Quantum Relative Entropy and the Mean-Field Limit

TL;DR

This paper introduces a quantum relative entropy approach to analyze the mean-field limit in quantum many-body systems, providing stability, propagation of chaos, and convergence results in both closed and open quantum systems.

Contribution

It develops a novel entropy-based method applicable to both closed and open quantum systems, extending mean-field analysis without relying on specific tensor-product structures.

Findings

Proves quantitative stability between N-body density matrix and Hartree solution.

Establishes propagation of chaos in trace norm for fixed marginals.

Derives uniform convergence estimates in the joint mean-field and semiclassical regime.

Abstract

We develop a quantum relative entropy method for the mean-field limit of quantum many-body systems. For closed systems governed by the von Neumann equation, we prove a quantitative stability estimate between the $N$-body density matrix and the tensorized solution of the Hartree equation. The argument is based on an entropy production identity, a cancellation mechanism for the centered two-body fluctuation, and a combinatorial estimate controlling the remaining mixed moments. As a consequence, we obtain propagation of chaos in trace norm for fixed marginals. We further combine the entropy estimate with known semiclassical Wasserstein bounds to derive a convergence estimate that is uniform in the Planck constant in an appropriate joint mean-field and semiclassical regime. Finally, we extend the method to finite-dimensional open quantum systems governed by Lindblad dynamics. In this setting, we establish an analogous relative entropy estimate for general bounded two-body interactions, where the mean-field potential is defined through partial trace. This shows that the entropy method does not rely on any special tensor-product decomposition of the interaction.