Approach to equilibrium in Markovian open quantum systems

TL;DR

The paper develops a rigorous framework for Markovian open quantum systems described by the von Neumann-Lindblad equation in the infinite-degree-of-freedom setting under quantum detailed balance. It builds the HL/ vNL dual form via generators $L=L_0+G$ and $L'=L_0'+G'$, analyzes their commutativity, and proves existence, uniqueness, and ergodic convergence to stationary states on appropriate state spaces $\mathcal{S}_{\tau}$, with the Gibbs state $\rho_{\tau}$ as a central stationary candidate. Under suitable conditions (notably (W),(QDB) and (U)), the paper shows that the stationary state is unique and characterized by explicit projections, yielding return-to-equilibrium results and a clear duality between observables and states. The results provide a mathematically rigorous basis for equilibration in quantum open systems with infinite degrees of freedom and offer explicit constructions of jump operators that satisfy quantum detailed balance, contributing to the understanding of quantum dissipative dynamics and its long-time behavior.

Abstract

In this paper, we study the evolution of Markovian open quantum systems, whose dynamics are governed by the von Neumann-Lindblad equations. Our goal is to prove the return-to-equilibrium property for systems of infinite degrees of freedom under quantum detailed balance condition.

Theorems & Definitions (126)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1
• proof