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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.

The structural properties of the completely positive semidefinite cone CSn +, consisting of all the n × n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size, are investigated. 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. Two main results are shown in this paper concerning the structure of the completely positive semidefinite cone, namely, about its interior and about its closure. On the one hand, a hierarchy of polyhedral cones covering the interior of CSn + is constructed, which is used for computing some variants of the quantum chromatic number by way of a linear program. On the other hand, an explicit description of the closure of the completely positive semidefinite cone is given, by showing that it consists of all matrices admitting a Gram representation in the tracial ultraproduct of matrix algebras.

The structural properties of the completely positive semidefinite cone CSn +, consisting of all the n × n symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size, are investigated. 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. Two main results are shown in this paper concerning the structure of the completely positive semidefinite cone, namely, about its interior and about its closure. On the one hand, a hierarchy of polyhedral cones covering the interior of CSn + is constructed, which is used for computing some variants of the quantum chromatic number by way of a linear program. On the other hand, an explicit description of the closure of the completely positive semidefinite cone is given, by showing that it consists of all matrices admitting a Gram representation in the tracial ultraproduct of matrix algebras.

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.

We investigate the completely positive semidefinite matrix cone CSn+, consisting of all n×n matrices that admit a Gram representation by positive semidefinite matrices (of any size). We use this new cone to model quantum analogues of the classical independence and chro- matic graph parameters α(G) and χ(G), which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0/1 scalars in the programs defining the classical parameters. We study relationships between the cone CSn+ and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. By using the truncated tracial quadratic module as sufficient condition for trace positivity, we can define hierarchies of cones aiming to approximate the dual cone of CSn+, which we then use to construct hierarchies of semidefinite programming bounds approximating the quantum graph parameters. Finally we relate their convergence properties to Connes’ embedding conjecture in operator theory.

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.

We investigate the completely positive semidefinite matrix cone CSn+, consisting of all n×n matrices that admit a Gram representation by positive semidefinite matrices (of any size). We use this new cone to model quantum analogues of the classical independence and chro- matic graph parameters α(G) and χ(G), which are roughly obtained by allowing variables to be positive semidefinite matrices instead of 0/1 scalars in the programs defining the classical parameters. We study relationships between the cone CSn+ and the completely positive and doubly nonnegative cones, and between its dual cone and trace positive non-commutative polynomials. By using the truncated tracial quadratic module as sufficient condition for trace positivity, we can define hierarchies of cones aiming to approximate the dual cone of CSn+, which we then use to construct hierarchies of semidefinite programming bounds approximating the quantum graph parameters. Finally we relate their convergence properties to Connes’ embedding conjecture in operator theory.