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arXiv:2610.01855

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

Augusto Gerolin, Zhiyi Lin

quant-phmath.APSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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Abstract

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.

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•Sep 17, 2026•0 citations
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.
Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.
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]).
Wasserstein distances and divergences of order ppp by quantum channels
We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce ppp-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction.
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.
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.
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.