I discuss a generalization of the Wasserstein distance of order 1 to quantum states. The proposed distance recovers the Hamming distance for the vectors of the canonical basis, and more generally the classical Wasserstein distance for quantum states diagonal in the canonical basis.
I discuss a continuity bound for the von Neumann entropy with respect to the proposed distance, which significantly strengthens the best continuity bound with respect to the trace distance. I also discuss applications, including proving tighter limitations on variational quantum algorithms and enhancing the learning of quantum data.
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