  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.
