  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 computational fluid
dynamical formulation of the Wasserstein-2 metric 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. This may have implications to
some key issues in quantum information theory.
