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

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 $n\in \mathbb{N}\mathrm{.}$ It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.