  In this paper, we extend the upper and lower bounds for the "pseudo-distance"
on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein)
distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343
(2016) 165-205] to positive quantizations defined in terms of the family of
phase space translates of a density operator, not necessarily of rank one as in
the case of the T\"oplitz quantization. As a corollary, we prove that the
uniform (for vanishing h) convergence rate for the mean-field limit of the
N-particle Heisenberg equation holds for a much wider class of initial data
than in [F. Golse, C. Mouhot, T. Paul, loc. cit.]. We also discuss the
relevance of the pseudo-distance compared to the Schatten norms for the purpose
of metrizing the set of quantum density operators in the semiclassical regime.
