Geometry of Grassmannians and optimal transport of quantum states

TL;DR

This paper develops a geometric framework for optimal transport of quantum states using the structure of Grassmannians, leading to a new Wasserstein distance compatible with quantum mechanics.

Contribution

It introduces a novel optimal transport theory for quantum states based on the geometry of Grassmannians and extends it to composite quantum systems.

Findings

Grassmannian of finite-dimensional subspaces is an Alexandrov space of nonnegative curvature.

Defines a Wasserstein distance for normal states compatible with the $w^*$-topology.

Provides a quantum interpretation of transport maps for composite systems.

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.