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.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.

arXiv:1209.2729Sep 12, 20129 citations

On symmetric nonlocal games

Andris Ambainis, Dmitry Kravchenko, Nikolay Nahimov +2 more

Theoretical Computer ScienceMar 25, 20132 citations

Playing games with bounded entropy

Mehrdad Valizadeh, Amin Gohari

arXiv:1702.05719Games and Economic BehaviorJun 1, 20177 citations

Binary Constraint System Games and Locally Commutative Reductions

Zhengfeng Ji

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 $Ω(\sqrt{n})$ EPR pairs in perfect strategies where $n$ is the number of variables in the constraint system.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 $Ω(\sqrt{n})$ EPR pairs in perfect strategies where $n$ is the number of variables in the constraint system.

arXiv:1310.3794Oct 14, 201328 citations

Algebraic and combinatorial aspects of incidence groups and linear system non-local games arising from graphs

Connor Paddock

To every linear binary-constraint system (LinBCS) non-local game, there is an associated algebraic object called the solution group. Cleve, Liu, and Slofstra showed that a LinBCS game has a perfect quantum strategy if and only if an element, denoted by $J$ , is non-trivial in this group. In this work, we restrict to the set of graph-LinBCS games, which arise from $\\mathbb{Z}_2$ -linear systems $Ax=b$ , where $A$ is the incidence matrix of a connected graph, and $b$ is a (non-proper) vertex $2$ -colouring. In this context, Arkhipov's theorem states that the corresponding graph-LinBCS game has a perfect quantum strategy, and no perfect classical strategy, if and only if the graph is non-planar and the $2$ -colouring $b$ has odd parity. In addition to efficient methods for detecting quantum and classical strategies for these games, we show that computing the classical value, a problem that is NP-hard for general LinBCS games can be done efficiently. In this work, we describe a graph-LinBCS game by a $2$ -coloured graph and call the corresponding solution group a graph incidence group. As a consequence of the Robertson-Seymour theorem, we show that every quotient-closed property of a graph incidence group can be expressed by a finite set of forbidden graph minors. Using this result, we recover one direction of Arkhipov's theorem and derive the forbidden graph minors for the graph incidence group properties: finiteness, and abelianness. Lastly, using the representation theory of the graph incidence group, we discuss how our graph minor criteria can be used to deduce information about the perfect strategies for graph-LinBCS games.To every linear binary-constraint system (LinBCS) non-local game, there is an associated algebraic object called the solution group. Cleve, Liu, and Slofstra showed that a LinBCS game has a perfect quantum strategy if and only if an element, denoted by $J$, is non-trivial in this group. In this work, we restrict to the set of graph-LinBCS games, which arise from $\\mathbb{Z}_2$-linear systems $Ax=b$, where $A$ is the incidence matrix of a connected graph, and $b$ is a (non-proper) vertex $2$-colouring. In this context, Arkhipov's theorem states that the corresponding graph-LinBCS game has a perfect quantum strategy, and no perfect classical strategy, if and only if the graph is non-planar and the $2$-colouring $b$ has odd parity. In addition to efficient methods for detecting quantum and classical strategies for these games, we show that computing the classical value, a problem that is NP-hard for general LinBCS games can be done efficiently. In this work, we describe a graph-LinBCS game by a $2$-coloured graph and call the corresponding solution group a graph incidence group. As a consequence of the Robertson-Seymour theorem, we show that every quotient-closed property of a graph incidence group can be expressed by a finite set of forbidden graph minors. Using this result, we recover one direction of Arkhipov's theorem and derive the forbidden graph minors for the graph incidence group properties: finiteness, and abelianness. Lastly, using the representation theory of the graph incidence group, we discuss how our graph minor criteria can be used to deduce information about the perfect strategies for graph-LinBCS games.

Jun 6, 20191 citation

Ω(\sqrt{n})

EPR pairs in perfect strategies where

n

is the number of variables in the constraint system.

J

, is non-trivial in this group. In this work, we restrict to the set of graph-LinBCS games, which arise from

\\mathbb{Z}_2

-linear systems

Ax=b

, where

A

is the incidence matrix of a connected graph, and

b

is a (non-proper) vertex

2

-colouring. In this context, Arkhipov's theorem states that the corresponding graph-LinBCS game has a perfect quantum strategy, and no perfect classical strategy, if and only if the graph is non-planar and the

2

-colouring

b

has odd parity. In addition to efficient methods for detecting quantum and classical strategies for these games, we show that computing the classical value, a problem that is NP-hard for general LinBCS games can be done efficiently. In this work, we describe a graph-LinBCS game by a

2

-coloured graph and call the corresponding solution group a graph incidence group. As a consequence of the Robertson-Seymour theorem, we show that every quotient-closed property of a graph incidence group can be expressed by a finite set of forbidden graph minors. Using this result, we recover one direction of Arkhipov's theorem and derive the forbidden graph minors for the graph incidence group properties: finiteness, and abelianness. Lastly, using the representation theory of the graph incidence group, we discuss how our graph minor criteria can be used to deduce information about the perfect strategies for graph-LinBCS games.