The completely positive semidefinite cone is a new matrix cone, which consists of all the symmetric matrices (of a given size) that admit a Gram factorization by positive semidefinite matrices of any size. This cone can thus be seen as a non-commutative analogue of the classical completely positive cone, replacing factorizations by nonnegative vectors (aka diagonal psd matrices) by arbitrary psd matrices. It permits to model bipartite quantum correlations and quantum analogues of classical graph parameters like colouring and stable set numbers, which arise naturally in the context of nonlocal games and the study of entanglement in quantum information.

We will consider various questions related to the structure of the completely positive semidefinite cone. In particular, we will discuss how semidefinite optimization and tracial non-commutative polynomial optimization can be used to design conic approximations, to model the dual cone, and to design lower bounds for matrix factorization ranks, also for the related notions of cp-rank and nonnegative rank.

Joint work with David de Laat (CWI, Amsterdam) and Sander Gribling (CWI, Amsterdam).

Presentation by: Monique Laurent. CWI, Amsterdam, and Tilburg University, Netherlands.