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Folded optimal transport and its application to separable quantum optimal transport

Thomas Borsoni

TL;DR

The paper develops folded optimal transport, a convex-extension framework that lifts distance data from the extreme boundary of a convex set to its interior via Choquet theory. It defines folded Kantorovich costs and folded Wasserstein distances, proving when they form genuine metrics, and analyzes their continuity and geodesic structure. The framework is then applied to finite-dimensional quantum state spaces to yield separable quantum Wasserstein distances on density matrices, with rich links to Beatty–França semidistances and Golse–Paul semiclassical costs. This unifies classical, semiclassical, and separable quantum OT under a single convex-roof-inspired construction, providing both structural results and computationally relevant reformulations.

Abstract

We introduce folded optimal transport, as a way of extending 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 theory and standard optimal transport, we introduce the so-called folded Kantorovich cost and folded Wasserstein distance, and study the metric properties it provides to the convex. 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.

Folded optimal transport and its application to separable quantum optimal transport

TL;DR

The paper develops folded optimal transport, a convex-extension framework that lifts distance data from the extreme boundary of a convex set to its interior via Choquet theory. It defines folded Kantorovich costs and folded Wasserstein distances, proving when they form genuine metrics, and analyzes their continuity and geodesic structure. The framework is then applied to finite-dimensional quantum state spaces to yield separable quantum Wasserstein distances on density matrices, with rich links to Beatty–França semidistances and Golse–Paul semiclassical costs. This unifies classical, semiclassical, and separable quantum OT under a single convex-roof-inspired construction, providing both structural results and computationally relevant reformulations.

Abstract

We introduce folded optimal transport, as a way of extending 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 theory and standard optimal transport, we introduce the so-called folded Kantorovich cost and folded Wasserstein distance, and study the metric properties it provides to the convex. 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.

Paper Structure

This paper contains 20 sections, 20 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Folded optimal transport and folded Wasserstein distances
  3. Preliminary tools
  4. Standard optimal transport and Wasserstein distance
  5. Choquet theory
  6. The folded Wasserstein (pseudo-)distance
  7. Construction
  8. Properties of folded Wasserstein
  9. Main theorems: summarized results
  10. Folded Kantorovich for a generic cost
  11. Application to separable quantum optimal transport
  12. Preliminary definitions
  13. Quantum folded Wasserstein
  14. Beatty--França beatty2025order and quantum folded Wasserstein
  15. On the subadditivity of the Beatty--França semi-distance .
  16. ...and 5 more sections

Key Result

Proposition 1

Consider the general Setting thesetting, and let $p \geqslant 1$. Then $D_p$ is a pseudo-distance on $C$ and $\widehat{D}_p$ is a semi-distance on $C$, with Moreover, we have the equivalence

Figures (1)

  • Figure 1: Illustration of convex state spaces.

Theorems & Definitions (45)

  • Definition 1: Standard optimal transport and Wasserstein distance
  • Definition 2: Representing probability
  • Definition 3: Pseudo/semi-distance terminology
  • Remark 1: Folded Wasserstein generalizes Wasserstein
  • Proposition 1: Basic properties of $D_p$ and $\widehat{D}_p$
  • proof
  • Proposition 2: Extension properties of $D_p$ and $\widehat{D}_p$
  • proof
  • Proposition 3: Monotonicity in $p$
  • Definition 4: Representing couplings
  • ...and 35 more