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arXiv:2605.06973 · Journal of Statistical Physics

Quantitative propagation of chaos for Lindblad dynamics

Nina H. Amini, Sofiane Chalal

math-phquant-phSubmitted May 7, 2026• Announced May 11, 2026• 0 citations
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Abstract

We consider an open quantum system governed $N$-body Lindblad equation and study mean-field limits in this setting. We prove that the $N$-particle dynamics converges, in the sense of quantum relative entropy, to the tensorized solution of the limiting nonlinear equation. More precisely, we establish explicit bounds of order $1/N$ on the relative entropy between the $N$-particle density operator and the corresponding product state, thereby providing a quantitative propagation of chaos.

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We consider an open quantum system governed NNN-body Lindblad equation and study mean-field limits in this setting. We prove that the NNN-particle dynamics converges, in the sense of quantum relative entropy, to the tensorized solution of the limiting nonlinear equation. More precisely, we establish explicit bounds of order 1/N1/N1/N on the relative entropy between the NNN-particle density operator and the corresponding product state, thereby providing a quantitative propagation of chaos.
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 NNN-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.
We use probabilistic and stochastic-analysis methods to prove trace-norm propagation of chaos for finite-dimensional quantum mean-field systems governed by Belavkin equations. The particles are density matrices, interact through a mean-field Hamiltonian, and are continuously monitored through independent diffusive observation channels. The limiting dynamics is a nonlinear matrix-valued McKean-Vlasov diffusion, random through its local observation record and coupled through the deterministic averaged state. The main result treats arbitrary one-particle density matrices, including mixed states, and both perfect and inefficient measurement regimes. Under strong tensorization of the initial data, every fixed marginal converges uniformly on compact time intervals to the tensor product of the nonlinear limiting filters, with an explicit quantitative bound. The proof combines purification, fully observed dilation, conditional expectation, relative entropy, and uniform stability of the associated Zakai equations. In the skew-adjoint measurement case, exterior observation noises disappear from the marginal equations and a stochastic BBGKY hierarchy is recovered. Under only marginal chaoticity of permutation-invariant initial states, we prove convergence of fixed marginals by an iteration of this hierarchy.
The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the density matrix is obtained from a mean field theory of interacting quantum systems or from a top-down control by a changing classical environment, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. We also discuss experimental situations for which our results could be relevant.
Abstract The Lindblad equation describes the dissipative time evolution of a density matrix that characterizes an open quantum system in contact with its environment. The widespread ensemble interpretation of a density matrix requires its time evolution to be linear. However, when the dynamics of the density matrix is of a quantum system results not only from the interaction with an external environment, but also with other quantum systems of the same type, the ensemble interpretation is inappropriate and nonlinear dynamics arise naturally. We therefore study the dynamical behavior of nonlinear Lindblad equations using the example of a two-level system. By using techniques developed for classical dynamical systems we show that various types of bifurcations and even chaotic dynamics can occur. As specific examples that display the various types of dynamical behavior, we suggest explicit models based on systems of interacting spins at finite temperature and exposed to a magnetic field that can change in dependence of the magnetization. Due to the interaction between spins, which is treated at mean-field level, the Hamiltonian as well as the transition rates of the Lindblad equation become dependent on the density matrix.
We study the quantum speed limit for open quantum systems described by the Lindblad master equation. The obtained inequality shows a trade-off relation between the operation time and the physical quantities such as the energy fluctuation and the entropy production. We further identify a quantity characterizing the speed of the state transformation, which appears only when we consider the open system dynamics in the quantum regime. When the thermal relaxation is dominant compared to the unitary dynamics of the system, we show that this quantity is approximated by the energy fluctuation of the counter-diabatic Hamiltonian which is used as a control field in the shortcuts to adiabaticity protocol. We discuss the physical meaning of the obtained quantum speed limit and try to give better intuition about the speed in open quantum systems.
We consider the dynamics of N interacting Bosons in three dimensions which are strongly confined in one or two directions. We analyze the mean field scaling and the nonlinear Schrödinger scaling of the interaction potential w and choose the initial wavefunction to be close to a product wavefunction. For both scalings we prove that in the mean field limit the dynamics of the N-particle system is described by a nonlinear equation in two or one dimensions. In the case of the mean field scaling this equation is the Hartree equation and for the second scaling the nonlinear Schrödinger equation. In both cases we obtain explicit bounds for the rate of convergence of the N-particle dynamics to the one-particle dynamics.