Estimating quantum chromatic numbers

Vern I. Paulsen, Simone Severini, Daniel Stahlke, Ivan G. Todorov, Andreas Winter

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We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lovász number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.
Original languageEnglish
Pages (from-to)2180-2222
Number of pages43
JournalJournal of Functional Analysis
Issue number6
Early online date02 Feb 2016
Publication statusPublished - 15 Mar 2016

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