  In the first part of this paper we generalize the result of Georgiou-Pavon
that a positive square matrix can be scaled uniquely to a column stochastic
matrix which maps a given positive probability vector to another given positive
probability vector. In the second part of this paper we prove that a positive
quantum channel can be scaled to another positive quantum channel which maps a
given positive definite density matrix to another given positive definite
density matrix using Brower's fixed point theorem. This result proves the
Georgiou-Pavon conjecture for two positive definite density matrices in their
recent paper. We show uniqueness of fixed points for certain two positive
definite density matrices.
