The operator layer cake theorem is equivalent to Frenkel's integral formula
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908693660958720 |
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| author | Cheng, Hao-Chung Gour, Gilad Lami, Ludovico Liu, Po-Chieh |
| author_facet | Cheng, Hao-Chung Gour, Gilad Lami, Ludovico Liu, Po-Chieh |
| contents | The operator layer cake theorem provides an integral representation for the directional derivative of the operator logarithm in terms of a family of projections [arXiv:2507.06232]. Recently, the related work [arXiv:2507.07065] showed that the theorem gives an alternative proof to Frenkel's integral formula for Umegaki's relative entropy [Quantum, 7:1102 (2023)]. In this short note, we find a converse implication, demonstrating that the operator layer cake theorem is equivalent to Frenkel's integral formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04345 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The operator layer cake theorem is equivalent to Frenkel's integral formula Cheng, Hao-Chung Gour, Gilad Lami, Ludovico Liu, Po-Chieh Quantum Physics Information Theory Mathematical Physics Functional Analysis The operator layer cake theorem provides an integral representation for the directional derivative of the operator logarithm in terms of a family of projections [arXiv:2507.06232]. Recently, the related work [arXiv:2507.07065] showed that the theorem gives an alternative proof to Frenkel's integral formula for Umegaki's relative entropy [Quantum, 7:1102 (2023)]. In this short note, we find a converse implication, demonstrating that the operator layer cake theorem is equivalent to Frenkel's integral formula. |
| title | The operator layer cake theorem is equivalent to Frenkel's integral formula |
| topic | Quantum Physics Information Theory Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2512.04345 |