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，作为量子互信息的逆向变体：$U(A;B)_\rho = \min_{\sigma_B} D(\rho_A\otimes \sigma_B\|\rho_{AB})$，其中 $D$ 表示量子相对熵。与经典情形类似 [Girardi et al., 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到量子信道，这也扩展了'oveloh信息'的概念 [Nuradha et al., arXiv:2404.16101]。我们证明了信道变音信息对经典-量子信道具有可加性，同时观察到全量子信道存在可加性破缺。受纠缠理论最新成果启发，我们随后证明主要结果：正则化变音信息构成了衡量量子信道中经典信息传输质量的基本度量——这与容量（量化可发送信息的数量）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 在经典-量子情形中获得的非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关联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-量子信道具有单字母形式，我们也针对全量子信道给出了基于变音信息'几何'版本的单字母界。