Quantum Wasserstein distances for quantum permutation groups

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.