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The operator layer cake theorem is equivalent to Frenkel's integral formula
arXiv
Authors: Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu
Year
2025
Paper ID
16220
Status
Preprint
Abstract Read
~2 min
Abstract Words
69
Citations
N/A
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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- 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].
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