Survey on the Bell nonlocality of a pair of entangled qudits

TL;DR

This survey analyzes how Bell nonlocality in bipartite entangled qudits diminishes with increasing dimension, comparing specific inequalities and probability spaces, revealing different decay patterns and the non-maximal entangled states' behavior.

Contribution

It provides a comprehensive comparison of Bell nonlocality in higher-dimensional systems using two approaches, highlighting the decay patterns and properties of various entangled states.

Findings

Nonlocality decreases with dimension, exponentially in CGLMP scenario.

Linear decay of nonlocality in the probability space approach.

Maximally entangled states do not always produce maximal violations.

Abstract

The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from $d=2$ to $d=7$. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective, by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension $d$ grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviours. Furthermore, in both cases the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison with maximally entangled states, so, no anomaly is observed. Finally, the nonlocality of states with non-maximal Schmidt rank is investigated.