arXiv:2602.08628

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

Gergely Bunth, Eszter Szabó, Dániel Virosztek

math-phmath.OAquant-phSubmitted Feb 9, 2026• Announced Feb 10, 2026• 0 citations• Revised Jul 28, 2026 (v2)

Abstract

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.

Comments: 13 pages, accepted manuscript version

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arXiv:2602.08628

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

Gergely Bunth, Eszter Szabó, Dániel Virosztek

math-phmath.OAquant-phSubmitted Feb 9, 2026• Announced Feb 10, 2026• 0 citations• Revised Jul 28, 2026 (v2)

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Abstract

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.

Comments: 13 pages, accepted manuscript version

Related papers

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

Gergely Bunth, Eszter Szabó, Dániel Virosztek

Proceedings of the American Mathematical SocietySep 2, 20260 citations

Quantum Wasserstein isometries with a single observable generating the transport cost

Eszter Szabó, Dániel Virosztek

Acta Scientiarum MathematicarumJun 27, 20260 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.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:2607.08298Jul 10, 20260 citations

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Quantum Wasserstein isometries on the qubit state space

György Pál Gehér, József Pitrik, Tamás Titkos +1 more

arXiv:2204.14134Journal of Mathematical Analysis and ApplicationsDec 28, 20229 citations

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.

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.

\mathcal{C}

by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in

\mathcal{C}

act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category

\mathcal{C}

being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.