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arXiv · 2512.04345

The operator layer cake theorem is equivalent to Frenkel's integral formula

Hao-Chung Cheng·Gilad Gour·Ludovico Lami·Po-Chieh Liu
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Abstract

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.

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Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu. 2025-12-04. The operator layer cake theorem is equivalent to Frenkel's integral formula. https://arxiv.org/abs/2512.04345

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