Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Metric convex extensions and optimal transport on classical and quantum structures

Thomas Borsoni

Published Sep 17, 2026• 0 citations
OpenAlex

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 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.

Related papers

Metric convex extensions and optimal transport on classical and quantum structures

Thomas Borsoni

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.

math.FACross-listedarXiv:2512.01722•Dec 2, 2025•0 citations

Quantum Optimal Transport Barycenters: Existence, Duality, and Gaussian Rigidity

Augusto Gerolin, Zhiyi Lin

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 $2$-quantum Wasserstein (QW) barycenters in both the quantum-state and quantum-channel formulations by specializing to the canonical quadratic cost operators associated with the $2$-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.

quant-pharXiv:2610.01855•Oct 2, 2026•0 citations

Towards optimal transport for quantum densities

Emanuele Caglioti, François Golse, Thierry Paul

An analogue of the quadratic Wasserstein (or Monge-Kantorovich) distance between Borel probability measures on $\mathbf{R}^d$ has been defined in [F. Golse, C. Mouhot, T. Paul: Commun. Math. Phys. 343 (2015), 165-205] for density operators on $L^2(\mathbf{R}^d)$, 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]).

arXiv:2101.03256•ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE•Jun 6, 2022•22 citations

Quantum Optimal Transport: Quantum Couplings and Many-Body Problems

François Golse

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 $2$ 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 $2$, 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.

arXiv:2308.11134•Aug 22, 2023•0 citations

Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost

Tomasz Miller

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.

quant-pharXiv:2609.03769•Sep 4, 2026•0 citations

Strong Kantorovich duality for quantum optimal transport with generic cost and optimal couplings on quantum bits

Gergely Bunth, József Pitrik, Tamás Titkos +1 more

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

arXiv:2510.26326•Linear Algebra and its Applications•Apr 1, 2026•2 citations
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.
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 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.
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.
We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.