Geometry of Grassmannians and optimal transport of quantum states

2021/04/06 by Paolo Antonini, Fabio Cavalletti, Antonini, Paolo +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Noncommutative and Quantum Gravity Theories #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2104.02616

openalex publication_date 2021/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H be a separable Hilbert space. We prove that the Grassmannian P_c(H) of the finite dimensional subspaces of 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 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. 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 H⊗ H. We provide indeed an interpretation of the pure normal states of B(H⊗ H) as families of transport maps. This also defines a Wasserstein cost for the pure normal states of B(H⊗ H), reconciling with our proposal.

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• A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems

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