Geometric decompositions of Bell polytopes with practical applications

Abstract

Abstract. In the well-studied (2, 2, 2) Bell experiment consisting of two parties, two measurement settings per party, and two possible outcomes per setting, it is known that if the experiment obeys no-signaling constraints, then the set of admissible experimental probability distributions is fully characterized as the convex hull of 24 distributions: 8 Popescu-Rohrlich (PR) boxes and 16 local deterministic distributions. Furthermore, it turns out that in the (2, 2, 2) case, any nonlocal nonsignaling distribution can always be uniquely expressed as a convex combination of exactly one PR box and (up to) eight local deterministic distributions. In this representation each PR box will always occur only with a fixed set of eight local deterministic distributions with which it is affiliated. In this paper, we derive multiple practical applications of this result: we demonstrate an analytical proof that the minimum detection efficiency for which nonlocality can be observed is η > 2/3 even for theories constrained only by the no-signaling principle, and we develop new algorithms that speed the calculation of important statistical functions of Bell test data. Finally, we enumerate the vertices of the no-signaling polytope for the (2, n, 2) "chained Bell" scenario and find that similar decomposition results are possible in this general case. Here, our results allow us to prove the optimality of a bound, derived in Barrett et al.[1], on the proportion of local theories in a local/nonlocal mixture that can be inferred from the experimental violation of a chained Bell inequality.

Geometry of the set of quantum correlations

“…The existence of a single type of (facet) Bell inequalities and the fact that any no-signalling probability point P P N S can violate at most one of these inequalities [49] means that we can interpret the CHSH violation as a measure of distance from the local set. More specifically, Bierhorst showed that the total variation distance from the local set and the local content [50] can be written as linear functions of the violation [51]. In Appendix E we show that the same property holds for various notions of visibility.…”

Geometry of the quantum set on no-signaling faces

“…By referring to the known symmetries, see the Ref. [27], of the no-signaling polytope in the CHSH scenario, without loss of generality, it is sufficient to consider nonlocal correlations in any one of the symmetric regions. Therefore, in this paper, we will focus on the nonlocal region defined by the convex hull of the canonical PR box (henceforth referred to simply as "the PR-box"), and the eight local vertices on the local face derived from "the CHSH inequality" given by condition (7).…”

“…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].…”

Certifying quantum randomness by probability estimation

“…According to Ref. [49], for any non-signaling distribution σ(ABXY ), if E σ (I CHSH ) > 2, then the distribution σ(ABXY ) can be decomposed as σ(ABXY ) = ω PR1 σ PR1 + i ω LRi σ LRi , where ω PR1 = (E σ (I CHSH ) − 2)/2, ω LRi ≥ 0, and i ω LRi = 1 − ω PR1 . Specializing to the distribution ν(ABXY ), we get that g 0 ≤ (Î − 2)/2 forÎ > 2.…”