  We consider the time evolution of quantum states by many-body Schr\"odinger
dynamics and study the rate of convergence of their reduced density matrices in
the mean field limit. If the prepared state at initial time is of coherent or
factorized type and the number of particles $n$ is large enough then it is
known that $1/n$ is the correct rate of convergence at any time. We show in the
simple case of bounded pair potentials that the previous rate of convergence
holds in more general situations with possibly correlated prepared states. In
particular, it turns out that the coherent structure at initial time is
unessential and the important fact is rather the speed of convergence of all
reduced density matrices of the prepared states. We illustrate our result with
several numerical simulations and examples of multi-partite entangled quantum
states borrowed from quantum information.
