Matrix Optimal Mass Transport: A Quantum Mechanical Approach

In this paper, we describe a possible generalization of the Wasserstein-2 metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one and to the space of matrix-valued probability densities. Our approach follows a control-theoretic optimization formulation of the Wasserstein-2 metric, having its roots in fluid dynamics, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric.