Asymptotic Properties for Markovian Dynamics in Quantum Theory and General Probabilistic Theories

TL;DR

This paper explores the long-term behavior of Markovian quantum and probabilistic systems, establishing equivalences between asymptotic decoupling, local mixing, and ergodicity, with implications for understanding system convergence.

Contribution

It demonstrates that asymptotic decoupling is equivalent to local mixing and that mixing is equivalent to ergodicity in the context of general probabilistic theories, providing new criteria for system convergence.

Findings

Asymptotic decoupling equals local mixing.

Mixing is equivalent to ergodicity for tensor product dynamics.

Provides linear equations for mixing criteria.

Abstract

We address asymptotic decoupling in the context of Markovian quantum dynamics. Asymptotic decoupling is an asymptotic property on a bipartite quantum system, and asserts that the correlation between two quantum systems is broken after a sufficiently long time passes. The first goal of this paper is to show that asymptotic decoupling is equivalent to local mixing which asserts the convergence to a unique stationary state on at least one quantum system. In the study of Markovian dynamics, mixing and ergodicity are fundamental properties which assert the convergence and the convergence of the long-time average, respectively. The second goal of this paper is to show that mixing for dynamics is equivalent to ergodicity for the two-fold tensor product of dynamics. This equivalence gives us a criterion of mixing that is a system of linear equations. All results in this paper are proved in the framework of general probabilistic theories (GPTs), but we also summarize them in quantum theory.