Polyhedral duality in Bell scenarios with two binary observables

TL;DR

This paper proves a duality between Bell inequalities and no-signaling boxes for any number of parties, introduces a symmetry-based extension technique, and explores the self-duality of quantum correlations.

Contribution

It establishes a general linear bijection linking Bell inequalities and no-signaling extremal boxes for any number of parties, extending known bipartite results.

Findings

Proves duality for any number of parties.

Introduces a symmetry-based extension method.

Shows quantum correlations are not generally self-dual.

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, \.Zukowski and Brukner, we prove this for any number of parties by describing a simple linear bijection mapping (tight) Bell inequalities to (extremal) no-signaling boxes and vice versa. Furthermore, a symmetry-based technique for extending Bell inequalities (resp. 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 tight Bell inequalities for 3 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 2 parties on the level of correlations. This self-duality implies Tsirelson's bound for the CHSH inequality.