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.

翻译：算子分层蛋糕定理通过一族投影给出了算子对数方向导数的积分表示[arXiv:2507.06232]。近期相关研究[arXiv:2507.07065]表明，该定理为Umegaki相对熵的Frenkel积分公式[Quantum, 7:1102 (2023)]提供了另一种证明。在这篇短文中，我们发现了逆向蕴含关系，证明算子分层蛋糕定理与Frenkel积分公式具有等价性。