On Schrodinger's bridge problem

TL;DR

This paper extends classical matrix scaling results to quantum channels, proving a conjecture about their ability to be scaled to map one density matrix to another, with implications for quantum information theory.

Contribution

It generalizes matrix scaling to quantum channels, proving the Georgiou-Pavon conjecture for positive definite density matrices using fixed point theorems.

Findings

Quantum channels can be scaled to map one density matrix to another.

Uniqueness of fixed points for certain pairs of density matrices.

Validation of the Georgiou-Pavon conjecture in the quantum setting.

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.