Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum (c-q) channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features. (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom Theorem. Namely, the error probability of sending $M$ messages through a c-q channel is upper bounded by the minimum error of distinguishing the joint channel input-output state against $(M-1)$ decoupled products states. (ii) Our bound directly yields asymptotic results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality are no longer needed. Hence, the derived one-shot bound sharpens existing results relying on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable $\epsilon$-one-shot capacity for c-q channel coding heretofore, improving the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot achievability for classical data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols.

翻译：信息理论中的可达性指展示一种编码策略，使其能够为特定任务达到预定的性能基准。在量子信息理论中，精妙的林-永尾算子不等式是证明多种单次可达性界的关键技术，因为它能有效模拟多种问题中的联合界。本文表明，优好测量本身同样天然具备联合界的作用。通过巧妙应用优好测量，我们能用简洁的三行证明大幅简化经典-量子信道编码的单次可达性推导。所提出的分析具有以下优点：(i) 建立的单次界可表示为如著名的霍列沃-赫尔斯特罗姆定理那样的闭式表达式，即通过经典-量子信道发送$M$条消息的错误概率，上界由区分联合信道输入-输出状态与$(M-1)$个解耦乘积状态的最小误差给出；(ii) 该界能统一导出大偏差、小偏差和中偏差渐近结果；(iii) 无需再用到林-永尾算子不等式中的系数，因此推导出的单次界优于依赖该不等式的现有结果。特别地，我们获得了迄今最紧的经典-量子信道编码ε-单次容量，改进了渐近场景中的三阶编码速率；(iv) 该结果适用于无限维希尔伯特空间；(v) 所提方法可推广至带量子边信息的经典数据压缩、量子信道的纠缠辅助经典通信以及多种量子网络信息处理协议的立足点可达性推导。