  We prove semiclassical estimates for the Schr\''odinger-von Neumann evolution
with $C^{1,1}$ potentials and density matrices whose square root have either
Wigner functions with low regularity independent of the dimension, or matrix
elements between Hermite functions having long range decay. The estimates are
settled in different weak topologies and apply to initial density operators
whose square root have Wigner functions $7$ times differentiable, independently
of the dimension. They also apply to the $N$ body quantum dynamics uniformly in
$N$. In a appendix, we finally estimate the dependence in the dimension of the
constant appearing on the Calderon-Vaillancourt Theorem.
