Burgdorf,S.Laurent,MoniquePiovesan,TeresaBeigi,S.Koenig,R.2025-01-312025-01-312015-11Burgdorf, 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.2015.1279783939897965ORCID: /0000-0001-8474-2121/work/12133409410.4230/LIPIcs.TQC.2015.127https://hdl.handle.net/20.500.14602/20483We 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.enginfo:eu-repo/semantics/closedAccessquantum graph parameterstrace nonnegative polynomialscopositive conechromatic numberquantum entanglementnonlocal gamesVon Neumann algebraOn the closure of the completely positive semidefinite cone and linear approximations to quantum coloringsConference contributionGeneral rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. - Users may download and print one copy of any publication from the public portal for the purpose of private study or research. - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal" Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.https://www.scopus.com/pages/publications/849590590519660232https://research.tilburguniversity.edu/en/publications/7096e1f5-c8f7-4fd1-be71-ed217b526870(c) Universiteit van TilburgBurgdorf, S.Laurent, Monique§0000-0001-8474-2121Piovesan, TeresaBeigi, S.Koenig, R.