Geometry of the skeleton graph of the no-signaling polytope for the 2223 Bell scenario and beyond

Abstract

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.