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arXiv:2605.03027

Relations between different definitions of the quantum Wasserstein distance for qubits

Géza Tóth, József Pitrik

quant-phmath-phSubmitted May 4, 2026• Announced May 6, 2026• 0 citations
v1 · May 4, 2026v2 · Sep 14, 2026v3 · Sep 16, 2026
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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 self-distance equals the Wigner-Yanase skew information in this case.
Comments
5 pages, revtex 4.2. arXiv admin note: substantial text overlap with arXiv:2506.14523, arXiv:2209.09925

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Quantum Wasserstein divergences are modified versions of quantum Wasserstein distances defined by channels and they have been conjectured to be genuine metrics on quantum state spaces by De Palma and Trevisan. We prove triangle inequality for quantum Wasserstein divergences for every quantum system described by a separable Hilbert space and any quadratic cost operator under the assumption that a particular state involved is pure and all the states have finite energy. We also provide strong numerical evidence suggesting that the triangle inequality holds in general for an arbitrary choice of states.
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