Concentration phenomena in the geometry of Bell correlations

Abstract

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 nonsignalling 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 nonsignalling correlations can behave very differently: in one case, the correlations are generically quantum and nonlocal while on the other quite the opposite happens as the correlations are generically classical and local. Second, by randomly sampling over nonsignalling 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 phenomena: their distribution is peaked at a distance from the local set that increases both with the number of parts or measurements being performed.

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].…”

Resource theory of contextuality

“…We point out that although the main application of a resource theory is to understand the role of a physical property as an operational resource, this construction can be interesting on its own and it can give insight into the physical property under consideration. For example, Duarte et al [43] use contextuality quantifiers to explore the geometry of the set of behaviours, finding the approximate relative volume of the non-contextual set in relation to the non-disturbing set.…”

Naturally restricted subsets of nonsignaling correlations: typicality and convergence

“…To this end, it is worth noting that, quantitative difference between the "size" of the Bell-local set L, the quantum set Q, and the nonsignaling set N S has been investigated in [47][48][49]. Specifically, in the simplest Bell scenario involving two parties, each performing two binary-outcome measurements, the volume of L and that of Q, relative to N S, in the subspace of "full" correlation functions [50] was first determined in [47].…”

“…Then, for the same Bell scenario, the analysis has been generalized [48] to include also the subspace spanned by marginal correlations. Beyond this, numerical estimation on the relative volume of L to N S was carried out in [49] for a few Bell scenarios with either two measurement settings or outcomes; some analytic results were also presented therein when restricted to the subspace of full correlations.…”

“…Here, we generalize the analysis of [47][48][49] in two directions. Firstly, we consider a more extensive list of Bell scenarios, including a few with both multiple measurement settings and multiple measurement outcomes, thereby allowing us to make observations that were not possible in prior works.…”