Polar Codes in Network Quantum Information Theory

Abstract

Abstract-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 classical-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 socalled quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.

Bounds on Information Combining With Quantum Side Information

“…also (61)) that H(X 1 + X 2 |B 1 B 2 ) = H 1 is possible only if H 1 = log 2 or H 2 = 0. Conversely, if H 1 = log 2 or H 2 = 0 then actually H(X 1 + X 2 |B 1 B 2 ) = H 1 since: (a) H 1 ≤ H(X 1 + X 2 |B 1 B 2 ) holds due to (55) along with strong subadditivity; (b) H(X 1 + X 2 |B 1 B 2 ) ≤ log 2 holds as X 1 + X 2 is a binary register; (c) and we have, similarly to (55) together with the fact that the conditional entropy H(X 2 |X 1 + X 2 , B 1 B 2 ) of a classical system is nonnegative: (54). The value of the bound along the diagonal line H 1 = H 2 is shown again as the purple curve in Fig.…”

“…Based on the classical setting, Polar codes were later generalized to channels with quantum outputs [7]. These quantum polar codes inherit many of the desirable features like the efficient encoder and the exponentially vanishing block error probability [7], [51], while especially the efficient decoder remains an open problem [52].…”

“…Since their introduction Polar codes have been investigated in many ways, like adaptations to many different settings in classical [53], [54] and quantum information theory [55], [56].…”

Preface to the Second Edition

“…A quantum interference channel has multiple senders and multiple receivers in which certain sender-receiver pairs are interested in communicating. Recent progress in this direction is in (Fawzi et al, 2012;Sen, 2011;Hirche et al, 2016). One could also consider distributed compression tasks, and various authors have contributed to this direction (Ahn et al, 2006;Abeyesinghe et al, 2009;Savov, 2008).…”

Strong secrecy on a class of Degraded Broadcast Channels using polar codes

“…Although recent literature has proven the existence of different (secrecy) capacity achieving polar coding schemes for multi-user scenarios (see, for example, [12]- [19]), practical polar codes for the two models on which this paper is focused are, as far as we know, not analyzed yet.…”