Wasserstein distances and divergences of order $p$ by quantum channels

• URL: http://arxiv.org/abs/2501.08066v2
• Date: Fri, 24 Jan 2025 15:19:07 GMT
• Title: Wasserstein distances and divergences of order $p$ by quantum channels
• Authors: Gergely Bunth, József Pitrik, Tamás Titkos, Dániel Virosztek,
• Abstract summary: We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem. Relying on this general machinery, we introduce $p$-Wasserstein distances and divergences. We prove triangle inequality for Wasserstein divergences under the sole assumption that an arbitrary one of the states involved is pure.
• Score: 1.8749305679160366
• License:
• Abstract: We introduce a non-quadratic generalization of the quantum mechanical optimal transport problem introduced in [De Palma and Trevisan, Ann. Henri Poincar\'e, {\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.

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