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- W3150571916 abstract "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." @default.
- W3150571916 created "2021-04-13" @default.
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- W3150571916 date "2021-04-06" @default.
- W3150571916 modified "2023-09-27" @default.
- W3150571916 title "Geometry of Grassmannians and optimal transport of quantum states" @default.
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