Lovász theta type norms and operator systems

We explore the concept of a graph homomorphism through the lens of C * -algebras and operator systems. We start by studying the various notions of a quantum graph homomorphism and examine how they are related to each other. We then define and study a C *algebra that encodes all the information about these homomorphisms and establish a connection between computational complexity and the representation of these algebras. We use this C * -algebra to define a new quantum chromatic number and establish some basic properties of this number. We then suggest a way of studying these quantum graph homomorphisms using certain completely positive maps and describe their structure. Finally, we use these completely positive maps to define the notion of a "quantum" core of a graph.

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“…Recall [12] that the operator system of a graph G on n vertices is the subspace of the n × n complex matrices M n given by…”

Section: Quantum Homomorphisms and Cp Mapsmentioning

confidence: 99%

Quantum graph homomorphisms via operator systems

Ortiz

Paulsen

2016

Linear Algebra and its Applications

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“…In this section, we examine inductive limits of graph operator systems, viewing them as the operator systems of topological graphs via the theory of topological equivalence relations [34]. We identify the C*-envelope of such an operator system, and prove an isomorphism theorem; these can be viewed as a topological version of recent results from [28]. We also establish a version of the Glimm Theorem for this class of operator systems.…”

Section: Inductive Limits Of Graph Operator Systemsmentioning

confidence: 99%

“…In Section 6, we consider inductive limits of graph operator systems. This class of operator systems was introduced in [11] and subsequently studied in [28], where the authors showed that the graph operator system is a complete isomorphism invariant for the corresponding graph, and identified its C*-envelope. In view of the importance of graph operator systems in Quantum Information Theory, where they correspond to confusability graphs of quantum channels [11], we establish inductive limit versions of the aforementioned results.…”

Section: Introductionmentioning

confidence: 99%

Inductive limits in the operator system and related categories

Mawhinney

Todorov

2018

Dissertationes Math.

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We present a systematic development of inductive limits in the categories of ordered *-vector spaces, Archimedean order unit spaces, matrix ordered spaces, operator systems and operator C*-systems. We show that the inductive limit intertwines the operation of passing to the maximal operator system structure of an Archimedean order unit space, and that the same holds true for the minimal operator system structure if the connecting maps are complete order embeddings. We prove that the inductive limit commutes with the operation of taking the maximal tensor product with another operator system, and establish analogous results for injective functorial tensor products provided the connecting maps are complete order embeddings. We identify the inductive limit of quotient operator systems as a quotient of the inductive limit, in case the involved kernels are completely biproximinal. We describe the inductive limit of graph operator systems as operator systems of topological graphs, show that two such operator systems are completely order isomorphic if and only if their underlying graphs are isomorphic, identify the C*-envelope of such an operator system, and prove a version of Glimm's Theorem on the isomorphism of UHF algebras in the category of operator systems. ContentsIn order to avoid excessive notation, we will sometimes denote the ordered *-vector space (V, V + , e) simply by V .Let us denote by OU the category whose objects are ordered *-vector spaces with order units and whose morphisms are unital positive maps, and by AOU the category whose objects are AOU spaces and whose morphisms are unital positive maps. Clearly, we have a forgetful functor F : AOU → OU. In [33, Section 3.2], the process of Archimedeanisation is discussed which provides us with a left adjoint to this functor. Let (V, V + , e) be an ordered *-vector space with order unit. Define D = {v ∈ V h : re + v ∈ V + for every r > 0} and (2) N = {v ∈ V : f (v) = 0 for all f ∈ S(V )}. Clearly, D is a cone, while N is a linear subspace of V . Equip V /N with the involution given by (v + N ) * = v * + N , and set (V /N ) + = {v + N : v ∈ D}.

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“…It is not known whether the converse is true or not. However, motivated by the discussions in [19,Section 3], we obtain the following characterization of the (min, max)-nuclear graph operator systems.…”

Section: Nuclearity Of Graph Operator Systemsmentioning

confidence: 99%

“…In Section 6, we characterize (min, max)-nuclear graph operator systems (as introduced in [15]) for finite graphs, purely in terms of graph-theoretic properties. We achieve this characterization using an identification of their C * -envelopes obtained in [19].…”

Section: Introductionmentioning

confidence: 99%

Operator System Nuclearity via -Envelopes

Gupta

Luthra

2016

J. Aust. Math. Soc.

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We prove that an operator system is (min, ess)-nuclear if its $C^{\ast }$-envelope is nuclear. This allows us to deduce that an operator system associated to a generating set of a countable discrete group by Farenick et al. [‘Operator systems from discrete groups’, Comm. Math. Phys.329(1) (2014), 207–238] is (min, ess)-nuclear if and only if the group is amenable. We also make a detailed comparison between ess and other operator system tensor products and show that an operator system associated to a minimal generating set of a finitely generated discrete group (respectively, a finite graph) is (min, max)-nuclear if and only if the group is of order less than or equal to three (respectively, every component of the graph is complete).

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Lovász theta type norms and operator systems

Cited by 20 publications

References 14 publications

Quantum graph homomorphisms via operator systems

Quantum graph homomorphisms via operator systems

Inductive limits in the operator system and related categories

Operator System Nuclearity via -Envelopes

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