We investigate structural properties of the completely positive semidefinite cone CS^n_+ , consisting of all the n×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.
Additional Metadata
Keywords Quantum graph parameters, trace nonnegative polynomials, copositive cone, chromatic number, quantum entanglement, nonlocal games, Von Neumann algebra
THEME Information (theme 2)
Publisher LIPICS - Leibniz International Proceedings in Informatics Schloss Dagstuhl -Dagstuhl Publishing
Editor S. Beigi , R. Koenig
Project Approximation Algorithms, Quantum Information and Semidefinite Optimization
Conference Conference on Theory of Quantum Computation, Communication and Cryptography
Grant This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/617.001.351 - Approximation Algorithms, Quantum Information and Semidefinite Optimization
Citation
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 Theory of Quantum Computation, Communication and Cryptography (TQC 2015) (pp. 127–146). LIPICS - Leibniz International Proceedings in Informatics Schloss Dagstuhl -Dagstuhl Publishing.