A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel $W$ and rate $R$: the constant $E(W,R)$ which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate $R$ to communicate over ever more (memoryless) instances of a given channel $W$. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai's sphere-packing upper bound [IEEE TIT 59, 8027 (2013)] for rates above a critical value, exactly analogous to the case of classical channels. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification [CMP 333, 335 (2015)]. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager's method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [IEEE TIT 69, 1680 (2023)], at least for linear randomness extractors.

翻译：在香农理论（无论是经典还是量子领域）中，一个基本关注量是给定信道 $W$ 和速率 $R$ 的误差指数：常数 $E(W,R)$，它决定了当使用固定速率 $R$ 的渐近最优码在给定信道 $W$ 的越来越多（无记忆）实例上进行通信时，解码误差的指数衰减速率。对于经典信道，近乎匹配的下界和上界是众所周知的。本文中，我给出了在任意经典-量子信道上通信的误差指数的一个下界，该下界在速率高于一个临界值时与 Dalai 的球堆积上界 [IEEE TIT 59, 8027 (2013)] 相匹配，这与经典信道的情况完全类似。然而，与经典情况不同，该论证并非通过对合适解码器的精细分析进行，而是通过利用 Hayashi 关于隐私放大这一密码学任务误差指数的界 [CMP 333, 335 (2015)]。然后，通过紧致的熵不确定关系以及 Gallager 为任意信道构造达到容量的奇偶校验码的方法，将此界与编码问题联系起来。在此过程中，我找到了相对于量子边信息的经典信息压缩任务的误差指数的一个下界，该下界与 Cheng 等人 [IEEE TIT 67, 902 (2021)] 的球堆积上界相匹配。反过来，Cheng 等人发现的球堆积界中的多项式前因子可以转化到隐私放大问题中，从而锐化了 Li、Yao 和 Hayashi 最近的一个结果 [IEEE TIT 69, 1680 (2023)]，至少对于线性随机性提取器是如此。