Propagation of chaos for Belavkin equations beyond pure states

We prove propagation of chaos for a finite-dimensional system of N interacting density matrix-valued diffusions. The interaction enters through a mean-field Hamiltonian, and each particle is driven by independent Brownian noises. The limiting equation is a nonlinear matrix-valued McKean-Vlasov diffusion whose coefficients depend on the averaged one-particle state.
Our first result treats arbitrary mixed initial states and all measurement efficiencies. We prove convergence, uniformly on compact time intervals, of every fixed marginal toward the corresponding tensor product of nonlinear limiting particles, with an explicit trace-norm bound depending on the initial tensorization error and on N. The proof combines a purification argument, a fully observed dilation, conditional expectation, a relative-entropy estimate, and a stability estimate for the associated linear Zakai equations. Our second result considers skew-adjoint measurement operators. In this case the evolution preserves permutation symmetry and admits a stochastic BBGKY hierarchy. Under the weaker assumption of chaotic initial data, we prove convergence of each fixed marginal, without an explicit rate, by compactness and uniqueness for the limiting hierarchy.