  Several extensions of the classical optimal transport distances to the
quantum setting have been proposed. In this paper, we investigate the
pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys
343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94,
2017]. These pseudometrics serve as a quantum analogue of the
Monge-Kantorovich-Wasserstein distances of order $2$ on the phase space. We
prove that they are comparable to negative Sobolev norms up to a small term due
to a positive "self-distance" in the semiclassical approximation, which can be
bounded above using the Wigner-Yanase skew information. This enables us to
improve the known results in the context of the mean-field and semiclassical
limits by requiring less regularity on the initial data.
