Probabilistic structure of bell inequalities in high dimension

Abstract

We reformulate the generalized Bell inequality for multipartite high-dimensional systems introduced by one of the authors in order to compare it with the Collins-Gisin-Linden-Massar-Popeseu (CGLMP) inequality. Quantum theory violates these inequalities that any local realistic theories must obey. We show that maximal entanglement leads to maximal violation of our Bell inequality whereas a non-maximally entangled state maximally violates the CGLMP inequality. In addition, we analyze the probabilistic structures of the two types by representing them in terms of joint probabilities and correlation weights. We show that both types have the equivalent probabilistic structure with respect to the joint probabilities, but they have different correlation weights for the measurement outcomes. We remark that the correlation weight plays a crucial role in determining the violation conditions.