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.
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.
Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.
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.
We show that if a graph has minimum vertex degree at least d and girth at least g, where (d, g) is (3, 6) or (4, 4), then the incidence system of the graph has a (possibly infinite-dimensional) quantum solution over
Zp for every choice of vertex weights and integer
p≥2. In particular, there are linear systems over
Zp, for
p an odd prime, such that the corresponding linear system nonlocal game has a perfect commuting-operator strategy, but no perfect classical strategy.
Abstract We investigate quantum and nonsignaling generalizations of perfect matchings in graphs using nonlocal games. Specifically, we introduce nonlocal games that test for L-perfect matchings in bipartite graphs, perfect matchings in general graphs and hypergraphs, and fractional perfect matchings. Our definitions come from the fact that these games are classical property tests for the corresponding matching conditions. We use the existence of perfect quantum and nonsignaling strategies for these games to define quantum and nonsignaling versions of perfect matchings. Finally, we provide characterizations of when graphs exhibit these extended properties: For nonsignaling matchings, we give a complete combinatorial characterization. In particular, a graph has a nonsignaling perfect matching if and only if it admits a fractional perfect matching that has bounded value on triangles. In bipartite graphs, the nonsignaling L-perfect matching property is achieved exactly when the left component of the graph can be split into two disjoint subgraphs: one with a classical L-perfect matching and another with left-degree 2. In the quantum setting, we show that complete graphs
Kn K n with odd
n≥7 n ≥ 7 have quantum perfect matchings. We prove that a graph has a quantum perfect matching if and only if the quantum independence number of its line graph is maximal, extending a classical relationship between perfect matchings and line graph independence numbers. For bipartite graphs, we establish that the L-perfect matching game does not exhibit quantum pseudotelepathy, but we characterize the quantum advantage for complete bipartite graphs
Kn,2 K n , 2 . Additionally, we prove that deciding quantum perfect matchings in hypergraphs is undecidable and leaves open the question of its complexity in graphs.
Abstract We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.