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arXiv:1209.2729

Characterization of Binary Constraint System Games

Richard Cleve, Mittal, Rajat

Published Sep 12, 2012• 9 citations
arXivHTMLPDF

Abstract

We consider a class of nonlocal games that are related to binary constraint systems (BCSs) in a manner similar to the games implicit in the work of Mermin [N.D. Mermin, "Simple unified form for the major no-hidden-variables theorems," Phys. Rev. Lett., 65(27):3373-3376, 1990], but generalized to n binary variables and m constraints. We show that, whenever there is a perfect entangled protocol for such a game, there exists a set of binary observables with commutations and products similar to those exhibited by Mermin. We also show how to derive upper bounds strictly below 1 for the the maximum entangled success probability of some BCS games. These results are partial progress towards a larger project to determine the computational complexity of deciding whether a given instance of a BCS game admits a perfect entangled strategy or not.

DOI
10.48550/arxiv.1209.2729

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We consider a class of nonlocal games that are related to binary constraint systems (BCSs) in a manner similar to the games implicit in the work of Mermin [N.D. Mermin, "Simple unified form for the major no-hidden-variables theorems," Phys. Rev. Lett., 65(27):3373-3376, 1990], but generalized to n binary variables and m constraints. We show that, whenever there is a perfect entangled protocol for such a game, there exists a set of binary observables with commutations and products similar to those exhibited by Mermin. We also show how to derive upper bounds strictly below 1 for the the maximum entangled success probability of some BCS games. These results are partial progress towards a larger project to determine the computational complexity of deciding whether a given instance of a BCS game admits a perfect entangled strategy or not.
First, we consider the problem of deciding whether a nonlocal game admits a perfect entangled strategy that uses projective measurements on a maximally entangled shared state. Via a polynomial-time Karp reduction, we show that independent set games are the hardest instances of this problem. Secondly, we show that if every independent set game whose entangled value is equal to one admits a perfect entangled strategy, then the same holds for all symmetric synchronous games. Finally, we identify combinatorial lower bounds on the classical and entangled values of synchronous games in terms of variants of the independence number of appropriate graphs. Our results suggest that independent set games might be representative of all nonlocal games when dealing with questions concerning perfect entangled strategies.
Mermin and Peres showed that there are Boolean constraint systems (BCSs) that are not satisfiable but are satisfiable with quantum observables. This has led to a burgeoning theory of quantum satisfiability for constraint systems, connected to nonlocal games and quantum contextuality. In this theory, different types of quantum satisfying assignments can be understood as representations of the BCS algebra of the system. This theory is closely related to the theory of synchronous games and algebras, and every synchronous algebra is a BCS algebra and vice versa. The purpose of this paper is to further develop the role of BCS algebras in this theory and tie up some loose ends: We give a new presentation of BCS algebras in terms of joint spectral projections and show that it is equivalent to the standard definition. We construct a constraint system that is C∗-satisfiable but not tracially satisfiable. We show that certain reductions between constraint systems lead to ∗-homomorphisms between the BCS algebras of the systems, and we use this to streamline and strengthen several results of Atserias, Kolaitis, and Severini on analogues of Schaefer’s dichotomy theorem. In particular, we show that the question of whether or not there is a nonhyperlinear group is linked to dichotomy theorems for RU-satisfiability.
A binary constraint system game is a two-player one-round non-local game defined by a system of Boolean constraints. The game has a perfect quantum strategy if and only if the constraint system has a quantum satisfying assignment [R. Cleve and R. Mittal, arXiv:1209.2729]. We show that several concepts including the quantum chromatic number and the Kochen-Specker sets that arose from different contexts fit naturally in the binary constraint system framework. The structure and complexity of the quantum satisfiability problems for these constraint systems are investigated. Combined with a new construct called the commutativity gadget for each problem, several classic NP-hardness reductions are lifted to their corresponding quantum versions. We also provide a simple parity constraint game that requires Ω(n)Ω(\sqrt{n})Ω(n​) EPR pairs in perfect strategies where nnn is the number of variables in the constraint system.
One of the longstanding open problems in quantum nonlocality is to determine if maximally entangled states are complete for bipartite pseudo-telepathic games: namely, if every nonlocal game which admits a perfect entangled strategy admits such a strategy which uses a maximally entangled state. We exhibit a counterexample to this in the form of a bipartite nonlocal game with input sets of size 4 and 3 and output sets both of size 6. This game is part of a new class of nonlocal games, which we call inner product games, which could be of independent interest.
We establish approximate rigidity results for several well-known families of nonlocal games. In particular, we show that near-perfect quantum strategies for boolean constraint system (BCS) games are approximate representations of the corresponding BCS algebra. Likewise, for the class of XOR nonlocal games, we show that near-optimal quantum strategies are approximate representations of the corresponding ∗*∗-algebra associated with optimal quantum values for the game. In both cases, the approximate representations are with respect to the normalized Hilbert-Schmidt norm and independent of the Hilbert space or quantum state employed in the strategy. We also show that approximate representation of the BCS (resp.~XOR-algebra) yields measurement operators for near-perfect (resp.~near-optimal) quantum strategies where the players employ a maximally entangled state in the game. As a corollary, every near-perfect BCS (resp.~near-optimal XOR) quantum strategy is close to a near-perfect (resp.~near-optimal) quantum strategy using a maximally entangled state. Lastly, we establish that every synchronous algebra is ∗*∗-isomorphic to a certain BCS algebra called the SynchBCS algebra. This allows us to apply our BCS rigidity results to the class of synchronous games as well.
We establish approximate rigidity results for several well-known families of nonlocal games. In particular, we show that near-perfect quantum strategies for boolean constraint system (BCS) games are approximate representations of the corresponding BCS algebra. Likewise, for the class of XOR nonlocal games, we show that near-optimal quantum strategies are approximate representations of the corresponding ∗-algebra associated with optimal quantum values for the game. In both cases, the approximate representations are with respect to the normalized Hilbert-Schmidt norm and independent of the Hilbert space or quantum state employed in the strategy. We also show that approximate representation of the BCS (resp. XOR-algebra) yields measurement operators for near-perfect (resp. near-optimal) quantum strategies where the players employ a maximally entangled state in the game. As a corollary, every near-perfect BCS (resp. near-optimal XOR) quantum strategy is close to a near-perfect (resp. near-optimal) quantum strategy using a maximally entangled state. Lastly, we establish that every synchronous algebra is ∗-isomorphic to a certain BCS algebra called the SynchBCS algebra. This allows us to apply our BCS rigidity results to the class of synchronous games as well.