A Coherent Approach to Recurrence and Transience for Quantum Markov Operators

TL;DR

This paper introduces a unified framework for analyzing recurrence and transience in quantum Markov operators, providing new theorems, classifications, and applications to quantum channels and operator algebras.

Contribution

It develops a coherent approach based on the Riesz decomposition, leading to new classifications of idempotent Markov operators and representations of quantum channels.

Findings

Classification of idempotent Markov operators

Introduction of the Choi-Effros product on their range

Abstract Poisson integral and representation theorem for entanglement breaking channels

Abstract

We present a coherent approach to recurrence and transience, starting from a version of the Riesz decomposition theorem for superharmonic elements. Our approach allows straightforward proofs of some known results, entails new theorems, and has applications to other aspects of completely positive operators: It leads to a classification of idempotent Markov operators, thereby identifying concretely the Choi-Effros product, which can be introduced on the range of these maps. We obtain an abstract Poisson integral and a representation theorem for idempotent entanglement breaking channels.