Quantum Wasserstein distances for quantum permutation groups

TL;DR

This paper introduces three quantum Wasserstein distances on the state space of the quantum permutation group, generalizing classical metrics and exploring their fundamental properties within noncommutative geometry.

Contribution

It defines and analyzes three new quantum Wasserstein distances on the quantum permutation group, extending classical metric concepts to the quantum setting.

Findings

Established basic metric properties of the distances

Proved subadditivity under convolution

Showed density of Lipschitz elements in the C*-algebra

Abstract

We seek an analog for the quantum permutation group $S_n^+$ of the normalized Hamming distance for permutations. We define three distances on the tracial state space of $C(S_n^+)$ that generalize the $L^1$-Wasserstein distance of probability measures on $S_n$ equipped with the normalized Hamming metric, for which we demonstrate basic metric properties, subadditivity under convolution, and density of the Lipschitz elements in the $\mathrm{C}^{\ast}$-algebra.