Open AccessOn the closure of the completely positive semidefinite cone and linear approximations to quantum colorings
Author(s) -
Sabine Burgdorf,
Monique Laurent,
Teresa Piovesan
Publication year - 2017
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/1081-3810.3201
Subject(s) - mathematics , positive definite matrix , cone (formal languages) , conic section , bipartite graph , convex cone , conic optimization , combinatorics , semidefinite programming , quantum , matrix (chemical analysis) , vertex (graph theory) , discrete mathematics , pure mathematics , regular polygon , graph , convex set , convex optimization , eigenvalues and eigenvectors , geometry , quantum mechanics , mathematical optimization , physics , materials science , algorithm , composite material
We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+$, consisting of all the $n \times n$ 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 $\mathcal Q$ 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 covering the interior of $\mathcal{CS}_+$, 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.