On Schrodinger's bridge problem

Understand

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.