Quantum graphs as quantum relations

Weaver, Nik

doi:10.48550/arxiv.1506.03892

Cited by 6 publications

(23 citation statements)

References 5 publications

(19 reference statements)

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“…The winning strategies for this game give rise to a notion of quantum graph homomorphism that consolidates and generalizes several notions of quantum graph homomorphism in the literature [34,44,52]. We also construct a game * -algebra for this and show that this game algebra extends the game algebra for graph homomorphisms given in [22].…”

mentioning

confidence: 83%

“…In this section, we define the quantum-to-classical game for quantum-classical graph homomorphisms. Throughout our discussion, we use the bimodule perspective of quantum graphs considered by N. Weaver [51,52] (which is a direct generalization of the non-commutative graphs considered by R. Duan, S. Severini and A. Winter in [13], and D. Stahlke in [44]).…”

Section: The Game For Quantum-to-classical Graph Homomorphismsmentioning

confidence: 99%

“…Traditionally, a quantum graph is viewed as an operator system that serves as a quantum generalization of the adjacency matrix. It was first introduced in [13] for studying a zero-error channel capacity problem and arose independently in the study of quantum relations [51,52] around the same time. An alternate approach was used in [34] to define a quantum graph using a quantum adjacency matrix acting on a finite-dimensional C * -algebra which plays the role of functions on the vertex set.…”

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confidence: 99%

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The quantum-to-classical graph homomorphism game

Brannan¹,

Ganesan²,

Harris³

2020

Preprint

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Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a "quantumclassical game"-that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game directly generalize the notion of non-commutative graph homomorphisms due to D. Stahlke [44]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by J.W. Helton, K. Meyer, V.I. Paulsen and M. Satriano [22]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the 4-coloring game for a quantum graph is always non-trivial, extending a result of [22].

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confidence: 83%

Section: The Game For Quantum-to-classical Graph Homomorphismsmentioning

confidence: 99%

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confidence: 99%

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The quantum-to-classical graph homomorphism game

Brannan¹,

Ganesan²,

Harris³

2020

Preprint

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“…The idea is that the edge structure of a classical graph can be encoded in an obvious way as a reflexive, symmetric relation on a set. This point of view was explicitly connected to the quantum error correction literature in [9]. Ramsey's theorem states that for any k there exists n such that every graph with at least n vertices contains either a k-clique or a k-anticlique, i.e., a set of k vertices among which either all edges are present or no edges are present.…”

Section: Introductionmentioning

confidence: 99%

“…As mentioned earlier, classical codes are taken to be independent sets, which is to say, anticliques. See also Section 4 of [9], where intuition for why P VP is correctly thought of as a "restriction" of V is given.…”

Section: Introductionmentioning

confidence: 99%

A “quantum” Ramsey theorem for operator systems

Weaver

2017

Proc. Amer. Math. Soc.

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Abstract. Let V be a linear subspace of Mn(C) which contains the identity matrix and is stable under the formation of Hermitian adjoints. We prove that if n is sufficiently large then there exists a rank k orthogonal projection P such that dim(P VP ) = 1 or k 2 . BackgroundAn operator system in finite dimensions is a linear subspace V of M n (C) with the propertieswhere I n is the n × n identity matrix and A * is the Hermitian adjoint of A. In this paper the scalar field will be complex and we will write M n = M n (C).Operator systems play a role in the theory of quantum error correction. In classical information theory, the "confusability graph" is a bookkeeping device which keeps track of possible ambiguity that can result when a message is transmitted through a noisy channel. It is defined by taking as vertices all possible source messages, and placing an edge between two messages if they are sufficiently similar that data corruption could lead to them being indistinguishable on reception. Once the confusability graph is known, one is able to overcome the problem of information loss by using an independent subset of the confusability graph, which is known as a "code". If it is agreed that only code messages will be sent, then we can be sure that the intended message is recoverable.When information is stored in quantum mechanical systems, the problem of error correction changes radically. The basic theory of quantum error correction was laid down in [3]. In [2] it was suggested that in this setting the role of the confusability graph is played by an operator system, and it was shown that for every operator system a "quantum Lovász number" could be defined, in analogy to the classical Lovász number of a graph. This is an important parameter in classical information theory. See also [5] for much more along these lines.The interpretation of operator systems as "quantum graphs" was also proposed in [8], based on the more general idea of regarding linear subspaces of M n as "quantum relations", and taking the conditions I n ∈ V and A ∈ V ⇒ A * ∈ V to respectively express reflexivity and symmetry conditions. The idea is that the edge structure of a classical graph can be encoded in an obvious way as a reflexive, symmetric relation on a set. This point of view was explicitly connected to the quantum error correction literature in [9].

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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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Quantum graphs as quantum relations

Cited by 6 publications

References 5 publications

The quantum-to-classical graph homomorphism game

The quantum-to-classical graph homomorphism game

A “quantum” Ramsey theorem for operator systems

The Morita Theory of Quantum Graph Isomorphisms

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