Categorical relations and bipartite entanglement in tensor cones for Toeplitz and Fejér-Riesz operator systems

Abstract: The present paper aims to understand separability and entanglement in tensor cones, in the sense of Namioka and Phelps, that arise from the base cones of operator system tensor products. Of particular interest here are the Toeplitz and Fej\'er-Riesz operator systems, which are, respectively, operator systems of Toeplitz matrices and Laurent polynomials of bounded degree (that is, trigonometric polynomials), and which are related in the operator system category through duality. Some additional categorical relationships established in this paper for Toeplitz and Fej\'er-Riesz operator systems. Of independent interest is a single matrix criterion, similar to the criterion involving the Choi matrix, for a linear map of the Fej\'er-Riesz operator system to be completely positive.