The operator layer cake theorem is equivalent to Frenkel's integral formula
Hao-Chung Cheng, Gilad Gour, Ludovico Lami, Po-Chieh Liu
TL;DR
The paper establishes the equivalence between the operator layer cake theorem for the derivative of the logarithm and Frenkel's integral formula for Umegaki relative entropy by proving (iii)⇒(ii) and (ii)⇒(i), thereby completing the cycle of implications with previously known (i)⇒(ii) and (i)⇒(iii). The methodology relies on differentiating spectral, projection-based representations, applying dominated convergence, and leveraging properties of the positive/negative parts and hockey-stick divergences E_γ. This unification provides a deeper understanding of how operator monotone derivatives relate to entropy representations, with potential implications for quantum information tasks such as error exponents and entropy bounds. Overall, the result shows Frenkel's integral formula and the operator layer cake theorem encode the same underlying structure for quantum divergences in finite dimensions.
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.