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