Random and Nonrandom Coding for Quantum Information

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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.