**"An Interpolation Problem for Completely Positive Maps
on Matrices"**

By

** Abstract:** We present certain existence
criteria and parametrizations 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.

**Date: ****Tuesday, October 1, 2013**

**Time: ****16.00**

**Place: ****Mathematics Seminar Room,
SA-141**

Tea and cookies will be served before the
seminar.