All tight correlation Bell inequalities have quantum violations

Llorenç Escolà,John Calsamiglia,Andreas Winter

doi:10.1103/physrevresearch.2.012044

Abstract

It is by now well-established that there exist non-local games for which the best entanglement-assisted performance is not better than the best classical performance. Here we show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e. a face of maximum dimension, of the local Bell polytope. Indeed, using semidefinite programming duality, we prove upper bounds on the dimension of these faces, bounding it far away from the maximum. In the special case of non-local computation games, it had been shown before that they are not facet-defining; our result generalises and improves this. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets.

Highlights

In 1964, Bell [1] proved that some predictions of quantum theory regarding the correlations between distant events cannot be explained by any classical, i.e., local realistic theory
Tsirelson [14] computed the maximal violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality [2] attainable by quantum mechanics; later, Popescu and Rohrlich [15] showed that quantum correlations belong to the set of no-signaling correlations, which are those not allowing for instantaneous communication, they do not attain the full strength allowed in principle by the no-signaling condition
Since XOR games fully characterize the correlation polytope, we answer Gill’s question in the affirmative for the correlation polytope: all nontrivial tight correlation Bell inequalities have quantum violations. The remainder of this Rapid Communication is structured as follows: (i) we first introduce the general formalism to describe the set of no-signaling, local classical and quantum correlations; (ii) we briefly present the XOR games and give general expressions for the winning probabilities under different locality scenarios; (iii) we present our main theorem and the main ideas of its proof; and (iv) we extend our result to the quantum set of correlations, and conclude with a discussion and outlook

Summary

All tight correlation Bell inequalities have quantum violations

Llorenç Escolà ,1,* John Calsamiglia ,1,† and Andreas Winter 1,2,‡ 1Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain 2Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. We show in contrast that any two-player XOR game, for which the corresponding Bell inequality is tight, has a quantum advantage. In geometric terms, this means that any correlation Bell inequality for which the classical and quantum maximum values coincide, does not define a facet, i.e., a face of maximum dimension, of the local (Bell) polytope. As a by-product of our analysis, we find a similar upper bound on the dimension of the faces of the convex body of quantum correlation matrices, showing that (except for the trivial ones expressing the non-negativity of probability) it does not have facets

Introduction

Published by the American Physical Society

Results

Discussion and outlook