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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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