Mathematical models of quantum mechanics can be studied and distinguished using nonlocal games. We discuss a class of nonlocal games called synchronous linear constraint system (syncLCS) games. We unify two algebraic approaches to studying syncLCS games and relate these games to nonlocal games played on graphs, known as graph isomorphism games. In more detail, syncLCS games are nonlocal games that verify whether or not two players share a solution to a given system of equations. Two algebraic objects associated with these games encode information about the existence of perfect strategies. They are called the game algebra and the solution group. Here, we show that these objects are essentially the same, i.e., the game algebra is a suitable quotient of the group algebra of the solution group. We also demonstrate that syncLCS games are equivalent to graph isomorphism games on a pair of graphs parameterized by the linear system.
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
mathbbZ2-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.
In the flavour of categorical quantum mechanics, we extend nonlocal games to allow quantum questions and answers, using quantum sets (special symmetric dagger Frobenius algebras) and the quantum functions of Musto, Reutter, and Verdon. Equations are presented using a diagrammatic calculus for tensor categories. To this quantum question and answer setting, we extend the standard definitions, including strategies, correlations, and synchronicity, and we use these definitions to extend results about synchronicity. We extend the graph homomorphism (isomorphism) game to quantum graphs, and show it is synchronous (bisynchronous) and connect its perfect (bi)strategies to quantum graph homomorphisms (isomorphisms). Our extended definitions agree with the existing quantum games literature, except in the case of synchronicity.
Abstract We develop an algebraic and operational framework for quantum isomorphisms of hypergraphs, using tools from compact quantum group theory. We introduce a new synchronous version of the hypergraph isomorphism game whose game algebra uniformly encodes multiple notions of quantum isomorphisms of hypergraphs. We show that there exist hypergraphs that are quantum isomorphic but not classically isomorphic. For graphs, we show that the ‐algebra of the hypergraph isomorphism game is a quotient of the ‐algebra of the graph isomorphism game. We further prove that the hypergraph game algebra forms a bi‐Galois extension over the quantum automorphism groups of the underlying hypergraphs. This allows us to deduce that the algebraic notion of a quantum isomorphism of hypergraphs coincides with the operational one coming from the existence of perfect quantum strategies. Viewing games themselves as hypergraphs, we analyze isomorphisms and the transfer of strategies within this setting. Finally, we construct a ‐algebra whose representation theory characterizes distinct classes of quantum isomorphisms between nonlocal games.
We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are
∗-isomorphic. We first show that the game algebra of any synchronous game on
n inputs and
k outputs is
∗-isomorphic to the game algebra of an associated bisynchronous game on
nk inputs and
nk outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in the quantum approximate model. Moreover, we show that there are bisynchronous games with equal question and answer sets that have non-zero game algebras, but no winning quantum commuting strategies, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a
∗-isomorphism between any synchronous game algebra with
n questions and
k>3 answers and a synchronous game algebra with
n(k−2) questions and
3 answers.
The perfect quantum strategies of a linear system game correspond to certain representations of its solution group. We study the solution groups of graph incidence games, which are linear system games in which the underlying linear system is the incidence system of a (non-properly) two-coloured graph. While it is undecidable to determine whether a general linear system game has a perfect quantum strategy, for graph incidence games this problem is solved by Arkhipov’s theorem, which states that the graph incidence game of a connected graph has a perfect quantum strategy if and only if it either has a perfect classical strategy, or the graph is nonplanar. Arkhipov’s criterion can be rephrased as a forbidden minor condition on connected two-coloured graphs. We extend Arkhipov’s theorem by showing that, for graph incidence games of connected two-coloured graphs, every quotient closed property of the solution group has a forbidden minor characterization. We rederive Arkhipov’s theorem from the group theoretic point of view, and then find the forbidden minors for two new properties: finiteness and abelianness. Our methods are entirely combinatorial, and finding the forbidden minors for other quotient closed properties seems to be an interesting combinatorial problem.
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.