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arXiv:2512.01722

Folded optimal transport and its application to separable quantum optimal transport

Thomas Borsoni

math.FAmath-phquant-phSubmitted Dec 1, 2025• Announced Dec 2, 2025• 0 citations• Revised Dec 18, 2025 (v2)
v1 · Dec 1, 2025v2 · Dec 18, 2025v3 · Jan 19, 2026v4 · Jun 8, 2026v5 · Sep 17, 2026
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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.

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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.
Convex roofs are natural ways to extend a scalar function defined on the extreme points of a convex set KKK to the whole set. In this work we focus on the case K=C×CK = C \times CK=C×C for some convex set CCC, and the extension of a distance ddd defined on the extreme points of CCC. 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 CCC with. More generally, we also consider products of different convex sets C1×C2C_1 \times C_2C1​×C2​ 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 Fran\c{c}a 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.
We develop a Quantum Optimal Transport (QOT) barycenter framework for quantum states, as an analog of Wasserstein barycenters~\cite{AguCar}, and establish existence and duality results for a broad class of possibly unbounded transport costs on separable Hilbert spaces. Our framework provides, in particular, a unified treatment of 222-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the 222-quantum Wasserstein distances of Caglioti--Golse--Mouhot--Paul~\cite{caglioti2021towards,Golse-Mouhot-Paul:2016} and De Palma--Trevisan~\cite{DPaTre19}. The central results concern Gaussian input states and, in particular, Gaussian rigidity: whether Gaussian input states force the 2-QW barycenter itself to be Gaussian and uniquely determined when the minimization is taken over all quantum states. We first show that the barycenter problem admits a Gaussian minimizer and reduces to a finite-dimensional convex optimization problem over covariance matrices. The main difficulty is that uniqueness of the optimal covariance does not, in general, imply uniqueness of the underlying quantum state. We bridge this gap through a state-reconstruction principle under covariance complementary slackness that upgrades uniqueness of the optimal covariance to uniqueness of the full quantum state, and thereby prove a global rigidity theorem: if at least one Gaussian input is faithful, then the barycenter is unique among all quantum states and is necessarily Gaussian. Faithfulness is sufficient but not necessary: families of pure inputs still determine a unique barycenter, whereas partially pure nonfaithful Gaussian inputs may admit multiple barycenters.
An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on Rd\mathbf{R}^dRd has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on L2(Rd)L^2(\mathbf{R}^d)L2(Rd), and used to estimate the convergence rate of various asymptotic theories in the context of quantum mechanics. The present work proves a Kantorovich type duality theorem for this quantum variant of the Monge-Kantorovich or Wasserstein distance, and discusses the structure of optimal quantum couplings. Specifically, we prove that optimal quantum couplings involve a gradient type structure similar to the Brenier transport map (which is the gradient of a convex function), or more generally, to the subdifferential of a l.s.c. convex function as in the Knott-Smith optimality criterion (see Theorem 2.12 in [C. Villani: "Topics in Optimal Transportation", Amer. Math. Soc. 2003]).
This text is a set of lecture notes for a 4.5-hour course given at the Erdös Center (Rényi Institute, Budapest) during the Summer School "Optimal Transport on Quantum Structures" (September 19th-23rd, 2023). Lecture I introduces the quantum analogue of the Wasserstein distance of exponent 222 defined in [F. Golse, C. Mouhot, T. Paul: Comm. Math. Phys. 343 (2016), 165-205], and in [F. Golse, T. Paul: Arch. Ration. Mech. Anal. 223 (2017) 57-94]. Lecture II discusses various applications of this quantum analogue of the Wasserstein distance of exponent 222, while Lecture III discusses several of its most important properties, such as the triangle inequality, and the Kantorovich duality in the quantum setting, together with some of their implications.
We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.
A quantum version of the Monge-Kantorovich optimal transport problem is analyzed. The transport cost is minimized over the set of all bipartite coupling states ρ^{AB} such that both of its reduced density matrices ρ^{A} and ρ^{B} of dimension N are fixed. We show that, selecting the quantum cost matrix to be proportional to the projector on the antisymmetric subspace, the minimal transport cost leads to a semidistance between ρ^{A} and ρ^{B}, which is bounded from below by the rescaled Bures distance and from above by the root infidelity. In the single-qubit case, we provide a semianalytic expression for the optimal transport cost between any two states and prove that its square root satisfies the triangle inequality and yields an analog of the Wasserstein distance of the order of 2 on the set of density matrices. We introduce an associated measure of proximity of quantum states, called swap fidelity, and discuss its properties and applications in quantum machine learning.