New quantum analogues of the Hamming Wasserstein distance are defined on traces of C(S_n^+) and proven to be metrics with subadditivity and exact classical recovery.
Carlen and Jan Maas, Gradient flow and entropy inequalities for quantum markov semigroups with detailed balance, Journal of Functional Analysis 273 (2017), no
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Quantum Wasserstein distances for quantum permutation groups
New quantum analogues of the Hamming Wasserstein distance are defined on traces of C(S_n^+) and proven to be metrics with subadditivity and exact classical recovery.