Wasserstein distances and divergences of order $p$ by quantum channels
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arXiv
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| Format: | Preprint |
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2025
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| author | Bunth, Gergely Pitrik, József Titkos, Tamás Virosztek, Dániel |
| author_facet | Bunth, Gergely Pitrik, József Titkos, Tamás Virosztek, Dániel |
| contents | We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wasserstein distances and divergences of order $p$ by quantum channels Bunth, Gergely Pitrik, József Titkos, Tamás Virosztek, Dániel Mathematical Physics Operator Algebras Quantum Physics Primary: 49Q22, 81P16. Secondary: 81Q10 We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] where quantum channels realize the transport. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences and study their fundamental geometric properties. Finally, we prove triangle inequality for quadratic Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure, which is a generalization of our previous result in this direction. |
| title | Wasserstein distances and divergences of order $p$ by quantum channels |
| topic | Mathematical Physics Operator Algebras Quantum Physics Primary: 49Q22, 81P16. Secondary: 81Q10 |
| url | https://arxiv.org/abs/2501.08066 |