Quantum Graph Homomorphisms via Operator Systems

Ortiz, Carlos M.; Paulsen, Vern I.

doi:10.48550/arxiv.1505.00483

Cited by 4 publications

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“…Various definitions of quantum graph homomorphisms were proposed in [8,9,11]. Here the term "homomorphism" conflicts somewhat with classical usage, where a homomorphism between graphs is usually taken to be an actual function between the vertex sets, not a channel which could map vertices to probability distributions.…”

Section: Pushforwardsmentioning

confidence: 99%

Quantum graphs as quantum relations

Weaver¹

2015

Preprint

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The "noncommutative graphs" which arise in quantum error correction are a special case of the quantum relations introduced in [12]. We use this perspective to interpret the Knill-Laflamme error-correction conditions [6] in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke's noncommutative graph homomorphisms [11] and Duan, Severini, and Winter's noncommutative bipartite graphs [3], and to realize the noncommutative confusability graph associated to a quantum channel [3] as the pullback of a diagonal relation.Our framework includes as special cases not only purely classical and purely quantum information theory, but also the "mixed" setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

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Section: Pushforwardsmentioning

confidence: 99%

Quantum graphs as quantum relations

Weaver¹

2015

Preprint

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“…Similarly, [19] also shows that there exists a quantum graph homomorphism, as defined by [13] and [25], from G to H if and only if A(G, H) has a non-zero *-homomorphism into the matrices, that is, a non-zero finite dimensional representation. A result of [8] shows that the problem of determining if χ q (G) ≤ 3 is an NP-hard problem.…”

Section: Introductionmentioning

confidence: 95%

“…Given graphs G and H, C. Ortiz and the third author [19] affiliated a *-algebra to this pair, denoted A(G, H), and characterized the existence of graph homomorphisms from G to H in terms of representations of this algebra: [19] proves that A(G, H) has a non-zero homomorphism into the complex numbers if and only if there exists a classical graph homomorphism from G to H. Because the problem of determining if χ(G) ≤ 3 is an NP-complete problem, the results of [19] allow us to deduce that the problem of determining if A(G, K 3 ) has a non-trivial homomorphism into the complex numbers is an NP-complete problem.…”

Section: Introductionmentioning

confidence: 99%

“…Currently, the computational complexity of these other variations of the quantum chromatic number is unknown. The results of [19] characterize the existence of these other types of quantum graph homomorphisms in terms of the existence of various types of traces on the algebra A(G, H). Thus, questions about the existence or non-existence of these various types of graph homomorphisms, and consequently quantum colorings, is reduced to questions about these algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Since all the earlier notions of quantum graph homomorphism and the corresponding quantum chromatic numbers, were defined in terms of the existence of perfect quantum strategies for certain games, we begin by identifying a certain family of games for which we can carry out the construction of [19] and affiliate an algebra to the game such that the existence of various types of representations of the algebra determines whether or not various types of perfect quantum strategies exist for the game. For this family of games it is possible that all of the notions of perfect quantum strategies coincide and are equivalent to a certain hereditary ideal in this algebra being proper.…”

Section: Introductionmentioning

confidence: 99%

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Algebras, Synchronous Games and Chromatic Numbers of Graphs

Helton,

Meyer,

Paulsen

et al. 2017

Preprint

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We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other strategies. When applied to the graph coloring game, this leads to characterizations in terms of properties of an algebra of various quantum chromatic numbers that have been studied in the literature. This allows us to develop a correspondence between various chromatic numbers of a graph and ideals in this algebra which can then be approached via Gröbner basis methods.

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Quantum and non-signalling graph isomorphisms

Atserias¹,

Mančinska²,

Roberson³

et al. 2016

Preprint

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We introduce a two-player nonlocal game, called the (G, H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G, H)-isomorphism game, respectively. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. We prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. On the basis of this correspondence, we show that quantum isomorphic graphs are necessarily cospectral. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to produce quantum isomorphic graphs that are nevertheless not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our construction is related to the FGLSS reduction from inapproximability literature, as well as the CFI construction.

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Quantum Graph Homomorphisms via Operator Systems

Cited by 4 publications

References 0 publications

Quantum graphs as quantum relations

Quantum graphs as quantum relations

Algebras, Synchronous Games and Chromatic Numbers of Graphs

Quantum and non-signalling graph isomorphisms

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