AI 中文总结
该研究通过“优良准则”从经典-量子信道推导二进制码的四个率-距离界，并用混合量子比特信道等新信道改进MRRW界，得到更优的码率性能。
AI 中文摘要
我们从一个定理“优良准则”（pretty good criterion）推导出二进制码的四个主要渐近率-距离权衡：Plotkin界、Elias-Bassalygo界以及两个McEliece-Rodemich-Rumsey-Welch（MRRW）界。若二进制输入输出对称的经典-量子（cq）信道在优良测量（PGM，后验采样的量子类比）下的误码率低于δ，则每个相对距离为δ、长度为n的二进制码（线性或非线性）的码率至多为该信道容量，误差为O(n^(-1/2))。由此，率-距离界可简化为信道设计问题，任务是在后验误码率约束下最小化信道容量。通过优良准则，二进制删除信道（BEC）对应Plotkin界，二进制对称信道（BSC）对应Elias-Bassalygo界，纯态信道（PSC）对应第一个MRRW界，掩码纯态信道对应第二个MRRW界。该框架随后用新信道实例化以改进MRRW界：混合量子比特信道（MQC，PSC的混合态版本）在0<δ<1/2的每个点严格改进第一个MRRW界，而掩码混合量子比特信道（2MQC）在相同区间严格改进第二个MRRW界。
英文摘要
We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.'' If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior sampling---of a binary-input output-symmetric classical--quantum (cq) channel lies below $δ$, then every length-$n$ binary code, linear or nonlinear, of relative distance $δ$ has rate at most the channel's capacity, up to an $O(n^{-1/2})$ correction. Rate--distance bounds thereby reduce to a channel design problem, wherein the task is to minimize channel capacity subject to the posterior bit error rate constraint. Via the pretty good criterion, the binary erasure channel (BEC) yields Plotkin, the binary symmetric channel (BSC) yields Elias--Bassalygo, the pure-state channel (PSC) yields the first MRRW bound, and a masked PSC yields the second MRRW bound exactly. This framework is then instantiated with new channels to improve upon the MRRW bounds. Specifically, the mixed-qubit channel (MQC), a mixed-state version of PSC, strictly improves the first MRRW bound at every $0 < δ< \frac{1}{2}$, while the masked mixed-qubit channel (2MQC) strictly improves the second MRRW bound throughout the same interval.
Comments77 pages, 4 figures
