Random and Nonrandom Coding for Quantum Information

量子信息理论是物理学和信息论的交叉学科。近年来，在许多操作量子信息任务中取得了巨大的进展：所有这些任务的基本问题都是在压缩、传输等的最佳速率之后，并且任何解决方案中的关键要素是编码方案的构造。几乎所有这些都是通过随机编码获得的，随机编码是信息论的关键方法，其中真实的艺术是一系列代码的建立和平均性能分析。这激发了拟议的研究，其起点是在最近的工作中成功使用的各种随机代码。然而，目前的知识状态仍然非常不完整：有许多基本问题无法用现有的随机码来解决，这意味着我们需要开发新的问题。另一方面，我们仍然不能计算信道的量子容量，尽管随机选择原则上描述了渐近最优码-这意味着我们仍然没有正确的码族用于量子纠错。覆盖和着色。在这些标题下，提出了对今后的进展至关重要的更具体的问题。

Quantum information theory represents a cross-disciplinary blend of physics and information theory. Recent years have seen dramatic progress in many operational quantum information tasks: the basic question in all of them is after the optimal rate of compression, transmission, etc., and the crucial element in any solution is the construction of a coding scheme. Virtually all of them are obtained by random coding'', the key method of information theory, where the real art is the setting-up and average performance analysis of a family of codes.This motivates the proposed research, at whose starting point are the various random codes used with such success in recent work. The present state of knowledge, however, is still sorely incomplete: there are a number of fundamental problems that cannot be approached with the available random codes, which means we need to develop new ones. And on the other hand we still cannot compute the quantum capacity of a channel, in spite of random selection in principle describing asymptotically optimal codes - which means that we still do not have the correct code family for quantum error correction.To focus the work, four larger themes are outlined: Characterisation of random codes; Failure of additivity, and nonrandom'' codes for channels; Reverse coding and normal forms; Covering and colouring. Under these headings more specific problems are formulated, which will be pivotal for future progress.