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Ergodic Properties of Quantum Markov Semigroups

Nicolas Mousset, Nina H. Amini

TL;DR

The paper develops a rigorous ergodic framework for infinite-dimensional quantum Markov semigroups, centering on the positive recurrent subspace ℛ₊ and the absorption operator A(ℛ₊). It provides a sufficient condition for exponential convergence toward ℛ₊ and demonstrates ergodicity across three physical models, including two-photon processes and a k-photon exchange oscillator. It then analyzes the internal structure of ℛ₊ via a decomposition into minimal enclosures, highlighting how GAS is governed by these invariant subspaces, and discusses finite-dimensional convergence rates through a Perron–Frobenius–type perspective. The work broadens the understanding of global asymptotic stability for open quantum systems and outlines avenues for extending ergodic criteria and stability analysis to broader classes of QMS.

Abstract

In this paper, we study the ergodic theorem for infinite-dimensional quantum Markov semigroups, originally introduced by Frigerio and Verri in 1982, and its latest version developed by Carbone and Girotti in 2021. We provide a sufficient condition that ensures exponential convergence towards the positive recurrent subspace, a well-known result for irreducible quantum Markov semigroups in finite-dimensional Hilbert spaces. Several illustrative examples are presented to demonstrate the application of the ergodic theorem. Moreover, we show that the positive recurrent subspace plays a crucial role in the study of global asymptotic stability.

Ergodic Properties of Quantum Markov Semigroups

TL;DR

The paper develops a rigorous ergodic framework for infinite-dimensional quantum Markov semigroups, centering on the positive recurrent subspace ℛ₊ and the absorption operator A(ℛ₊). It provides a sufficient condition for exponential convergence toward ℛ₊ and demonstrates ergodicity across three physical models, including two-photon processes and a k-photon exchange oscillator. It then analyzes the internal structure of ℛ₊ via a decomposition into minimal enclosures, highlighting how GAS is governed by these invariant subspaces, and discusses finite-dimensional convergence rates through a Perron–Frobenius–type perspective. The work broadens the understanding of global asymptotic stability for open quantum systems and outlines avenues for extending ergodic criteria and stability analysis to broader classes of QMS.

Abstract

In this paper, we study the ergodic theorem for infinite-dimensional quantum Markov semigroups, originally introduced by Frigerio and Verri in 1982, and its latest version developed by Carbone and Girotti in 2021. We provide a sufficient condition that ensures exponential convergence towards the positive recurrent subspace, a well-known result for irreducible quantum Markov semigroups in finite-dimensional Hilbert spaces. Several illustrative examples are presented to demonstrate the application of the ergodic theorem. Moreover, we show that the positive recurrent subspace plays a crucial role in the study of global asymptotic stability.

Paper Structure

This paper contains 14 sections, 19 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Quantum Markov semigroup
  4. Transience and recurrence of QMS
  5. Ergodic theorem
  6. Applications of the ergodic theorem
  7. Two-photon absorption and emission model
  8. Generic QMS
  9. Quantum harmonic oscillator
  10. Decomposition of the positive recurrent subspace
  11. Decomposition into minimal enclosures
  12. Link with the NFD and global asymptotic stability
  13. Convergence speed for finite-dimensional Hilbert spaces
  14. Conclusion

Key Result

Theorem 2.1

A bounded operator $\mathcal{L}$ on $\mathcal{B(H)}$ is the infinitesimal generator of a uniformly continuous QMS iff there exists a complex separable Hilbert space $\mathcal{K}$, a bounded operator $L : \mathcal{H} \to \mathcal{H} \otimes \mathcal{K}$ and an operator $G$ in $\mathcal{H}$ such that: for all $A \in \mathcal{B(H)}$.

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 2.1: Lindblad
  • Theorem 2.2: fagnola1999quantum, Theorem 3.22
  • Theorem 3.1: frigerio1982long, Theorem 2.1
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • ...and 31 more