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

TL;DR

This paper characterizes the isometries of the n-qubit state space under the quantum Wasserstein distance, showing they are exactly the Wigner symmetries, which are unitary or anti-unitary conjugations.

Contribution

It establishes that the isometry group of the quantum Wasserstein distance for n-qubits coincides with the Wigner symmetry group, extending understanding of quantum state space symmetries.

Findings

Isometry group is exactly the Wigner symmetry group.

Quantum Wasserstein distance's isometries are unitary or anti-unitary conjugations.

Results hold for all natural numbers n.

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.