Bell nonlocality measure of density matrix based on trace distance

Abstract Quantum information technology exhibits advantages in terms of algorithm efficiency, communication security, and other aspects compared to classical information technology, partially due to quantum nonlocality, which is a characteristic that distinguishes quantum mechanics from classical mechanics. Compared to quantum entanglement and quantum steering, Bell nonlocality exhibits stronger quantum properties in a certain sense. However, research on efficient estimation of Bell nonlocality of general quantum states remains rare. This work proposes a generalized Bell nonlocality measure of density matrices. Our method generalizes the fundamental matrix for classical correlation and applies it to experimental scenarios of various dimensions. We prove that this measure of Bell nonlocality is well-defined. We also propose a relaxation algorithm for numerically estimating this measure. Applying the see-saw method, the algorithm obtains an upper bound on this measure. Numerical experiments show the effectiveness of our method in low dimensions and suggest that it could be used in high dimensions.