Communications in Mathematical Physics

, Volume 365, Issue 2, pp 797–845 | Cite as

The Morita Theory of Quantum Graph Isomorphisms

  • Benjamin Musto
  • David Reutter
  • Dominic VerdonEmail author
Open Access


We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.



We are grateful to Jamie Vicary for many useful discussions and to David Roberson for sending us an early draft of [34]. We thank an anonymous referee for helpful comments on an earlier draft of this work. This work was supported by the Engineering and Physical Sciences Research Council and the Clarendon Trust.


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Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

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