A new framework defines channel cost and contraction via Lipschitz seminorms, proving duality, tensor properties, and applications to word length, Carnot-Carathéodory distance, and mixing times.
Quantum Wasserstein distances for quantum permutation groups
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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.
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Transportation cost and contraction coefficient for channels on von Neumann algebras
A new framework defines channel cost and contraction via Lipschitz seminorms, proving duality, tensor properties, and applications to word length, Carnot-Carathéodory distance, and mixing times.