Quantum graph homomorphisms via operator systems

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• 作者：Carlos M. Ortiz^a ; ^{carlos.ortiz@pnnl.gov" class="auth_mail" title="E-mail the corresponding author} ; Vern I. Paulsen^b ; ^{vpaulsen@uwaterloo.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46L15 ; secondary ; 47L25
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 May 2016
• 年：2016
• 卷：497
• 期：Complete
• 页码：23-43
• 全文大小：403 K

文摘

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.