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

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 Navascués, Pironio and Acín (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 Lovász 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 B Antonios Varvitsiotis 123 432 J. Sikora, A. Varvitsiotis 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.

Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

“…The completely positive semidefinite cone was first studied in [50] to describe quantum analogues of the stability number and of the chromatic number of a graph. This was later extended to general graph homomorphisms in [72] and to graph isomorphism in [3]. In addition, as shown in [54,72], there is a close connection between the completely positive semidefinite cone and the set of quantum correlations.…”

“…This was later extended to general graph homomorphisms in [72] and to graph isomorphism in [3]. In addition, as shown in [54,72], there is a close connection between the completely positive semidefinite cone and the set of quantum correlations. This also gives a relation between the completely positive semidefinite rank and the minimal entanglement dimension necessary to realize a quantum correlation.…”

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

“…By Artin-Wedderburn theory [53,2], a finite dimensional C * -algebra is isomorphic to a matrix algebra. So the above reformulations of χ q (G) and α q (G) can be seen as feasibility problems of systems of equations in matrix variables of unspecified (but finite) dimension; such formulations are given in [9,28,46]. Restricting to scalar solutions (1 × 1 matrices) in these feasibility problems recovers the classical graph parameters χ(G) and α(G).…”

“…Completely positive semidefinite matrices are used in [25] to model quantum graph parameters and the completely positive semidefinite rank is investigated in [43,16,44,15]. By combining the proofs from [46] (see also [28]) and [41] one can show the following link between synchronous correlations and completely positive semidefinite matrices. 1 Proposition A.1.…”

Correlation matrices, Clifford algebras, and completely positive semidefinite rank

“…The cpsd cone was introduced recently to provide linear conic formulations for the quantum analogues of various classical graph parameters [14,20]. Subsuming these results, it was shown in [21] that the set of joint probability distributions that can be generated using quantum resources can be expressed as the projection of an affine section of the CS n + cone.…”