### 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 language | English |
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Title of host publication | Proceedings of the 10th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2015) |

Editors | S. Beigi, R. Koenig |

Place of Publication | Leibniz |

Publisher | Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik |

Pages | 127-146 |

Volume | 44 |

ISBN (Print) | 9783939897965 |

DOIs | |

Publication status | Published - Nov 2015 |

Event | 10th Conference on the Theory of Quantum Computation, Communication and Cryptography - Brussels, Belgium Duration: 20 May 2015 → 22 May 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics |
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Volume | 44 |

### Conference

Conference | 10th Conference on the Theory of Quantum Computation, Communication and Cryptography |
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Country | Belgium |

City | Brussels |

Period | 20/05/15 → 22/05/15 |

### Keywords

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

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## 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). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. https://doi.org/10.4230/LIPIcs.TQC.2015.127