A compositional approach to quantum functions

Musto, Benjamin; Reutter, David; Verdon, Dominic

doi:10.1063/1.5020566

Cited by 56 publications

(92 citation statements)

References 61 publications

(235 reference statements)

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“…Nonetheless, such "genuinely quantum" quantum permutations arise naturally in a variety of contexts. For example, in quantum information theory, quantum permutation matrices arise naturally in the framework of non-local games and go under the name "projective permutation matrices" [4,36,34,35]. From the perspective of non-commutative geometry and quantum group theory, N × N quantum permutation matrices were discovered by Wang [46] to be precisely the structure that encode the quantum symmetries of a finite set of N points.…”

Section: Introductionmentioning

confidence: 99%

Topological generation and matrix models for quantum reflection groups

Brannan

Chirvăsitu

Freslon

2020

Advances in Mathematics

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We establish several new topological generation results for the quantum permutation groups S + N and the quantum reflection groups H s+ N . We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for S + N are inner faithful.

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Section: Introductionmentioning

confidence: 99%

Topological generation and matrix models for quantum reflection groups

Brannan

Chirvăsitu

Freslon

2020

Advances in Mathematics

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“…Quantum pseudo-telepathy is exhibited by graphs that are quantum but not classically isomorphic. This work builds on two recent articles, in which Lupini, Mančinska and Roberson [34] and the present authors [37] independently discovered a connection between these quantum isomorphisms and the quantum automorphism groups of graphs [6,9,10,11,14] studied in the framework of compact quantum groups [51]. This connection has already proven to be fruitful, introducing new quantum information-inspired techniques to the study of quantum automorphism groups [8,34].…”

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confidence: 59%

The Morita Theory of Quantum Graph Isomorphisms

Musto

Reutter

Verdon

2018

Commun. Math. Phys.

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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 vertextransitive 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.

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“…Remark 3.5 In the special case where O(X) is equipped its unique tracial δ-form, then the definition of a quantum graph given here is equivalent to the one given in [19]. In addition, as explained in [19], a quantum graph X = (O(X), ψ X , A X ), where O(X) is a commutative C * -algebra, captures precisely the notion of a classical graph. Indeed, in this case the spectrum X of O(X) is a finite set and ψ X is the uniform probability measure on X.…”

Section: Quantum Sets and Graphsmentioning

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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A compositional approach to quantum functions

Cited by 56 publications

References 61 publications

Topological generation and matrix models for quantum reflection groups

Topological generation and matrix models for quantum reflection groups

The Morita Theory of Quantum Graph Isomorphisms

Bigalois Extensions and the Graph Isomorphism Game

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