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arXiv:2607.07764

Comment on 'Quantum Monge-Kantorovich Problem and Transport Distance between Density Matrices'

Tomasz Miller

quant-phmath-phSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
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Abstract

Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the $p$-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.

Comments
Comment on arXiv:2102.07787, 7 pages, 1 figure

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Friedland et al. [PRL 129, 110402 (2022)] proposed and studied a quantum analogue of the ppp-Wasserstein distance based on quantum cost matrices and quantum couplings. They conjectured that, despite being only a semidistance in general, this quantity is a true distance for a particular quantum cost matrix and for cost matrices in a small neighborhood of it. We disprove these conjectures by exhibiting an explicit family of triples of states for which the triangle inequality fails.
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.
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 show that the quantum generalization of the 222-Wasserstein distance proposed by Chakrabarti et al. is not monotone under partial traces. This disproves a recent conjecture by Friedland et al. Finally, we propose a stabilized version of the original definition, which we show to be monotone under the application of general quantum channels.
Abstract We study a quantum analogue of the 2-Wasserstein distance as a measure of proximity on the set Ω N of density matrices of dimension N. We show that such (semi-)distances do not induce Riemannian metrics on the tangent bundle of Ω N and are typically not unitarily invariant. Nevertheless, we prove that for N = 2 dimensional Hilbert space the quantum 2-Wasserstein distance (unique up to rescaling) is monotonous with respect to any single-qubit quantum operation and the solution of the quantum transport problem is essentially unique. Furthermore, for any N ⩾ 3 and the quantum cost matrix proportional to a projector we demonstrate the monotonicity under arbitrary mixed unitary channels. Finally, we provide numerical evidence which allows us to conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps for dimension N = 3 and 4.