Bigalois Extensions and the Graph Isomorphism Game

Abstract

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].

Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups

“…As an analogue of the fact that simple undirected classical graphs are irreflexive symmetric relations, Weaver [19] formulated quantum graphs as reflexive symmetric quantum relations on a von Neumann algebra, which extends [10] and quantum relations were introduced by Kuperberg, Weaver [12]. Following those works, Musto, Reutter, Verdon [13] formulated finite quantum graphs as adjacency operators on tracial finite quantum sets, and Brannan et al [7] generalized them for nontracial setting.…”

“…The key tool of [13] is the string diagrams formulated by Vicary [17], but it should be treated with care if applied to nontracial quantum graphs in [7]. So in the former part of this paper we discuss the diagramatic formulation of nontracial quantum graphs.…”

“…Brannan et al [7] also introduced the quantum automorphism groups and bigalois extensions of quantum graphs in order to refine the notion of quantum isomorphisms between quantum graphs. The quantum automorphism group of classical graphs was first introduced by Bichon [6, Definition 3.1] in a slightly different way from [7].…”

“…Brannan et al [7] also introduced the quantum automorphism groups and bigalois extensions of quantum graphs in order to refine the notion of quantum isomorphisms between quantum graphs. The quantum automorphism group of classical graphs was first introduced by Bichon [6, Definition 3.1] in a slightly different way from [7]. The origin of the formulation in [7] is due to Banica [4,Definition 3.2], following the quantum symmetry group of finite spaces introduced by Wang [18,Definition 2.3].…”

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Motivated by string diagramatic approach to undirected tracial quantum graphs by Musto, Reutter, Verdon [13], in the former part of this paper we diagramatically formulate directed nontracial quantum graphs by Brannan, Chirvasitu, Eifler, Harris, Paulsen, Su, Wasilewski [7]. In the latter part we supply a concrete classification of undirected reflexive quantum graphs on M2 and their quantum automorphism groups in both tracial and nontracial settings.

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