Polyhedral duality in Bell scenarios with two binary observables

Abstract

For the Bell scenario with two parties and two binary observables per party, it is known that the no-signaling polytope is the polyhedral dual (polar) of the Bell polytope. Computational evidence suggests that this duality also holds for three parties. Using ideas of Werner, Wolf, Żukowski, and Brukner, we prove this for any number of parties by describing a simple linear bijection mapping (facet) Bell inequalities to (extremal) no-signaling boxes and vice versa. Furthermore, a symmetry-based technique for extending Bell inequalities (respectively, no-signaling boxes) with two binary observables from n parties to n + 1 parties is described; the Mermin–Klyshko family of Bell inequalities arises in this way, as well as 11 of the 46 classes of facet Bell inequalities for three parties. Finally, we ask whether the set of quantum correlations is self-dual with respect to our transformation. We find this not to be the case in general, although it holds for two parties on the level of correlations. This self-duality implies Tsirelson's bound for the CHSH inequality.