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arXiv:1511.04127 · Journal of Physics A Mathematical and Theoretical

Geometric decompositions of Bell polytopes with practical applications

Peter Bierhorst

Published Apr 22, 2016• 19 citations
arXivHTMLPDF

Abstract

In the well-studied 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: eight Popescu–Rohrlich (PR) boxes and 16 local deterministic distributions. Furthermore, it turns out that in the 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 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 '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 (2006 Phys. Rev. Lett. 97 170409 ), on the proportion of local theories in a local/nonlocal mixture that can be inferred from the experimental violation of a chained Bell inequality.

DOI
10.1088/1751-8113/49/21/215301

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In the well-studied 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: eight Popescu–Rohrlich (PR) boxes and 16 local deterministic distributions. Furthermore, it turns out that in the 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 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 '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 (2006 Phys. Rev. Lett. 97 170409 ), on the proportion of local theories in a local/nonlocal mixture that can be inferred from the experimental violation of a chained Bell inequality.
FINDING: The Popescu-Rohrlich (PR) box defines the maximal nonlocal vertex of the Bell polytope; its extremal structure is characterized by convex geometry, and recent work shows PR-box fraction cannot uniquely single out quantum theory among generalized nonsignaling theories. | MATH: PR box correlations: P(a,b∣x,y)=12P(a,b|x,y) = \frac{1}{2}P(a,b∣x,y)=21​ if a⊕b=xya \oplus b = xya⊕b=xy, else 0. Bell polytope B\mathcal{B}B: convex hull of local deterministic strategies; its extremal points in (2,2,2) scenario are the 8 local vertices plus 4 PR-box vertices (the "nonlocal" extreme points). The PR box saturates the CHSH inequality: ⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩=4\langle A_0B_0\rangle + \langle A_0B_1\rangle + \langle A_1B_0\rangle - \langle A_1B_1\rangle = 4⟨A0​B0​⟩+⟨A0​B1​⟩+⟨A1​B0​⟩−⟨A1​B1​⟩=4 (Tsirelson bound for quantum: 22≈2.8282\sqrt{2} \approx 2.82822​≈2.828). | CONNECTION: The ratio 22/4=1/2≈0.70712\sqrt{2}/4 = 1/\sqrt{2} \approx 0.707122​/4=1/2​≈0.7071 — not directly in the golden set, but note 0.70712=0.50.7071^2 = 0.50.70712=0.5, and the quantum-to-PR gap 4−22≈1.17164 - 2\sqrt{2} \approx 1.17164−22​≈1.1716 relates to \(2(\sqrt{2}-1) \approx 0.82 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The study of no-signaling polytopes is addressed under an alternative approach, focusing on the skeleton graph of the polytopes rather than on the Bell inequalities solely. The paper starts with the study of the Bell scenario where one of the parties (Bob) has three output options (2223 scenario). We found that in this polytope the Popescu-Rohrlich (PR) vertices are interconnected, making up two networks around the polytope of local correlations, which is a new feature in relation to the 2222 polytope. We also found that in this scenario each CHSH inequality is violated by five PR vertices: the PR vertex associated to that inequality and its four adjacent PR vertices. Later, we study the facet structure of the corresponding polytope of local correlations. What we obtain is a one-to-one correspondence between some two-dimensional faces of the no-signaling polytope defined by specific loops in the adjacency networks of PR vertices, the lifting Bell functions, and the nontrivial facets of the local polytope. Finally, by studying the skeleton graphs of the polytopes for the general 222vB222{v}_{B}222vB​ scenarios, where vB{v}_{B}vB​ is the number of possible Bob's outcomes, a striking high degeneracy is unveiled. We show that the set of local vertices exhibits a much higher degeneracy than the PR vertices, which in turn are more degenerate than a regular vertex.
Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.
Characterizing the set of quantum nonlocal correlations through information-theoretic constraints is a central problem in quantum foundations. We show that the convex hull of no-signalling behaviors in a bipartite Bell experiment satisfying a common-overlap no-disturbance-without-uncertainty condition is exactly the finite-dimensional quantum set whenever one party has two binary measurements and the other has an arbitrary finite number of settings, each with an arbitrary finite number of outcomes. In the simplest binary input-output Bell scenario, this characterizes the full eight-dimensional set of quantum behaviors, including the local marginals. The proof constructs a quantum realization directly from the conditional probabilities and no-signalling. The characterization also reduces optimization of any linear Bell expression in these scenarios to maximization over a one-parameter family of second-order cone programs. The results establish a connection between local uncertainty-disturbance constraints and quantum nonlocality, mediated by no-signalling and shared classical randomness.
Bell showed 50 years ago that quantum theory is non-local via his celebrated inequalities, turning the issue of quantum non-locality from a matter of taste into a matter of test. Years later, Hardy proposed a test for non-locality without inequality, which is a kind of 'something-versus-nothing' argument. Hardyʼs test for n particles induces an n -partite Bellʼs inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell–Hardy inequality arising form Hardyʼs non-locality test is tight for an arbitrary number of parties, i.e., it defines a facet of the Bell polytope in the given scenario. On the other hand quantum theory is not that non-local since it forbids signaling and even not as non-local as allowed by non-signaling conditions, i.e., quantum mechanical predictions form a strict subset of the so called non-signaling polytope. In the scenario of each observer measuring two dichotomic observables, Fritz established a duality between the Bell polytope and the non-signaling polytope: tight Bellʼs inequalities, the facets of the Bell polytope, are in a one-to-one correspondence with extremal non-signaling boxes, the vertices of the non-signaling polytope. Our second result is to provide an alternative and more direct formula for this duality. As an example, the tight Bell–Hardy inequality gives rise to an extremal non-signaling box that serves as a natural multipartite generalization of Popescu–Rohrlich box. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.