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).
[BPTV25b] Gergely Bunth, József Pitrik, Tamás Titkos, and Dániel Virosztek.Wasserstein Distances and Diver- gences of Orderpby Quantum Channels
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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).