A quantum version of the Monge-Kantorovich optimal transport problem aims at optimizing a Hermitian cost matrix $C$ over the set of all bipartite coupling states $\rho_{AB}$, s.t. both of its reduced density matrices $\rho_A$ and $\rho_B$ are fixed. I'm interested in its "applications". The classical optimal transport theory can be either used as theoretical tools to study important problems arising in PDE (Schrödinger equation/e Boltzmann equation), metric geometry, random matrices, etc., or used to study problems in practice, e.g. urban planning, economics, data science, etc. So I wish to know whether the quantum optimal transport has similar relevance, from both theoretical and practical perspective. 

Any answer, references (survey summarizing its development) and comments are highly appreciated.