Optimal transport pseudometrics for quantum and classical densities

This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric of exponent 2 introduced in Golse et al. (2016) [13] to compare two density operators, and in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator. The argument differs noticeably from the classical proof of the triangle inequality for Wasserstein metrics, which is based on a disintegration theorem for probability measures, and uses in particular an analogue of the Kantorovich duality for the functional defined in Golse and Paul (2017) [14]. Finally, this duality theorem is used to define an analogue of the Brenier transport map for the functional defined in Golse and Paul (2017) [14] to compare a phase space probability measure and a density operator.