Matricial Wasserstein-1 Distance

We propose an extension of the Wasserstein 1-metric (W
₁
) for density matrices, matrix-valued density measures, and an unbalanced interpretation of mass transport. We use duality theory and, in particular, a “dual of the dual” formulation of W
₁
. This matrix analog of the Earth Mover's Distance has several attractive features including ease of computation.