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

arXiv:2408.09879 · Linear Algebra and its Applications

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

Richárd Simon, Dániel Virosztek

Published Mar 13, 2025• 3 citations
arXivHTMLPDF

Abstract

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 .

DOI
10.1016/j.laa.2025.03.004

Related papers

Quantum Wasserstein isometries with a single observable generating the transport cost

Eszter Szabó, Dániel Virosztek

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.

Acta Scientiarum Mathematicarum•Jun 27, 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

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

Metric property of quantum Wasserstein divergences

Gergely Bunth, József Pitrik, Tamás Titkos +1 more

Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.

arXiv:2402.13150•Physical Review A•Aug 7, 2024•7 citations

Shape space in terms of Wasserstein geometry and application to quantum physics

Bernadette Lessel

This thesis offers a mathematical framework to treat quantum dynamics without reference to a background structure, but rather by means of the change of the shape of the state. For this, Wasserstein geometry is used. The so called Shape space, then, is defined as the quotient space of a Wasserstein space modulo the action of the isometry group of the background space. On Shape space we find a natural metric distance, the Shape distance, of which we investigate topological, metric and geodesic properties. Canonically mapped solutions of the Schrödinger equation turn out to be naturally nice curves in Shape space and some even constitue geodesics there. To also be able to speak about infinitesimal change of shapes, a definition for a tangent space at a point in Shape space is defined and applied. Finally a notion of differentiability for maps between Wasserstein spaces is proposed.

•Jan 1, 2018•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.

arXiv:2009.04469•IEEE Transactions on Information Theory•May 3, 2021•91 citations
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 .
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.
Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.
This thesis offers a mathematical framework to treat quantum dynamics without reference to a background structure, but rather by means of the change of the shape of the state. For this, Wasserstein geometry is used. The so called Shape space, then, is defined as the quotient space of a Wasserstein space modulo the action of the isometry group of the background space. On Shape space we find a natural metric distance, the Shape distance, of which we investigate topological, metric and geodesic properties. Canonically mapped solutions of the Schrödinger equation turn out to be naturally nice curves in Shape space and some even constitue geodesics there. To also be able to speak about infinitesimal change of shapes, a definition for a tangent space at a point in Shape space is defined and applied. Finally a notion of differentiability for maps between Wasserstein spaces is proposed.
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.