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DOI: 10.4230/LIPIcs.TQC.2015.127
On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings

Authors: Sabine Burgdorf, Monique Laurent, and Teresa Piovesan

Published in: LIPIcs, Volume 44, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)

Abstract
We investigate structural properties of the completely positive semidefinite cone CS^n_+, consisting of all the n x n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This cone has been introduced to model quantum graph parameters as conic optimization problems. Recently it has also been used to characterize the set Q of bipartite quantum correlations, as projection of an affine section of it. We have two main results concerning the structure of the completely positive semidefinite cone, namely about its interior and about its closure. On the one hand we construct a hierarchy of polyhedral cones which covers the interior of CS^n_+, which we use for computing some variants of the quantum chromatic number by way of a linear program. On the other hand we give an explicit description of the closure of the completely positive semidefinite cone, by showing that it consists of all matrices admitting a Gram representation in the tracial ultraproduct of matrix algebras.
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Sabine Burgdorf, Monique Laurent, and Teresa Piovesan. On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings. In 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 44, pp. 127-146, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{burgdorf_et_al:LIPIcs.TQC.2015.127,
  author =	{Burgdorf, Sabine and Laurent, Monique and Piovesan, Teresa},
  title =	{{On the Closure of the Completely Positive Semidefinite Cone and Linear Approximations to Quantum Colorings}},
  booktitle =	{10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)},
  pages =	{127--146},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-96-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{44},
  editor =	{Beigi, Salman and K\"{o}nig, Robert},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2015.127},
  URN =		{urn:nbn:de:0030-drops-55537},
  doi =		{10.4230/LIPIcs.TQC.2015.127},
  annote =	{Keywords: Quantum graph parameters, Trace nonnegative polynomials, Copositive cone, Chromatic number, Quantum Entanglement, Nonlocal games, Von Neumann algebra}
}
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