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arXiv:1903.12369 · Journal of Mathematical Physics

Quantum semigroups from synchronous games

P. M. Sołtan

Published Apr 1, 2019• 6 citations
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Abstract

We show that the C*-algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game. In particular situations (e.g., for graph endomorphism games), these quantum families have a natural quantum semigroup structure, and if the condition of preservation of a natural state is added, they are in fact compact quantum groups.

DOI
10.1063/1.5085118

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We show that the C*-algebras associated with synchronous games give rise to certain quantum families of maps between the input and output sets of the game. In particular situations (e.g., for graph endomorphism games), these quantum families have a natural quantum semigroup structure, and if the condition of preservation of a natural state is added, they are in fact compact quantum groups.
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 nnn inputs and kkk outputs is ∗*∗-isomorphic to the game algebra of an associated bisynchronous game on nknknk inputs and nknknk 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 nnn questions and k>3k>3k>3 answers and a synchronous game algebra with n(k−2)n(k-2)n(k−2) questions and 333 answers.
We study a class of nonlocal games, called transitive games, for which the set of perfect strategies forms a semigroup. We establish several interesting correspondences of bisynchronous transitive games with the theory of compact quantum groups. In particular, we associate a quantum permutation group with each bisynchronous transitive game and vice versa. We prove that the existence of a C*-strategy, the existence of a quantum commuting strategy, and the existence of a classical strategy are all equivalent for bisynchronous transitive games. We then use some of these correspondences to establish necessary and sufficient conditions for some classes of correlations, that arise as perfect strategies of transitive games, to be nonlocal.
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.
We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via various Grobner basis methods.
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.