We study a quantum analogue of the 2-Wasserstein distance as a measure of
    proximity on the set $\Omega_N$ of density matrices of dimension $N$. We show
    that such (semi-)distances do not induce Riemannian metrics on the tangent
    bundle of $\Omega_N$ 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 $N \geq 3$ 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 $N$.

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