Linear conic formulations for two-party correlations and values of nonlocal games

TL;DR

This paper introduces conic programming frameworks for analyzing two-party correlations and nonlocal game values, providing new bounds, hierarchies, and decision procedures within quantum information theory.

Contribution

It presents novel conic formulations for classical, quantum, and no-signaling correlations, linking nonlocal game values to linear conic programs and hierarchies.

Findings

Spectrahedral outer approximation to quantum correlations.

Feasible conditions for quantum correlation sets.

Semidefinite programming bounds for nonlocal game values.

Abstract

In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the Navascu\'es, Pironio and Ac\'in (NPA) hierarchy and also a sufficient condition for the set of quantum correlations to be closed. Furthermore, by our conic formulations, the value of a nonlocal game over the sets of classical, quantum, no-signaling and unrestricted correlations can be cast as a linear conic program. This allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lov\'asz is in fact an upper bound to the quantum value of the game and moreover, it is at least as strong as optimizing over the first level of the NPA hierarchy. Lastly, we show that deciding the existence of a perfect quantum (resp. classical) strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices (resp. completely positive matrices). By specializing the results to synchronous nonlocal games, we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature.