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 $222{v}_{B}$ scenarios, where ${v}_{B}$ 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.
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 222vB scenarios, where 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.
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.
We study the faces of the set of quantum correlations, i.e., the Bell and noncontextuality inequalities without any quantum violation. First, we investigate the question of whether every proper (facet-defining) Bell inequality for two parties, other than the trivial ones from positivity, normalization, and no-signaling, can be violated by quantum correlations, i.e., whether the classical Bell polytope or the smaller correlation polytope share any facets with their respective quantum sets. To do this, we develop a recently derived bound on the quantum value of linear games based on the norms of game matrices to give a simple sufficient condition to identify linear games with no quantum advantage. Additionally we show how this bound can be extended to the general class of unique games. We then show that the paradigmatic examples of correlation Bell inequalities with no quantum violation, namely the nonlocal computation games, do not constitute facet-defining Bell inequalities, not even for the correlation polytope. We also extend this to an arbitrary prime number of outcomes for a specific class of these games. We then study the faces in the simplest Clauser-Horne-Shimony-Holt Bell scenario of binary dichotomic measurements, and identify edges in the set of quantum correlations in this scenario. Finally, we relate the noncontextual polytope of single-party correlation inequalities with the cut polytope $\text{CUT}(\mathbf{\ensuremath{\nabla}}G)$, where G denotes the compatibility graph of observables in the contextuality scenario and $\mathbf{\ensuremath{\nabla}}G$ denotes the suspension graph of G. We observe that there exist facet-defining noncontextuality inequalities with no quantum violation, and furthermore that this set of inequalities is beyond those implied by the consistent exclusivity principle.
In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator in- equalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural prop- erties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.
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)=21 if a⊕b=xy, else 0. Bell polytope 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 (Tsirelson bound for quantum: 22≈2.828). | CONNECTION: The ratio 22/4=1/2≈0.7071 — not directly in the golden set, but note 0.70712=0.5, and the quantum-to-PR gap 4−22≈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
We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario (X,Y,A,B) is defined by the numbers of settings and outcomes per party, X,A, and Y,B for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of (6,3,2,2), (3,3,3,2), (3,2,3,3), and (2,2,3,5). We provide extensive lists of facets for (2,2,4,4), (3,3,4,2), and (4,3,3,2). For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits, and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to the Clauser-Horne-Shimony-Holt inequality. Such scenarios could find important applications in quantum communication.
We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of simplicial distributions, a pair of simplicial sets encoding measurement and outcome spaces determines a convex polytope of compatible probability assignments. Our first results give geometric criteria for detecting vertices of fiber products in terms of support data. These results are obtained in the more general framework of inverse limits of diagrams of polytopes in standard form, and they translate to corresponding criteria for simplicial distributions on arbitrary colimits of measurement spaces. We then focus on one-dimensional measurement spaces, where simplicial distributions recover and generalize local marginal polytopes in graphical models. In this setting, our sharpest results concern dipole graphs, for which we obtain a complete characterization of vertices and refine it to a graph-theoretic criterion. These characterizations are reminiscent of the classical support-graph criteria for transportation polytopes, but they arise in a richer class of polytopes in which vertex structure depends not only on support acyclicity but also on additional geometric compatibility data. Using the collapsing method from simplicial topology, we transfer the dipole characterization to rose graphs and obtain analogous results there. Finally, we apply collapsing to complete bipartite graphs, which encode physically relevant bipartite Bell scenarios, and more generally to arbitrary connected graphs, yielding lower bounds on the number of vertices.