Title:
Completely positive semidefinite matrices: some properties and applications

Abstract:
We consider the completely positive semidefinite cone CS_+, which consists of all symmetric matrices admitting a Gram representation by positive semidefinite matrices (of any size). This cone permits to model some optimization problems in quantum information as conic optimization over affine sections of CS_+ and its dual cone relates to tracial nonnegative quadratic polynomials. We will discuss hierarchies of inner and outer approximations for CS_+, and show classes of matrices in CS_+ having exponentially large CS_+-rank (aka the smallest dimension of the psd matrices in a Gram representation).