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arXiv · 1505.00483

Quantum Graph Homomorphisms via Operator Systems

Carlos M. Ortiz·Vern I. Paulsen
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Abstract

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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Carlos M. Ortiz, Vern I. Paulsen. 2016-02-20. Quantum Graph Homomorphisms via Operator Systems. https://arxiv.org/abs/1505.00483

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