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) to compare two density operators, and in Golse and Paul (2017) 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). Finally, this duality theorem is used to define an analogue of the Brenier transport map for the functional defined in Golse and Paul (2017) to compare a phase space probability measure and a density operator.