Attraction Domain Analysis for Steady States of Markovian Open Quantum Systems

TL;DR

This paper investigates the attraction domains of steady states in Markovian open quantum systems, providing conditions to determine whether initial states evolve towards specific steady states, with implications for understanding quantum system stability.

Contribution

It introduces necessary and sufficient conditions for identifying attraction domains of steady states and analyzes their measure properties in open quantum systems.

Findings

Steady states without uniqueness have measure-zero attraction domains.

Provided criteria to determine if a state belongs to a steady state's attraction domain.

Illustrated results with an example of an open Heisenberg XXZ spin chain.

Abstract

This article concerns the attraction domain analysis for steady states in Markovian open quantum systems. The central question is proposed as: given a steady state, which part of the state space of density operators does it attract and which part does it not attract? We answer this question by presenting necessary and sufficient conditions that determine, for any steady state and initial state, whether the latter belongs to the attraction domain of the former. Moreover, we show that steady states without uniqueness in the set of density operators have attraction domains with measure zero under some translation invariant and locally finite measures. Finally, an example regarding an open Heisenberg XXZ spin chain is presented.