Quantum no-signalling correlations and non-local 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.

Synchronicity for quantum non-local games

“…Our main results in this direction (cf. Theorem 3.2, Corollary 3.7, Theorem 4.1, Corollary 4.4) provide an operational interpretation of the tracial QNS correlations introduced in [30] in terms of perfect strategies of concurrent and quantum mirror games, and their associated game algebras.…”

“…zero-error quantum information theory, quantum error correction, quantum groups, quantum teleportation schemes, and subfactor theory) [3,4,21,29,33]. In Section 5, we study the quantum graph homomorphism game in detail, extending previous work of the authors [5,30] in the classical-quantum hybrid setting, and also making connections with the work of Stahlke [29] and the algebraic work of Musto-Reutter-Verdon [21] on quantum graph homomorphisms.…”

“…The purpose of the present paper is to introduce and study generalisations of synchronous non-local games within the framework of quantum non-local games -non-local games where the questions and answers are allowed to be quantum states, or possibly mixtures of classical and quantum states. In this paper, we use the language of quantum no-signalling (QNS) correlations and quantum non-local games recently introduced by two of the present authors [30]. Classically, in the course of a non-local game G = (X, Y, A, B, λ), Alice and Bob's behaviour is described by a family p = (p(a, b|x, y)) (a,b,x,y)∈A×B×X×Y of conditional probability distributions, which can, in a canonical way, be viewed as a noisy information channel N : X × Y → A × B with well-defined marginal channels.…”

Morita equivalence for operator systems

“…Arveson in [1] -have played a cornerstone role in building quantised functional analysis [25,49], and are currently enjoying a surge of interest, both from a purely theoretical perspective [21,44,45,46] and in applications to the area of quantum information theory, where they are studied as non-commutative graphs [24,57,14,58,15]. Stable isomorphism of operator systems has further proved essential in recent developments in non-commutative geometry and mathematical physics [20].…”

“…The introduction of bihomomorphism contexts can be seen as a first step towards the development of an abstract, representation-free, view on non-commutative graph homomorphisms. The latter notion, introduced in [57], has gained importance both as an early sample of non-commutative combinatorics [59] and as a first step towards the quantisation of graph homomorphisms carried out in [58,15].…”