Within quadratic costs from ≤ d^{2}-1 observables, Wasserstein isometries are precisely the Wigner symmetries iff the cost is isotropic (a positive multiple of the identity on the traceless subspace).
Isometries of the qubit state space with respect to quantum Wasserstein distances
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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 $\sigma_z$.
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Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions
Within quadratic costs from ≤ d^{2}-1 observables, Wasserstein isometries are precisely the Wigner symmetries iff the cost is isotropic (a positive multiple of the identity on the traceless subspace).