An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.

翻译：信息论领域长期存在一个重要议题，即关于达到信道容量的码的输出统计特性，以及这些经验分布与信道容量问题中最优输入所诱导的输出分布之间的比较。文献中已对多种优秀码的经验输出分布进行了研究，例如在错误概率趋近于零与非零两种情况下，将码的输出分布与最优输出分布进行比较。受此启发，我们旨在为用于经典通信的量子码实现类似结果，即通过量子码字经噪声量子信道传输经典信息的场景。我们首先证明了最优输出分布的唯一性，以便更具体地讨论最优输出分布。随后，我们运用与经典情形思想相近的技术，将错误概率趋近于零的结果推广至量子情形。此外，基于量子广义去极化半群的超压缩性结果所导出的二阶逆定理，我们还将非零错误概率的结果推广至量子分组码的情形。