# On the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan equation

## TL;DR

This paper reviews theoretical conditions for the uniqueness of steady states in Lindblad equations and provides practical criteria to ensure their attractivity in open quantum systems.

## Contribution

It offers new, simple criteria for guaranteeing the uniqueness and attractivity of steady states in open quantum systems, applicable to various physical models.

## Key findings

- Derived simple criteria for steady-state uniqueness
- Validated criteria on spin and bosonic systems
- Enhanced understanding of dissipative quantum dynamics

## Abstract

The aims of this paper are two. The first is to give a brief review of the most relevant theoretical results concerning the uniqueness of the steady-state solution of the Lindblad-Gorini-Kossakowski-Sudarshan master equation and the criteria which guarantee relaxingness and irreducibility of dynamical semigroups. In particular, we test and discuss their physical meaning by considering their applicability to the characterisation of the simplest open quantum system \emph{i.e.} a two-level system coupled to a bath of harmonic oscillators at zero temperature. The second aim is to provide a set of sufficient conditions which guarantees the uniqueness of the steady-state solution and its attractivity. Starting from simple assumptions, we derive simple criteria that can be efficiently exploited to characterise the behavior of dissipative systems of spins and bosons (with truncated Fock space), and a wide variety of other open quantum systems recently studied.