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Physical Review A

Bell-nonlocality quantifiers and their persistent mismatch with the entropy of entanglement

Ari Patrick, Giulio Camillo, Fernando Parisio, Bárbara Amaral

Published Apr 7, 2023• 1 citation
Publisher

Abstract

For some commonly used nonlocality quantifiers, the maximally nonlocal state may not coincide with the maximally entangled state, what became known as the anomaly of nonlocality. More recently, nonanomalous quantifiers have been defined, for which maximal entanglement and maximal nonlocality do coincide. In this work we investigate in detail how these quantifiers behave for nonmaximal-resource states and show that some mismatch remains between the studied figures of merit. Besides the nonlocal volume (a quantifier that counts the volume of the set of behaviors accessible to a specific state that gives rise to nonlocality), we present an alternative quantity referring to states and based on behaviors, the trace-weighted nonlocal volume. The construction is based on the nonlocal volume integral, but weighted by a quantifier of nonlocality for behaviors (the shortest distance between the behavior and the local set). Although in all investigated scenarios the anomaly is not present, the nonlocality quantifiers as compared to the entropy of entanglement are inversely related for some set of states within some Bell scenarios. In the investigated situations this phenomenon occurs when the considered states go through a change of rank. This fact is discussed in general and illustrated through computations for the (2,3,2) scenario.

DOI
10.1103/physreva.107.042410

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For some commonly used nonlocality quantifiers, the maximally nonlocal state may not coincide with the maximally entangled state, what became known as the anomaly of nonlocality. More recently, nonanomalous quantifiers have been defined, for which maximal entanglement and maximal nonlocality do coincide. In this work we investigate in detail how these quantifiers behave for nonmaximal-resource states and show that some mismatch remains between the studied figures of merit. Besides the nonlocal volume (a quantifier that counts the volume of the set of behaviors accessible to a specific state that gives rise to nonlocality), we present an alternative quantity referring to states and based on behaviors, the trace-weighted nonlocal volume. The construction is based on the nonlocal volume integral, but weighted by a quantifier of nonlocality for behaviors (the shortest distance between the behavior and the local set). Although in all investigated scenarios the anomaly is not present, the nonlocality quantifiers as compared to the entropy of entanglement are inversely related for some set of states within some Bell scenarios. In the investigated situations this phenomenon occurs when the considered states go through a change of rank. This fact is discussed in general and illustrated through computations for the (2,3,2) scenario.
We define an alternative way of quantifying nonlocality of states based on Bell nonlocality of behaviors, called the trace-weighted nonlocal volume. The construction is based on the nonlocal volume, a quantifier of nonlocality for states that counts the volume of the set of measurements that give rise to nonlocal behaviors when applied to this state, plus the trace distance, a quantifier of nonlocality for behaviors based on the distance between the behavior and the local set. The key difference from preceding candidates was the introduction of a quantifier of nonlocality to weight each contribution from behaviors in the nonlocal volume. We list some interesting properties of this quantifier and investigate the (2, 2, 2) and (2, 3, 2) scenarios. We show that the weak anomaly of nonlocality for the (2, 2, 3) scenario persists, but the local minimum for nonlocality with the trace-weighted nonlocal volume occurs in a different state as compared to the minimum for the non-weighted version, showing that the weak anomaly is not an intrinsic characteristic of the scenario, but is dependent of the choice of quantifier.
There is no doubt about the fact that entanglement and nonlocality are distinct resources. It is acknowledged that a clear illustration of this point is the difference between maximally entangled states and states that maximally violate a Bell inequality. We give strong evidence that this anomaly may be an artifact of the measures that have been used to quantify nonlocality. By reasoning that the numeric value of a Bell function is akin to a witness rather than a quantifier, we define a measure of nonlocality and show that, for pairs of qutrits and of four-level systems, maximal entanglement does correspond to maximal nonlocality in the same scenario that gave rise to the discrepancy.
Motivated by the recent proposal for the quantification of the Bell nonlocality [Phys. Rev. A 92, 030101(R) (2015)], we provide an extensive analysis of this concept, namely, the volume of violation. This new measure of nonlocality has been proposed to deliver an evidence that the anomaly between maximally entangled states and states that maximally violate a Bell inequality is caused by the method, which has been applied to quantify nonlocality and the anomaly disappears when the volume of violation in used. We prove that such conclusion is not true for all bipartite quantum systems with dimension $d\ensuremath{\bigotimes}d$. In fact, if one assumes the same condition as in Phys. Rev. A 92, 030101(R) (2015), it is limited to $d\ensuremath{\le}7$. Furthermore, we discuss several type of local measurements and their influence on the quantification of nonlocality by mean of volume of violation. In particular, we propose a set of local observers that significantly enhance the volume of violation, which may have important meaning in a real-world application. Finally, we present a comparison between the volume of violation and the maximal violation of a Bell inequality.
Measurements performed on distant parts of an entangled quantum state can generate correlations incompatible with classical theories respecting the assumption of local causality. This is the phenomenon known as quantum nonlocality that, apart from its fundamental role, can also be put to practical use in applications such as cryptography and distributed computing. Clearly, developing ways of quantifying nonlocality is an important primitive in this scenario. Here, we propose to quantify the nonlocality of a given probability distribution via its trace distance to the set of classical correlations. We show that this measure is a monotone under the free operations of a resource theory and, furthermore, that it can be computed efficiently with a linear program. We put our framework to use in a variety of relevant Bell scenarios also comparing the trace distance to other standard measures in the literature.
Ever since the work of Bell, it has been known that entangled quantum states can rise non-local correlations. However, for almost forty years, it has been assumed that the most non-local states would be the maximally entangled ones. Surprisingly it is not the case: non-maximally entangled states are generally more non-local than maximally entangled states for all the measures of non-locality proposed to date: Bell inequalities, the Kullback-Leibler distance, entanglement simulation with communication or with non-local boxes, the detection loophole and efficiency of cryptography. In fact, one can even find simple examples in low dimensions, confirming that it is not an artefact of a specifically constructed Hilbert space or topology. This anomaly shows that entanglement and non-locality are not only different concepts, but also truly different resources. We review the present knowledge on this anomaly, point out that Hardy's theorem has the same feature, and discuss the perspectives opened by these discoveries.
Bell's theorem shows that local measurements on entangled states give rise to correlations incompatible with local-hidden-variable models. The degree of quantum nonlocality is not maximal though, as there are even more nonlocal theories beyond quantum theory still compatible with the nonsignaling principle. In spite of decades of research, we still have a very fragmented picture of the whole geometry of these different sets of correlations. Here we employ both analytical and numerical tools to ameliorate that. First, we identify two different classes of Bell scenarios where the nonsignaling correlations can behave very differently: In one case, the correlations are generically quantum and nonlocal while in the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignaling correlations, we compute the distribution of a nonlocality quantifier based on the trace distance to the local set. With that we conclude that the nonlocal correlations can show a concentration phenomenon: Their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.