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

Sabine Burgdorf(burgdorf***at***cwi.nl)
Monique Laurent(monique***at***cwi.nl)
Teresa Piovesan(piovesan***at***cwi.nl)

Abstract: We investigate structural properties of the completely positive semidefinite cone, consisting of all the nxn 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 the completely positive semidefinite cone, 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.

Keywords: Quantum graph parameters, Trace nonnegative polynomials, Copositive cone, Chromatic number, Quantum Entanglement, Nonlocal games, Von Neumann algebra

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Combinatorial Optimization (Polyhedra )


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Entry Submitted: 03/02/2015
Entry Accepted: 03/02/2015
Entry Last Modified: 03/02/2015

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