Discrete-time classical and quantum Markovian evolutions: Maximum entropy problems on path space

TL;DR

This paper extends the Schrödinger bridge theory to classical and quantum discrete-time Markov processes, providing solutions to maximum entropy problems and exploring quantum time reversal and space-time harmonic processes.

Contribution

It introduces a unified approach to maximum entropy problems for both classical and quantum Markov evolutions using multiplicative functional transformations.

Findings

Solution of maximum entropy problems via multiplicative transformations

Extension of Schrödinger bridge theory to quantum Markov processes

Discussion of quantum time reversal and space-time harmonic processes

Abstract

The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path space maximum entropy problems is obtained from the a priori model in both cases via a suitable multiplicative functional transformation. In the quantum case, nonequilibrium time reversal of quantum channels is discussed and space-time harmonic processes are introduced.