Monotonicity of a quantum 2-Wasserstein distance

Abstract We study a quantum analogue of the 2-Wasserstein distance as measure proximity on set ? N density matrices dimension N . show that such (semi-)distances do not induce Riemannian metrics tangent bundle and are typically unitarily invariant. Nevertheless, we prove for = 2 dimensional Hilbert space (unique up to rescaling) is monotonous with respect any single-qubit operation solution transport problem essentially unique. Furthermore, $N \geqslant 3$?> ? 3 cost matrix proportional projector demonstrate monotonicity under arbitrary mixed unitary channels. Finally, provide numerical evidence which allows us conjecture invariant semi-distance all CPTP maps 3 4.