Metric convex extensions and optimal transport on classical and quantum structures

Abstract

Convex roofs are natural ways to extend a scalar function defined on the extreme points of a convex set $K$ to the whole set. In this work we focus on the case $K = C \times C$ for some convex set $C$, and the extension of a distance $d$ defined on the extreme points of $C$. In general, the triangle inequality may not survive the extension process. We therefore consider the metric envelope of such extensions and study the metric properties they endow $C$ with. More generally, we also consider products of different convex sets $C_1 \times C_2$ and costs defined on their extreme points, in which case no metric envelope is involved. We then show a two-way correspondence between convex extensions and Kantorovich optimal transport formulations. On the one hand, these extensions can be recovered as quotients of standard, classical Kantorovich costs or Wasserstein distances by the barycenter map. On the other hand, various Kantorovich formulations re-write as such convex extensions. Classical optimal transport is recovered as the case of the simplex, and both the semiclassical optimal transport cost of Golse and Paul and the entanglement-free quantum Wasserstein objects of Beatty and Stilck França are recovered by considering the non-simplex convex set of quantum states. This work provides a clear bridge between the theory of convex extensions and that of optimal transport, with a unifying framework for classical, semiclassical and entanglement-free quantum instances of Kantorovich formulations.