Survey on the Bell nonlocality of a pair of entangled qudits

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.

Multipartite entanglement analysis from random correlations

“…However, there exist bi-separable four-qubit states that violate this hypothetical bound. We found by a numerical study that the inequality satisfied by bi-separable four qubit states has the following dependence on the purity, M 4 8 81 ð1 À P 2 Þ: (11) This bound is also tight for P ! 5 8 and achieved by, e.g., the state p ϕ þ j i 12 hϕ þ j jϕ þ i 34 hϕ þ j þ ð1 À pÞjϕ þ i 13 hϕ þ j jϕ þ i 24 ϕ þ h j.…”

“…Under the same constraints, also adaptive methods for entanglement detection have been developed 8,9 . In the absence of any reference frames Bell violations can be measured with some probability 10,11 and entanglement can be detected by evaluating the second moment of the distribution of correlations obtained by measuring random observables on each subsystem [12][13][14][15][16][17] . Furthermore, it has been shown recently that higher-order moments of this distribution allow discrimination of very specific types of multipartite entanglement 18 .…”

Experimentally friendly approach towards nonlocal correlations in multisetting N-partite Bell scenarios

“…The probability of violation of local realism under random measurements, proposed in [12,13], has gained considerable attention as an operational measure of nonclassicality of quantum states [14]. It has been demonstrated both numerically [15][16][17][18] and analytically [14,19] that this quantity is a good candidate for a nonlocality measure. Furthermore, in [19] it was proved that this quantifier satisfies some natural properties and expectations for an operational measure of nonclassicality, e.g., invariance under local unitaries.…”

Strength and typicality of nonlocality in multisetting and multipartite Bell scenarios

“…The probability of violation of local realism under random measurements, proposed in [1], has gained considerable attention as an operational measure of nonclassicality of quantum states [2]. It has been demonstrated both numerically [3][4][5] and analytically [2,6] that this quantity is a good candidate for a nonlocality measure. What is more, in [6] it was proved that this quantifier satisfies some natural properties and expectations for an operational measure of nonclassicality.…”