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The operator layer cake theorem is equivalent to Frenkel's integral formula
Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu·December 4, 2025
Quantum Physicscs.ITMathematical Physicsmath.FA
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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.