We study a doubly minimized variant of the lautum information - a reversed analogue of the mutual information - defined as the minimum relative entropy between any product state and a fixed bipartite quantum state; we refer to this measure as the tumula information. In addition, we introduce the corresponding Petz Renyi version, which we call the doubly minimized Petz Renyi lautum information (PRLI). We derive several general properties of these correlation measures and provide an operational interpretation in the context of hypothesis testing. Specifically, we show that the reverse direct exponent of certain binary quantum state discrimination problems is quantified by the doubly minimized PRLI of order $α\in (0,1/2)$, and that the Sanov exponent is determined by the tumula information. Furthermore, we investigate the extension of the tumula information to channels and compare its properties with previous results on the channel umlaut information [Girardi et al., arXiv:2503.21479].

翻译：我们研究了一种双重最小化的lautum信息变体——作为互信息反向模拟的该量定义为任意乘积态与固定二分量子态之间的最小相对熵；我们将此度量称为tumula信息。此外，我们引入了相应的Petz Renyi版本，称之为双重最小化Petz Renyi lautum信息（PRLI）。我们推导了这些关联度量的若干一般性质，并在假设检验背景下提供了操作解释。具体而言，我们证明特定二元量子态判别问题的反向直接指数由阶数$α\in (0,1/2)$的双重最小化PRLI量化，且Sanov指数由tumula信息决定。进一步，我们研究了tumula信息向信道的扩展，并将其性质与先前关于信道分音信息的结果[Girardi et al., arXiv:2503.21479]进行了比较。