On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings

S. Burgdorf, Monique Laurent, Teresa Piovesan

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

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.
Original languageEnglish
Title of host publicationProceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015)
EditorsS. Beigi, R. Koenig
Place of PublicationLeibniz
PublisherSchloss Dagstuhl-Leibniz-Zentrum fuer Informatik
Pages127-146
Volume44
ISBN (Print)9783939897965
DOIs
Publication statusPublished - Nov 2015
Event10th Conference on the Theory of Quantum Computation, Communication and Cryptography - Brussels, Belgium
Duration: 20 May 201522 May 2015

Publication series

NameLeibniz International Proceedings in Informatics
Volume44

Conference

Conference10th Conference on the Theory of Quantum Computation, Communication and Cryptography
CountryBelgium
CityBrussels
Period20/05/1522/05/15

Fingerprint

Positive semidefinite
Linear Approximation
Colouring
Closure
Cone
Interior
Conic Optimization
Ultraproduct
Quantum Graphs
Polyhedral Cones
Positive Semidefinite Matrix
Matrix Algebra
Chromatic number
Symmetric matrix
Linear Program
Structural Properties
Projection
Cover
Optimization Problem
Computing

Keywords

  • quantum graph parameters
  • trace nonnegative polynomials
  • copositive cone
  • chromatic number
  • quantum entanglement
  • nonlocal games
  • Von Neumann algebra

Cite this

Burgdorf, S., Laurent, M., & Piovesan, T. (2015). On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. In S. Beigi, & R. Koenig (Eds.), Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015) (Vol. 44, pp. 127-146). (Leibniz International Proceedings in Informatics; Vol. 44). Leibniz: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TQC.2015.127
Burgdorf, S. ; Laurent, Monique ; Piovesan, Teresa. / On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). editor / S. Beigi ; R. Koenig. Vol. 44 Leibniz : Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. pp. 127-146 (Leibniz International Proceedings in Informatics).
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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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Burgdorf, S, Laurent, M & Piovesan, T 2015, On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. in S Beigi & R Koenig (eds), Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). vol. 44, Leibniz International Proceedings in Informatics, vol. 44, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Leibniz, pp. 127-146, 10th Conference on the Theory of Quantum Computation, Communication and Cryptography, Brussels, Belgium, 20/05/15. https://doi.org/10.4230/LIPIcs.TQC.2015.127

On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. / Burgdorf, S.; Laurent, Monique; Piovesan, Teresa.

Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). ed. / S. Beigi; R. Koenig. Vol. 44 Leibniz : Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. p. 127-146 (Leibniz International Proceedings in Informatics; Vol. 44).

Research output: Chapter in Book/Report/Conference proceedingConference contributionScientificpeer-review

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N2 - 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.

AB - 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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Burgdorf S, Laurent M, Piovesan T. On the closure of the completely positive semidefinite cone and linear approximations to quantum colorings. In Beigi S, Koenig R, editors, Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015). Vol. 44. Leibniz: Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. 2015. p. 127-146. (Leibniz International Proceedings in Informatics). https://doi.org/10.4230/LIPIcs.TQC.2015.127