Attractor Subspace and Decoherence-Free Algebra of Quantum Dynamics

TL;DR

This paper reviews the asymptotic behavior of finite-dimensional open quantum systems, exploring spectral and algebraic methods in both discrete and continuous settings, and discusses challenges in infinite-dimensional cases.

Contribution

It provides a comprehensive comparison of spectral and algebraic approaches to quantum dynamics and introduces examples involving complex von Neumann algebras.

Findings

Relationship between spectral and algebraic methods clarified

Analysis of Markovian quantum dynamics in finite and infinite dimensions

Example of decoherence-free algebra as a type III von Neumann algebra

Abstract

In this review we discuss some results on the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. Both the spectral and algebraic approaches to this topic are addressed, with particular emphasis on their relationship. The analysis is conducted in both the discrete-time and the continuous-time Markovian settings. In the final part of the work, some issues emerging in the infinite-dimensional case are also discussed. In particular, we provide an example of a Markovian evolution whose decoherence-free algebra is a type III von Neumann algebra.