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arXiv:1409.7246 · IEEE Transactions on Information Theory

Polar Codes in Network Quantum Information Theory

Christoph Hirche, Ciara Morgan, Mark M. Wilde

Published Jan 4, 2016• 23 citations
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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 this paper, 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 so-called quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.

DOI
10.1109/tit.2016.2514319

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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 this paper, 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 so-called quantum simultaneous decoder. Thus, this paper paves the way for developing network classical-quantum information theory further without requiring a quantum simultaneous decoder.
Polar codes are the first capacity achieving and efficiently implementable codes for classical communication. Recently they have also been generalized to communication over classical-quantum and quantum channels. In this work we present our recent results for polar coding in quantum information theory, including applications to classical-quantum multiple access channels, interference channels and compound communication settings, including the first proof of channel coding achieving the Han-Kobayashi rate region of the interference channel without the need of a simultaneous decoder. Moreover we add to the existing framework by extending polar codes to achieve the asymmetric capacity and improving the block error probability for classical-quantum channels. In addition we use polar codes to prove a new achievable rate region for the classical-quantum broadcast channel. We also discuss polar codes for quantum communication over quantum channels and state results towards codes for compound quantum channels in this setting. We conclude by stating a list of interesting open questions to invite further research on the topic.
We construct new polar coding schemes for the transmission of quantum or private classical information over arbitrary quantum channels. In the former case, our coding scheme achieves the symmetric coherent information, and in the latter, the symmetric private information. Both schemes are built from a polar coding construction capable of transmitting classical information over a quantum channel. Appropriately merging two such classical-quantum schemes, one for transmitting amplitude information and the other for transmitting phase, leads to the new private and quantum coding schemes, similar to the construction for Pauli and erasure channels of Renes et al. The encoding is entirely similar to the classical case, and thus efficient. The decoding can also be performed by successive cancellation, as in the classical case, but no efficient successive cancellation scheme is yet known for arbitrary quantum channels. An efficient code construction is unfortunately still unknown. Generally, our two coding schemes require entanglement or secret-key assistance, respectively, but we extend two known conditions under which the needed assistance rate vanishes. Finally, although our results are formulated for qubit channels, we show how the scheme can be extended to multiple qubits. This then demonstrates a near-explicit coding method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels, which individually have zero quantum capacity can have a nonzero quantum capacity when used together.
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of this paper is to fill this gap by constructing near-explicit “polar” codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization is an effect in which one can synthesize a set of channels, by “channel combining” and “channel splitting,” in which a fraction of the synthesized channels are perfect for data transmission, while the other channels are completely useless for data transmission, with the good fraction equal to the capacity of the channel. The channel polarization effect then leads to a simple scheme for data transmission: send the information bits through the perfect channels and “frozen” bits through the useless ones. The main technical contributions of this paper are threefold. First, we leverage several known results from the quantum information literature to demonstrate that the channel polarization effect occurs for channels with classical inputs and quantum outputs. We then construct linear polar codes based on this effect, and the encoding complexity isO(NlogN), whereNis the blocklength of the code. We also demonstrate that a quantum successive cancellation decoder works well, in the sense that the word error rate decays exponentially with the blocklength of the code. For this last result, we exploit Sen's recent “noncommutative union bound” that holds for a sequence of projectors applied to a quantum state.
We discuss two techniques for transmitting classical information over quantum broadcast channels. The first technique is a quantum generalization of the superposition coding scheme for the classical broadcast channel. We use a quantum simultaneous nonunique decoder and obtain a simpler proof of the rate region recently published by Yard et al. in independent work. Our second result is a quantum Marton coding scheme, which gives the best known achievable rate region for quantum broadcast channels. Both results exploit recent advances in quantum simultaneous decoding developed in the context of quantum interference channels.
Polar coding, introduced 2008 by Arıkan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for classical discrete memoryless channels in the asymptotic limit of large block sizes. Here, we study the use of polar codes for the transmission of quantum information. Focusing on the case of qubit Pauli channels and qubit erasure channels, we use classical polar codes to construct a coding scheme that asymptotically achieves a net transmission rate equal to the coherent information using efficient encoding and decoding operations and code construction. Our codes generally require preshared entanglement between sender and receiver, but for channels with a sufficiently low noise level we demonstrate that the rate of preshared entanglement required is zero.