Attraction domain analysis for steady states of Markovian open quantum systems

Abstract

This article concerns the attraction domain analysis for steady states in Markovian open quantum systems, which are mathematically described by Lindblad master equations. 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. Furthermore, it is found that the attraction domain of a steady state is the intersection between the set of density operators and an affine space which contains that steady state. 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. We pick two of the system’s steady states with different magnetization profiles and analyse their attraction domains.