Analysis of quantum semigroups with GKS-Lindblad generators I. Simple generators

TL;DR

This paper analyzes simple quantum semigroup generators with Lindblad form, providing explicit formulas for stationary states and methods to construct evolutions with desired stationary states in finite-dimensional open quantum systems.

Contribution

It offers a detailed characterization of simple Lindblad generators and explicit formulas for stationary states, enhancing understanding of quantum dynamical semigroups.

Findings

Complete set of stationary states derived

Formula for stationary state from generator provided

Method to construct evolution with a given stationary state

Abstract

Semigroups describing the time evolution of open quantum systems in finite dimensional spaces have generators of a special form, known as Lindblad generators. The simple generators, characterized by only one operator, are analyzed. The complete set of all the stationary states is presented in detail, including a formula to calculate a stationary state from the generating operator. Also the opposite task can be fulfilled, to construct an evolution leading to a prescribed stationary state.