How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Polar codes in network quantum information theory

Polar coding is a method for communication over noisy classical channels which is provably capacity-achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In the present work, we apply the polar coding method to network quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, our work paves the way for developing network quantum information theory further without requiring a quantum simultaneous decoder.Comment: 18 pages, 2 figures, v2: 10 pages, double column, version accepted for publicatio