On Schrodinger's bridge problem

Abstract

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.

16 pages, slightly revised version

References in corpus (1)

• Positive contraction mappings for classical and quantum Schrodinger systems

Cited by in corpus (2)

• A review of matrix scaling and Sinkhorn's normal form for matrices and positive maps
• Partial minimization of strict convex functions and tensor scaling