Towards an equivalence between maximal entanglement and maximal quantum nonlocality

Abstract

While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality.In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.

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

“…Consequently, for a scenario with 3 × 3 × 3 × 2 measurement settings the nonlocal fraction is close to unity. It means that our results satisfy the theorem of Lipinska et al [19] that P V → 1 with infinitely many settings. The explanation of this fact once again has a purely statistical background, i.e.…”

“…(ii) As shown in Ref. [15,19] the nonlocal fraction increases rapidly when the number of measurement settings grows. This property might arise either by the increase of the number of equivalent Bell inequalities or by the emergence of new types of Bell inequalities.…”

“…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. The probability of violation is also often called volume of violation [14] or nonlocal fraction [18,19]. In this paper we will use the latter name.…”

“…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.…”

“…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. The probability of violation is also often called volume of violation [14] or nonlocal fraction [18,19].…”