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

TL;DR

This paper establishes the equivalence between the operator layer cake theorem and Frenkel's integral formula, revealing a deep connection in quantum information theory through integral representations and derivatives.

Contribution

It proves that the operator layer cake theorem and Frenkel's integral formula are mathematically equivalent, providing new insights into their relationship.

Findings

Proves the equivalence between the operator layer cake theorem and Frenkel's integral formula.

Shows that the theorem provides an alternative proof for Frenkel's integral formula.

Highlights the deep connection between integral representations and quantum entropy measures.

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.