Geometry of Grassmannians and optimal transport of quantum states

Let H be a separable Hilbert space. Building on the metric geometry of the Grassmannian P-c(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 P-c(H) (via the spectral theorem) we define an optimal transport cost and the Wasserstein distance for normal states. We prove that the transport cost induces the w*-topology and satisfies 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 circle times H) as families of transport maps. In this way we assign a Wasserstein cost to each pure normal state of B(H circle times H), consistently with the transport cost defined via P-c (H).