Tight Lower Bound on the Error Exponent of Classical-Quantum Channels

IF 2.2 3区 计算机科学 Q3 COMPUTER SCIENCE, INFORMATION SYSTEMS
Joseph M. Renes
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Abstract

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 for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. 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. 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. 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, at least for linear randomness extractors.
经典量子信道误差指数的紧下界
香侬理论(经典或量子)的一个基本量是给定信道W和速率R的误差指数:常数$E(W,R)$,当使用固定速率R的更大的最佳代码在给定信道W的更多(无内存)实例上进行通信时,它控制解码错误的指数衰减。经典信道中几乎匹配的下界和上界是众所周知的。这里,我展示了任意经典量子(CQ)信道上通信误差指数的下界,它与达赖的球填充上界相匹配,其速率高于临界值,完全类似于经典信道的情况。这证明了Holevo在研究这个问题时的一个猜想。然而,与经典案例不同的是,该论点并没有通过对合适解码器的精确分析来进行,而是通过利用Hayashi对隐私放大加密任务的错误指数的限制来进行。然后,通过紧熵不确定性关系和Gallager构造任意信道的容量实现奇偶校验码的方法,将该界与编码问题联系起来。在此过程中,我发现经典信息压缩任务相对于量子侧信息的误差指数的下界与Cheng等人的球体填充上界相匹配。反过来,Cheng等人发现的球体填充界的多项式前因子可以转化为隐私放大问题,强化了Li, Yao和Hayashi最近的结果,至少对于线性随机提取器。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory 工程技术-工程:电子与电气
CiteScore
5.70
自引率
20.00%
发文量
514
审稿时长
12 months
期刊介绍: The IEEE Transactions on Information Theory is a journal that publishes theoretical and experimental papers concerned with the transmission, processing, and utilization of information. The boundaries of acceptable subject matter are intentionally not sharply delimited. Rather, it is hoped that as the focus of research activity changes, a flexible policy will permit this Transactions to follow suit. Current appropriate topics are best reflected by recent Tables of Contents; they are summarized in the titles of editorial areas that appear on the inside front cover.
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