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

Acta Scientiarum Mathematicarum

Quantum Wasserstein isometries with a single observable generating the transport cost

Eszter Szabó, Dániel Virosztek

Published Jun 27, 2026• 0 citations
Publisher

Abstract

Abstract We consider quantum Wasserstein distances where the transport cost is generated by a single observable quantity, and study the isometries of quantum state spaces with respect to these distances. The main result is an extension of Theorem 2 in [R. Simon, D. Virosztek, Linear Algebra Appl. 714 (2025), 1–14], which describes the qubit Wasserstein isometries induced by a single observable, to finite quantum systems of arbitrary dimensions.

DOI
10.1007/s44146-026-00254-5

Related papers

Quantum Wasserstein isometries of the 𝑛-qubit state space: A Wigner-type result

Gergely Bunth, Eszter Szabó, Dániel Virosztek

We determine the isometry group of the n n -qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all n ∈ N n \in \mathbb {N} . It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.

Proceedings of the American Mathematical Society•Sep 2, 2026•0 citations

Quantum Wasserstein isometries of the nnn-qubit state space: a Wigner-type result

Gergely Bunth, Eszter Szabó, Dániel Virosztek

We determine the isometry group of the $n$-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all $n \in \mathbb{N}.$ It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.

math-phCross-listedarXiv:2602.08628•Feb 10, 2026•0 citations

Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions

Gergely Bunth

We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost $C$ on a $d$-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator $Θ(C)$, and is therefore at most $d^2-1$. Nonzero isotropic costs require exactly $d^2-1$ generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.

math-phCross-listedarXiv:2607.08298•Jul 10, 2026•0 citations

Isometries of the qubit state space with respect to quantum Wasserstein distances

Richárd Simon, Dániel Virosztek

In this paper we study isometries of quantum Wasserstein distances and divergences on the quantum bit state space. We describe isometries with respect to the symmetric quantum Wasserstein divergence d sym , the divergence induced by all of the Pauli matrices. We also give a complete characterization of isometries with respect to D z , the quantum Wasserstein distance corresponding to the single Pauli matrix σ z .

arXiv:2408.09879•Linear Algebra and its Applications•Mar 13, 2025•3 citations

Quantum Wasserstein distance for Gaussian states

Anaelle Hertz, Mohammad Ahmadpoor, Oleksandr Dzhenzherov +2 more

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.

quant-pharXiv:2512.17809•Dec 22, 2025•0 citations

The Quantum Wasserstein Distance of Order 1

Giacomo De Palma, Milad Marvian, Dario Trevisan +1 more

We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.

•Jan 1, 2021•74 citations
Abstract We consider quantum Wasserstein distances where the transport cost is generated by a single observable quantity, and study the isometries of quantum state spaces with respect to these distances. The main result is an extension of Theorem 2 in [R. Simon, D. Virosztek, Linear Algebra Appl. 714 (2025), 1–14], which describes the qubit Wasserstein isometries induced by a single observable, to finite quantum systems of arbitrary dimensions.
We determine the isometry group of the nnn-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all n∈N.n \in \mathbb{N}.n∈N. It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.
We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost CCC on a ddd-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator Θ(C)Θ(C)Θ(C), and is therefore at most d2−1d^2-1d2−1. Nonzero isotropic costs require exactly d2−1d^2-1d2−1 generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.
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.
We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits. The proposal recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis. The proposed distance is invariant with respect to permutations of the qudits and unitary operations acting on one qudit and is additive with respect to the tensor product. Our main result is a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. We also propose a generalization of the Lipschitz constant to quantum observables. The notion of quantum Lipschitz constant allows us to compute the proposed distance with a semidefinite program. We prove a quantum version of Marton's transportation inequality and a quantum Gaussian concentration inequality for the spectrum of quantum Lipschitz observables. Moreover, we derive bounds on the contraction coefficients of shallow quantum circuits and of the tensor product of one-qudit quantum channels with respect to the proposed distance. We discuss other possible applications in quantum machine learning, quantum Shannon theory, and quantum many-body systems.