An Interpolation Problem for Completely Positive Maps on Matrix Algebras: Solvability and Parametrisation

TL;DR

This paper develops criteria and parametrizations for solving an interpolation problem involving completely positive maps on matrix algebras, with applications to quantum channels and states.

Contribution

It introduces new existence criteria and a parametrization method for the interpolation problem, connecting density matrices and linear functionals in a novel way.

Findings

Necessary and sufficient conditions for solution existence

Parametrization of all solutions via convex sets

Inclusion of linear restrictions like trace preservation

Abstract

We present certain existence criteria and parameterisations 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 *-subspaces of matrices, inspired by the Smith-Ward linear functional and Arveson's Hahn-Banach type Theorem. We perform a careful investigation on the intricate relation between the positivity of the density matrix and the positivity of the corresponding linear functional. A necessary and sufficient condition for the existence of solutions and a parametrisation 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.