Reducibility Theory and Ergodic Theorems for Ergodic Quantum Processes

TL;DR

This paper develops a Perron-Frobenius theory for ergodic quantum processes, unifying various models and establishing ergodic theorems with refinements for i.i.d. cases.

Contribution

It introduces a unifying Perron-Frobenius framework for ergodic quantum channels, covering multiple models and providing new ergodic theorems and refinements.

Findings

Unified framework for ergodic quantum processes.

Characterizations of irreducibility leading to ergodic theorems.

Refined results specifically for i.i.d. quantum processes.

Abstract

We develop a Perron-Frobenius type theory for products of random quantum channels acting on finite-dimensional matrix algebras sampled from a stationary and ergodic stochastic process, which, in keeping with the literature, we call ergodic quantum processes. This serves as a unifying framework for many models, including i.i.d., Markovian, periodic, and quasiperiodic models. We establish various characterizations of irreducibility, from which we recover a number of general ergodic theorems. We then analyze some specific examples, and, in particular, give a refinement of our theory in the i.i.d. case.