Geometry of Grassmannians and optimal transport of quantum states

Let H be a separable Hilbert space. Building on the metric geometry of the Grassmannian Pc(H) of finite dimensional subspaces of H, we develop a theory of optimal transport for the normal states of the von Neumann algebra of linear and bounded operators B(H). Seeing density matrices as discrete probability measures on Pc(H) (via the spectral theorem) we dene an optimal transport cost and the Wasserstein distance for normal states. We prove that the transport cost induces the w-topology and satises the triangular inequality on a dense subset of normal states. Our construction is compatible with the quantum mechanics approach of composite systems as tensor products. We provide a natural interpretation of the pure normal states of B(H H) as families of transport maps. In this way we assign a Wasserstein cost to each pure normal state of B(H H) consistently with the transport cost defined via Pc(H).