Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum space-time Poincaré inequality for Lindblad dynamics

Bowen Li, Jianfeng Lu

TL;DR

The paper addresses the mixing of primitive quantum Markov semigroups with a coherent drift that breaks detailed balance, focusing on how the spectral gap of the generator governs $L^2$-mixing and how a Hamiltonian term can accelerate convergence. It develops a hypocoercivity framework for quantum dynamics, including a quantum space-time Poincaré inequality and a time-averaged $L^2$-decay method that yields explicit exponential rates in terms of the dissipative gap $\lambda_D$ and a macroscopic coercivity parameter $s_H$. It also analyzes the large coherent limit, showing the asymptotic gap $\lambda(\alpha)$ converges to $\inf_{\nu\in B_H} \lambda_\nu$ and provides conditions under which the gap increases with the coherent strength. The results are illustrated through Davies generators and quantum Gibbs-sampling-type models, with detailed unital and non-unital examples offering concrete rate bounds and pathways to achieve rapid quantum equilibration in noise-assisted state preparation and sampling tasks.

Abstract

We investigate the mixing properties of primitive Markovian Lindblad dynamics (i.e., quantum Markov semigroups), where the detailed balance is disrupted by a coherent drift term. It is known that the sharp $L^2$-exponential convergence rate of Lindblad dynamics is determined by the spectral gap of the generator. We show that incorporating a Hamiltonian component into a detailed balanced Lindbladian can generically enhance its spectral gap, thereby accelerating the mixing. In addition, we analyze the asymptotic behavior of the spectral gap for Lindblad dynamics with a large coherent contribution. However, estimating the spectral gap, particularly for a non-detailed balanced Lindbladian, presents a significant challenge. In the case of hypocoercive Lindblad dynamics, we extend the variational framework originally developed for underdamped Langevin dynamics to derive fully explicit and constructive exponential decay estimates for convergence in the noncommutative $L^2$-norm. This analysis relies on establishing a quantum analog of space-time Poincaré inequality. Furthermore, we provide several examples with connections to quantum noise and quantum Gibbs samplers as applications of our theoretical results.

Quantum space-time Poincaré inequality for Lindblad dynamics

TL;DR

The paper addresses the mixing of primitive quantum Markov semigroups with a coherent drift that breaks detailed balance, focusing on how the spectral gap of the generator governs -mixing and how a Hamiltonian term can accelerate convergence. It develops a hypocoercivity framework for quantum dynamics, including a quantum space-time Poincaré inequality and a time-averaged -decay method that yields explicit exponential rates in terms of the dissipative gap and a macroscopic coercivity parameter . It also analyzes the large coherent limit, showing the asymptotic gap converges to and provides conditions under which the gap increases with the coherent strength. The results are illustrated through Davies generators and quantum Gibbs-sampling-type models, with detailed unital and non-unital examples offering concrete rate bounds and pathways to achieve rapid quantum equilibration in noise-assisted state preparation and sampling tasks.

Abstract

We investigate the mixing properties of primitive Markovian Lindblad dynamics (i.e., quantum Markov semigroups), where the detailed balance is disrupted by a coherent drift term. It is known that the sharp -exponential convergence rate of Lindblad dynamics is determined by the spectral gap of the generator. We show that incorporating a Hamiltonian component into a detailed balanced Lindbladian can generically enhance its spectral gap, thereby accelerating the mixing. In addition, we analyze the asymptotic behavior of the spectral gap for Lindblad dynamics with a large coherent contribution. However, estimating the spectral gap, particularly for a non-detailed balanced Lindbladian, presents a significant challenge. In the case of hypocoercive Lindblad dynamics, we extend the variational framework originally developed for underdamped Langevin dynamics to derive fully explicit and constructive exponential decay estimates for convergence in the noncommutative -norm. This analysis relies on establishing a quantum analog of space-time Poincaré inequality. Furthermore, we provide several examples with connections to quantum noise and quantum Gibbs samplers as applications of our theoretical results.
Paper Structure (14 sections, 19 theorems, 332 equations)

This paper contains 14 sections, 19 theorems, 332 equations.

Table of Contents

  1. Introduction
  2. Main results and discussions
  3. Related works
  4. Quantum Markov semigroup and its mixing
  5. Structures of quantum Markov semigroup
  6. Spectral gap and relaxation time
  7. Spectral gap under large coherent limit
  8. Convergence rate estimate via hypocoercivity
  9. Setup and main result
  10. Scaling of the convergence rate
  11. Proofs and Lemmas
  12. Applications and examples
  13. Unital case
  14. Non-unital case

Key Result

Proposition 2.1

Let $\mathcal{L}$ be a Lindbladian satisfying $\sigma$-KMS DBC. Then there exist $\{V_j\}_{j = 1}^{J_D} \subset \mathcal{B}(\mathcal{H})$ such that $\{V_j\}_{j = 1}^{J_D} = \{V_j^\dag\}_{j = 1}^{J_D}$ and the associated Dirichlet form can be written as where $[A,B] = AB - BA$ denotes the commutator.

Theorems & Definitions (46)

  • Proposition 2.1
  • Lemma 2.2: wolf5quantum*Theorem 7.2
  • Lemma 2.3
  • Remark 1: Quantum analog of kinetic Fokker-Planck equation
  • Remark 2: Generalized detailed balance and time reversal
  • Remark 3
  • Definition 2.4
  • Remark 4: Coercivity and symmetrized Lindbladian
  • Lemma 2.5
  • proof
  • ...and 36 more