Reexamination of a multisetting Bell inequality for qudits

Abstract

The class of $d$-setting, $d$-outcome Bell inequalities proposed by Ji and co-workers [Phys. Rev. A 78, 052103 (2008)] are reexamined. For every positive integer $d>2$, we show that the corresponding nontrivial Bell inequality for probabilities provides the maximum classical winning probability of the Clauser-Horne-Shimony-Holt-like game with $d$ inputs and $d$ outputs. We also demonstrate that the general classical upper bounds given by Ji et al. are underestimated, which invalidates many of the corresponding correlation inequalities presented thereof. We remedy this problem, partially, by providing the actual classical upper bound for $d\ensuremath{\le}13$ (including nonprime values of $d$). We further determine that for prime value $d$ in this range, most of these probability and correlation inequalities are tight, i.e., facet-inducing for the respective classical correlation polytope. Stronger lower and upper bounds on the quantum violation of these inequalities are obtained. In particular, we prove that once the probability inequalities are given, their correlation counterparts given by Ji and co-workers are no longer relevant in terms of detecting the entanglement of a quantum state.