Abstract
We prove a new version of Egorov's theorem formulated in the Schr\"{o}dinger picture of quantum mechanics, using the p-Wasserstein metric applied to the Husimi functions of quantum states. The special case p=1 corresponds to a "low-regularity" Egorov theorem, while larger values p>1 yield progressively stronger estimates. As a byproduct of our analysis, we prove an optimal transport inequality analogous to a result of Golse and Paul in the context of mean-field many-body quantum mechanics.