Synchronous correlation matrices and Connes’ embedding conjecture

Journal Article · Fri Jan 15 04:00:00 EST 2016 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4936751· OSTI ID:22479618

Paulsen, Vern ^[1]

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1. Department of Mathematics, University of Houston, Houston, Texas 77204 (United States)

In the work of Paulsen et al. [J. Funct. Anal. (in press); preprint arXiv:1407.6918], the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of various versions of quantum chromatic numbers of graphs and other quantum versions of graph theoretic parameters. In this paper, we develop these ideas further, focusing on the relations between synchronous correlation matrices and microstates. We prove that Connes’ embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices. We prove that if Connes’ embedding conjecture has a positive answer, then the tracial rank and projective rank are equal for every graph. We then apply these results to more general non-local games.

• OSTI ID: 22479618
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 1 Vol. 57; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGEBRA
CORRELATIONS
DIAGRAMS
GRAPH THEORY
MATRICES