Folded optimal transport and its application to separable quantum optimal transport

We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal transport, found to be the particular case of the convex being a simplex. Relying on Choquet's theory and standard optimal transport, we introduce the folded Kantorovich cost and folded Wasserstein distances, and study their induced metric properties. We then apply the construction to the quantum setting, and obtain an actual separable quantum Wasserstein distance on the set of density matrices from a distance on the set of pure states, closely related to the semi-distance of Beatty and Stilck-Franca [4], and of which we obtain a variety of properties. We also find that the semiclassical Golse-Paul [16] cost writes as a folded Kantorovich cost. Folded optimal transport therefore provides a unified framework for classical, semiclassical and separable quantum optimal transport.