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This research establishes rigorous mathematical bounds between competing definitions of quantum Wasserstein distance, clarifying their equivalence for qubits and proving inequality relationships for higher-dimensional systems. By linking these distances to the Wigner-Yanase skew information, the work advances foundational theory in quantum optimal transport and information geometry. Such theoretical refinements are critical for developing efficient quantum algorithms that rely on precise metric structures for state comparison and control.
Relations between different definitions of the quantum Wasserstein distance
arXiv:2605.03027v2 Announce Type: replace Abstract: The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the square of the self-distance equals the Wigner-Yanase skew information in this case. Moreover, for finite-dimensional systems of dimension $d\ge2$ and any number of operators, we show that the Golse-Mouhot-Paul-Caglioti self-distance is bounded from below by the De Palma-Trevisan self-distance.
Abstract:The quantum Wasserstein distances defined by Golse, Mouhot, Paul, and Caglioti and by De Palma and Trevisan coincide for qubits when a single operator appears in the cost function. As a consequence, the square of the self-distance equals the Wigner-Yanase skew information in this case. Moreover, for finite-dimensional systems of dimension $d\ge2$ and any number of operators, we show that the Golse-Mouhot-Paul-Caglioti self-distance is bounded from below by the De Palma-Trevisan self-distance.
6 pages, revtex 4.2; v2: further Theorems about N>1 and d>2 added
arXiv:2605.03027 [quant-ph]
(or arXiv:2605.03027v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2605.03027 Focus to learn more arXiv-issued DOI via DataCite
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