Non-closure of the Set of Quantum Correlations via Graphs

Dykema, Ken; Paulsen, Vern I.; Prakash, J.

doi:10.1007/s00220-019-03301-1

Cited by 86 publications

(83 citation statements)

References 12 publications

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“…According to it, the measurements of the two parties act on a single (infinite dimensional) Hilbert space; this setting was studied in [19] and subsequently in [28], where the author showed that every correlation from C qc can be approximated by ones from C q if and only if the Connes Embedding Problem in operator algebra theory [8] has an affirmative answer. Deep results about the inequality between those classes of correlations, answering questions left open by Tsirelson (see [39] and [40]) were recently Date: 9 April 2018. obtained by Slofstra in [36] and [37] when the number of inputs is large and in [11] similar results are shown for a small number of inputs. The relevance of operator algebraic techniques in the study of correlation sets became also apparent through [13] and, subsequently, [32], where operator systems and their tensor products were used to describe some correlation classes.…”

Section: Introductionsupporting

confidence: 76%

Perfect Strategies for Non-Local Games

Lupini

Mančinska

Paulsen

et al. 2020

Math Phys Anal Geom

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We unify and consolidate various results about non-signalling games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are reflexive games, which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce imitation games, in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and unique games. We associate a C*-algebra C * (G) to any imitation game G, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of G can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call mirror games, and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.

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Section: Introductionsupporting

confidence: 76%

Perfect Strategies for Non-Local Games

Lupini

Mančinska

Paulsen

et al. 2020

Math Phys Anal Geom

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“…we explicitly include all the limit points, which makes the quantum set Q compact. 2 The fundamental result that Q ‰ Q finite for some finite Bell scenarios was only recently established by Slofstra [43] (see also the recent work of Dykema et al [44]).…”

Section: The Quantum Set Qmentioning

confidence: 97%

Geometry of the set of quantum correlations

et al. 2018

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It is well known that correlations predicted by quantum mechanics cannot be explained by any classical (local-realistic) theory. The relative strength of quantum and classical correlations is usually studied in the context of Bell inequalities, but this tells us little about the geometry of the quantum set of correlations. In other words, we do not have good intuition about what the quantum set actually looks like. In this paper we study the geometry of the quantum set using standard tools from convex geometry. We find explicit examples of rather counter-intuitive features in the simplest non-trivial Bell scenario (two parties, two inputs and two outputs) and illustrate them using 2-dimensional slice plots. We also show that even more complex features appear in Bell scenarios with more inputs or more parties. Finally, we discuss the limitations that the geometry of the quantum set imposes on the task of self-testing.

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“…Moreover, for n, k ≥ 2, it is known that C loc (n, k) = C q (n, k). While for n ≥ 5, k ≥ 2, we have C qs (n, k) = C qa (n, k) by [14], and for n ≥ 5, k ≥ 3, we have C q (n, k) = C qs (n, k) [10]. The most famous question is whether or not C qa (n, k) = C qc (n, k), ∀n, k ≥ 2, since this is known to be equivalent to Connes' embedding conjecture [23].…”

Section: Games and Strategiesmentioning

confidence: 99%

“…14 The Hopf * -algebra A = O(G) of representative functions on a compact quantum group G is a natural example of an A − A-bigalois extension admitting a bi-invariant state. Indeed, just take ω = h, the Haar state on A.…”

mentioning

confidence: 99%

Bigalois Extensions and the Graph Isomorphism Game

Brannan

Chirvăsitu

Eifler

et al. 2019

Commun. Math. Phys.

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We study the graph isomorphism game that arises in quantum information theory from the perspective of bigalois extensions of compact quantum groups. We show that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises as a quotient of a certain measured bigalois extension for the quantum automorphism groups G X and G Y of the graphs X and Y . In particular, this implies that the quantum groups G X and G Y are monoidally equivalent. We also establish a converse to this result, which says that every compact quantum group G monoidally equivalent to G X is of the form G Y for a suitably chosen quantum graph Y that is quantum isomorphic to X. As an application of these results, we deduce that the * -algebraic, C * -algebraic, and quantum commuting (qc) notions of a quantum isomorphism between classical graphs X and Y all coincide. Using the notion of equivalence for non-local games, we deduce the same result for other synchronous non-local games, including the synBCS game and certain related graph homomorphism games. Universiteit Leuven, mateusz.wasilewski@kuleuven.be 2. Conversely, for any compact quantum group G monoidally equivalent to G X , one can construct from this monoidal equivalence a quantum graph Y , an isomorphism of quantum groups G ∼ = G Y , and an algebraic quantum isomorphism X ∼ = A * Y .Recasting all of the above in the context of the (classical) graph isomorphism game, our results show that the condition A(Iso(X, Y )) = 0 is sufficient to ensure the existence of perfect quantum strategies for this game (Corollary 4.8 and Theorem 4.9):Theorem Two classical graphs X and Y are algebraically quantum isomorphic if and only if the graph isomorphism game has a perfect quantum-commuting (qc)-strategy.We mention that a weaker version of the above theorem (that assumed the existence of a non-zero C * -algebra representation of A(Iso(X, Y ))) was recently proved in [18].

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Non-closure of the Set of Quantum Correlations via Graphs

Abstract: We prove that the set of quantum correlations for a bipartite system of 5 inputs and 2 outputs is not closed. Our proof relies on computing the correlation functions of a graph, which is a concept that we introduce.2010 Mathematics Subject Classification. Primary 46L05; Secondary 47L90.

Cited by 86 publications

References 12 publications

Perfect Strategies for Non-Local Games

Perfect Strategies for Non-Local Games

Geometry of the set of quantum correlations

Bigalois Extensions and the Graph Isomorphism Game

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