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

TL;DR

This paper explores the structure of the completely positive semidefinite cone, providing a hierarchy of polyhedral approximations for its interior and characterizing its closure via tracial ultraproducts, with applications to quantum graph parameters.

Contribution

It introduces a hierarchy of polyhedral cones approximating the cone’s interior and characterizes the cone’s closure using tracial ultraproducts, advancing quantum graph parameter analysis.

Findings

Hierarchy of polyhedral cones covers the cone's interior.

Explicit description of the cone's closure via tracial ultraproducts.

Linear programming methods for quantum chromatic number variants.

Abstract

We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+^n$, 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 which covers the interior of $\mathcal{CS}_+^n$, 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.