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.
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January 2016
Research Article|
December 10 2015
Synchronous correlation matrices and Connes’ embedding conjecture
Kenneth J. Dykema;
1Department of Mathematics,
Texas A&M University
, College Station, Texas 77843-3368, USA
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Vern Paulsen
2Department of Mathematics,
University of Houston
, Houston, Texas 77204, USA
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a)
E-mail address: [email protected]
b)
Current address: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. E-mail address: [email protected]
J. Math. Phys. 57, 015214 (2016)
Article history
Received:
March 26 2015
Accepted:
November 16 2015
Connected Content
A correction has been published:
Publisher’s Note: “Synchronous correlation matrices and Connes’ embedding conjecture” [J. Math. Phys. 57, 015214 (2016)]
Citation
Kenneth J. Dykema, Vern Paulsen; Synchronous correlation matrices and Connes’ embedding conjecture. J. Math. Phys. 1 January 2016; 57 (1): 015214. https://doi.org/10.1063/1.4936751
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