Monotonicity of the quantum 2-Wasserstein distance

TL;DR

This paper investigates a quantum analogue of the 2-Wasserstein distance, revealing its properties, limitations, and monotonicity under certain quantum operations, with numerical evidence suggesting broader monotonicity in higher dimensions.

Contribution

It establishes the monotonicity of the quantum 2-Wasserstein distance for qubits and certain higher-dimensional cases, and provides numerical evidence for its general monotonicity under CPTP maps.

Findings

Quantum 2-Wasserstein distance is not a Riemannian metric.

Monotonous under single-qubit quantum operations.

Numerical evidence suggests monotonicity under all CPTP maps.

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 $N$.