An interpolation problem for completely positive maps on matrix algebras: solvability and parametrization

Abstract

We present certain existence criteria and parameterizations for an interpolation problem for completely positive maps that take given matrices from a finite set into prescribed matrices. Our approach uses density matrices associated to linear functionals on (Formula presented.) -subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson’s Hahn-Banach Type Theorem. A necessary and sufficient condition for the existence of solutions and a parametrization of the set of all solutions of the interpolation problem in terms of a closed and convex set of an affine space are obtained. Other linear affine restrictions, like trace preserving, can be included as well, hence covering applications to quantum channels that yield certain quantum states at prescribed quantum states. We also perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional.