Quantifying Bell nonlocality with the trace distance

Abstract

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 non-locality 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 non-locality is an important primitive in this scenario. Here, we propose to quantify the non-locality 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 that furthermore 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.

Concentration phenomena in the geometry of Bell correlations

“…2) as how far a given dcorrelation is from the local polytope defining the correlations (2), with the quantitative aspect that the degree of nonlocality of q that is equal to zero if and only if q is local. Beyond its geometrical and quantitative content, we point out that this distance has computational and numerical appeal, as it can be evaluated efficiently via a linear program [62]. Considering a number of different scenarios, we have uniformly sampled over the NS correlations and computed the distance of each sampled correlation to the set of local correlations.…”

“…The distribution of our chosen quantifier for nonlocality -based on the trace distance between the probability distribution under test and the set of local correlations introduced in Ref. [62]-has unveiled interesting features in various scenarios. Of particular relevance we have seen that in the scenarios (2, m, 2) and (N, 2, 2) not only the volume of the local set decreases very rapidly but also that the nonlocal points concentrate at a distance from the local set that increases with both m and N. We believe that such a surprising behaviour reflects the signature of the complicated geometry of the nonsignalling and local sets, giving further insight on the relation between local and nonsignalling set.…”

“…Sec. V discusses the distribution of the nonlocality measure introduced in [62] in a variety of Bell scenarios and points out to the emergence of concentration phenomena. Finally, we discuss our results as well as potential venues for future work in Sec.…”

Geometry of the quantum set on no-signaling faces

“…Covering both foundational and applied perspectives, a crucial aspect to better understand quantum correlations, their potential advantages over classical resources but also their limitations in the processing of information, relies on understanding their geometry [21]. Many more works [22][23][24][25][26][27][28][29][30] have also revealed a number of interesting geometrical aspects of the set of quantum correlations. Perhaps the best available tool for studying the quantum-postquantum boundary is the Navascues-Pironio-Acin (NPA) hierarchy [31,32], which gives a series of outer approximations converging to a set of quantum correlations Q. Interestingly, in a more recent development [33], it was shown that any nonlocal correlation which belongs to the set of almost quantum correlations Q (1+ab) , the set determined by the (1 + ab) level of the NPA hierarchy [31,32], satisfies all physical principles proposed so far, with two possible exceptions: (i) the information causality principle [11,12] and its generalization [13], and (ii) the recently proposed principle of many-box locality [16].…”

Nonlocality distillation and quantum voids

“…Different measures have been considered before, the violation of the CHSH inequality itself [20] and the so called EPR-2 decomposition [23]. Here we employ the trace distance measure introduced in [37], basically quantifying the minimum distance of the nonlocal point in question to the set of local correlations. In the CHSH scenario the trace distance quantifier has been shown to be equivalent to the CHSH inequality violation (up to a constant factor), reason why we consider here as a quantifier NL(p) of the nonlocality of a given distribution p = p(ab|xy) the following quantity:…”