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Proceedings of the American Mathematical Society

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

Gergely Bunth, Eszter SzabĂł, DĂĄniel Virosztek

Published Sep 2, 2026• 0 citations
Publisher

Abstract

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.
DOI
10.1090/proc/17907

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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.
In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries C\mathcal{C}C by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in C\mathcal{C}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 C\mathcal{C}C being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.
We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.