Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions

We study the quantum Wasserstein distances introduced by De Palma and Trevisan associated with quadratic cost operators generated by families of self-adjoint observables. We first show that an arbitrary positive semidefinite cost operator is completely determined by the restriction of the corresponding Wasserstein distance to pairs of pure states. This allows geometric invariance of the pure-state distance to be translated directly into invariance of the cost operator.
Within the class of nonzero quadratic costs generated by at most $d^2-1$ observables on a $d$-dimensional Hilbert space, we prove that the Wasserstein isometry monoid consists exactly of the Wigner symmetries, that is, unitary and antiunitary conjugations, if and only if the distance is invariant under unitary conjugations on pure states. Equivalently, the cost operator intertwines the adjoint representation of the unitary group and is a positive scalar multiple of the identity on the traceless subspace.
We further construct explicit mutually inverse maps between quadratic cost operators generated by observables and Hilbert--Schmidt frame-type operators formed from their traceless parts. Under this correspondence, isotropy of the cost is equivalent to the tight frame property of the associated Hilbert--Schmidt operator. Consequently, a nonzero isotropic cost requires at least $d^2-1$ self-adjoint generators, and equality holds precisely when their traceless parts form, up to a common scale, a Hilbert--Schmidt orthonormal basis. Thus the geometric, representation-theoretic, operator-theoretic, and frame-theoretic notions of symmetry all determine the same one-parameter family of quantum Wasserstein distances.