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.
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.
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.