  Let $\mathsf{H}$ be a separable Hilbert space. We prove that the Grassmannian
$\mathsf{P}_c(\mathsf{H})$ of the finite dimensional subspaces of $\mathsf{H}$
is an Alexandrov space of nonnegative curvature and we employ its metric
geometry to develop the theory of optimal transport for the normal states of
the von Neumann algebra of linear and bounded operators
$\mathsf{B}(\mathsf{H})$. Seeing density matrices as discrete probability
measures on $\mathsf{P}_c(\mathsf{H})$ (via the spectral theorem) we define an
optimal transport cost and the Wasserstein distance for normal states. In
particular we obtain a cost which induces the $w^*$-topology.
  Our construction is compatible with the quantum mechanics approach of
composite systems as tensor products $\mathsf{H}\otimes \mathsf{H}$. We provide
indeed an interpretation of the pure normal states of
$\mathsf{B}(\mathsf{H}\otimes \mathsf{H})$ as families of transport maps. This
also defines a Wasserstein cost for the pure normal states of
$\mathsf{B}(\mathsf{H}\otimes \mathsf{H})$, reconciling with our proposal.
