Reexamination of a multisetting Bell inequality for qudits

TL;DR

This paper reexamines a class of Bell inequalities for qudits, correcting classical bounds, analyzing their tightness, and exploring quantum violations to better understand quantum entanglement detection.

Contribution

It provides corrected classical bounds, identifies which inequalities are facet-inducing, and assesses quantum violations, advancing the understanding of Bell inequalities for qudits.

Findings

Classical bounds for d ≤ 13 are explicitly calculated.

Most inequalities are tight for prime d in the specified range.

Quantum violations are bounded, impacting entanglement detection.

Abstract

The class of d-setting, d-outcome Bell inequalities proposed by Ji and collaborators [Phys. Rev. A 78, 052103] are reexamined. For every positive integer d > 2, we show that the corresponding non-trivial 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 less than or equal to 13 (including non-prime 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.