We study the quantum umlaut information, a correlation measure defined for bipartite quantum states $\rho_{AB}$ as a reversed variant of the quantum mutual information: $U(A;B)_\rho = \min_{\sigma_B} D(\rho_A\otimes \sigma_B\|\rho_{AB})$ in terms of the quantum relative entropy $D$. As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of `oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel -- as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling--assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the `geometric' version of umlaut information.

翻译：我们研究量子变音信息，这是一种针对二分量子态 $\rho_{AB}$ 定义的相关性度量，作为量子互信息的反向变体：$U(A;B)_\rho = \min_{\sigma_B} D(\rho_A\otimes \sigma_B\|\rho_{AB})$，其中 $D$ 表示量子相对熵。与经典情形类似[Girardi 等人，arXiv:2503.18910]，该定义允许闭式表达，并具有操作解释：在判断给定二分态是否为乘积态的假设检验任务中，它表征了渐近错误指数。我们将变音信息推广至量子信道，这也扩展了'oveloh 信息'的概念[Nuradha 等人，arXiv:2404.16101]。我们证明了对于经典-量子信道，信道变音信息具有可加性，而对于完全量子信道，我们观察到了可加性违反。受纠缠理论近期成果的启发，我们随后证明，作为主要结果，正则化变音信息构成了衡量经典信息通过量子信道传输质量的基本度量——这与容量（量化可发送信息量）形成对比。这一解释适用于由激活的非信号关联辅助的编码，且信道变音信息通常大于 Dalai 针对经典-量子情形所得的无辅助通信对应表达式[IEEE Trans. Inf. Theory 59, 8027 (2013)]。结合先前关于非信号辅助零误差信道容量的研究，我们的发现意味着零速率错误指数与零误差通信两种场景之间存在二分性。虽然我们的结果仅对经典-量子信道是单字母形式的，我们也针对完全量子信道给出了一个基于变音信息'几何'版本的单字母界。