoretic techniques such as the Lovász number and is limited by spacelike-separated measurement constraints that prevent the upper bound from reaching the Lovász number, our method bypasses the graph theory. It offers a simpler, more efficient approach to determining the maximum quantum violations of Bell-type inequalities, while naturally incorporating measurement constraints and enabling direct determination of Tsirelson bounds, making it more versatile for a broader range of complex Bell-type inequalities.