The Morita Theory of Quantum Graph Isomorphisms

Musto, Benjamin; Reutter, David; Verdon, Dominic

doi:10.1007/s00220-018-3225-6

Cited by 34 publications

(37 citation statements)

References 53 publications

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“…Concerning the Paley graph P 17 , observe that 0, 2,4,8,9,13,15,16 are the squares in F 17 . Let (i, k), (j, l) ∈ E. Since P 17 is distance-transitive, we can choose j = 1, l = 2 by Lemma 3.3.…”

Section: 5mentioning

confidence: 99%

On the quantum symmetry of distance-transitive graphs

Schmidt

2020

Advances in Mathematics

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In this article, we study quantum automorphism groups of distancetransitive graphs. We show that the odd graphs, the Hamming graphs H(n, 3), the Johnson graphs J(n, 2) and the Kneser graphs K(n, 2) do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order ≤ 20 has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the 4 × 4 rook's graph), where one of them has quantum symmetry and the other one does not. The author is supported by the DFG project Quantenautomorphismen von Graphen. He thanks his supervisor Moritz Weber for proofreading the article and many helpful comments and suggestions. This article is part of the author's PhD thesis.Theorem 1.4. For n ≥ 5, the Johnson graphs J(n, 2) and the Kneser graphs K(n, 2) do not have quantum symmetry.

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Section: 5mentioning

confidence: 99%

On the quantum symmetry of distance-transitive graphs

Schmidt

2020

Advances in Mathematics

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“…All of the definitions that follow are quite standard in the operator algebra literature [28,2,3,11]. The idea of a quantum set or a quantum graph also appears in [19,20] using the language of special symmetric dagger Frobenius algebras.…”

Section: Quantum Sets Graphs and Their Quantum Automorphism Groupsmentioning

confidence: 99%

“…As is the case classically, every quantum graph X has a quantum automorphism group G X . The recent papers [18,19,20] uncover further remarkable connections between graph isomorphism games and quantum automorphism groups. Moreover, while [18] focuses on classical graphs, [19,20] consider a more general categorical quantum mechanical framework which leads naturally to the notion of a quantum graphs and the generalization of the graph isomorphism game to that framework.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, [20] obtains a characterization of (finite-dimensional) quantum isomorphic quantum graphs X, Y in terms of simple dagger Frobenius monoids in the category of finite dimensional representations of the Hopf * -algebra O(G X ) of the corresponding quantum automorphism group G X . On the other hand, [18] uses ideas from quantum group theory to establish the equivalence between the existence of C * -quantum isomorphisms for graphs and the existence of perfect strategies for the isomorphism game within the so-called quantum commuting framework.…”

Section: Introductionmentioning

confidence: 99%

“…Here, we continue in the same vein investigating connections between quantum automorphism groups of graphs and the graph isomorphism game, taking a somewhat dual approach to the one in [19,20]:…”

Section: Introductionmentioning

confidence: 99%

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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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Covariant Quantum Combinatorics with Applications to Zero-Error Communication

Verdon

2024

Commun. Math. Phys.

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We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $$C^*$$ C ∗ -algebras) carry an action of a compact quantum group G, and all channels (completely positive maps preserving a certain G-invariant functional) are covariant with respect to the G-actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: (1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. (2) We show that every quantum confusability graph with a G-action (which we call a quantum G-graph) arises as the confusability graph of a covariant channel. (3) We show that a covariant channel is reversible precisely when its confusability G-graph is discrete. (4) When G is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability G-graphs.

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The Morita Theory of Quantum Graph Isomorphisms

Cited by 34 publications

References 53 publications

On the quantum symmetry of distance-transitive graphs

On the quantum symmetry of distance-transitive graphs

Bigalois Extensions and the Graph Isomorphism Game

Covariant Quantum Combinatorics with Applications to Zero-Error Communication

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