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Unravelling mean-field Lindblad equation

Sofiane Chalal, Nina H. Amini

TL;DR

This work addresses the challenge of simulating the N-body Lindblad equation for open quantum systems by introducing a mean-field particle Monte Carlo method. It derives a nonlinear Lindblad dynamics via a McKean–Vlasov-type mean-field limit and proves propagation of chaos for a system of interacting quantum trajectories, connecting the N-particle dynamics to a nonlinear mean-field equation. The main results provide $O(1/N)$ convergence rates for the mean-field approximation and a trajectory-level corollary, validating the use of independent-particle mean-field trajectories as accurate representatives of the nonlinear dynamics and enabling scalable, parallelizable simulations. The approach lays groundwork for future improvements in numerical discretization and extensions to more complex interactions in open quantum systems.

Abstract

We propose a mean-field particle Monte Carlo method for simulating the N-body Lindblad equation. We provide a convergence result showing that a system of interacting particles converges to the corresponding nonlinear Lindblad equation in the large N limit.

Unravelling mean-field Lindblad equation

TL;DR

This work addresses the challenge of simulating the N-body Lindblad equation for open quantum systems by introducing a mean-field particle Monte Carlo method. It derives a nonlinear Lindblad dynamics via a McKean–Vlasov-type mean-field limit and proves propagation of chaos for a system of interacting quantum trajectories, connecting the N-particle dynamics to a nonlinear mean-field equation. The main results provide convergence rates for the mean-field approximation and a trajectory-level corollary, validating the use of independent-particle mean-field trajectories as accurate representatives of the nonlinear dynamics and enabling scalable, parallelizable simulations. The approach lays groundwork for future improvements in numerical discretization and extensions to more complex interactions in open quantum systems.

Abstract

We propose a mean-field particle Monte Carlo method for simulating the N-body Lindblad equation. We provide a convergence result showing that a system of interacting particles converges to the corresponding nonlinear Lindblad equation in the large N limit.
Paper Structure (7 sections, 5 theorems, 51 equations, 1 figure, 1 algorithm)

This paper contains 7 sections, 5 theorems, 51 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Mean-field interaction operator
  2. Example :
  3. Matrix Formulation :
  4. Function Formulation :
  5. Lemma proof
  6. Theorem proof
  7. Corollary proof

Key Result

Proposition 1

If the initial state is pure state then it remains pure state at later times i.e :

Figures (1)

  • Figure 1: Overview of the method

Theorems & Definitions (9)

  • Proposition 1
  • Lemma 1: Theorem 1.4.8 khanfer24applied
  • Lemma 2
  • Theorem 1: Monte-Carlo Approximation
  • Corollary 1: Classical Propagation of chaos
  • Remark 1
  • proof
  • proof
  • proof