Monotonicity of the quantum 2-Wasserstein distance

Abstract

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 $N\ge 3$ 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 $N$.

arXiv comment30 pages, 5 figures

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• Isometries of the qubit state space with respect to quantum Wasserstein distances
• A perspective on Lindblad's Non-Equilibrium Entropy