Complementarity and nonlocality in two-qudit systems

Complementarity and nonlocality are two characteristic traits of quantum physics that distinguish it from classical physics. In this paper, for the two-qubit case, we see that the complementarity between global and local observables in Bell's experiment sets a decisive foundation for the nonlocality of composite systems. We use the Hilbert-Schmidt norm on the commutator of two observables to quantify complementarity between them. Based on the CHSH experiment, we define a measure of complementarity Md for the two-qubit case, and extend it to two-qudit systems. Furthermore, we obtain an upper bound on Md that scales linearly in the Hilbert space dimension of the qudit.