Monotonicity of the quantum 2-Wasserstein distance

• URL: http://arxiv.org/abs/2204.07405v2
• Date: Fri, 9 Sep 2022 07:26:50 GMT
• Title: Monotonicity of the quantum 2-Wasserstein distance
• Authors: Rafa{\l} Bistro\'n, Micha{\l} Eckstein, Karol \.Zyczkowski
• Abstract summary: We show that for $N=2$ dimensional Hilbert space the quantum 2-Wasserstein distance is monotonous with respect to any single-qubit quantum operation. We conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps in any dimension $N$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• 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 $N \geq 3$ 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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