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

Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu (Dec 05 2025).

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.

Arxiv: https://arxiv.org/abs/2512.04345