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

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.Comment: 20 page

Approximate Completely Positive Semidefinite Rank

In this paper we provide an approximation for completely positive semidefinite (cpsd) matrices with cpsd-rank bounded above (almost) independently from the cpsd-rank of the initial matrix. This is particularly relevant since the cpsd-rank of a matrix cannot, in general, be upper bounded by a function only depending on its size. For this purpose, we make use of the Approximate Carath\'eodory Theorem in order to construct an approximate matrix with a low-rank Gram representation. We then employ the Johnson-Lindenstrauss Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.Comment: v2: clarified and corrected some citation

Quantum Bilinear Optimization

We study optimization programs given by a bilinear form over noncommutative variables subject to linear inequalities. Problems of this form include the entangled value of two-prover games, entanglement-assisted coding for classical channels, and quantum-proof randomness extractors. We introduce an asymptotically converging hierarchy of efficiently computable semidefinite programming (SDP) relaxations for this quantum optimization. This allows us to give upper bounds on the quantum advantage for all of these problems. Compared to previous work of Pironio, Navascués, and Acín [SIAM J. Optim., 20 (2010), pp. 2157-2180], our hierarchy has additional constraints. By means of examples, we illustrate the importance of these new constraints both in practice and for analytical properties. Moreover, this allows us to give a hierarchy of SDP outer approximations for the completely positive semidefinite cone introduced by Laurent and Piovesan

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

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

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

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

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

We investigate 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. 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 CSn+, 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

Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples