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Frontiers in Optics

Concatenated Capacity-Achieving Polar Codes for Optical Quantum Channels

László Gyöngyösi, Sándor Imre

Published Jan 1, 2012• 0 citations
Publisher

Abstract

We construct concatenated capacity-achieving quantum codes for noisy optical quantum channels. We demonstrate that the error-probability of capacity-achieving quantum polar encoding can be reduced by the proposed low-complexity concatenation scheme.
DOI
10.1364/fio.2012.fw3a.47

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Quantum reading provides a general framework to formulate the statistical discrimination of quantum channels. Applying classical codes to this task gives at least two open questions: how to construct low complexity encoding schemes that are interesting for channel discrimination and, how to develop capacity-achieving protocols. The aim of this paper is to present a solution to these questions using polar codes. We characterize the rate and reliability of the channels under polar encoding. Additionally, it is shown that the error probability of the proposed scheme decays exponentially with respect to the code length. A full version of this paper is accessible at https://arxiv.org/abs/2012.07198v2
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Quantum readout provides a general framework for formulating statistical discrimination of quantum channels. Several paths have been taken for such this problem. However, there is much to be done in the avenue of optimizing channel discrimination using classical codes. At least two open questions can be pointed out: how to construct low complexity encoding schemes that are interesting for channel discrimination and, more importantly, how to develop capacity-achieving protocols. This paper aims at presenting a solution to these questions using polar codes. Firstly, we characterize the information rate and reliability parameter of the channels under polar encoding. We also show that the error probability of the scheme proposed decays exponentially with the square root of the code length. Secondly, an analysis of the optimal quantum states to be used as probes is given.
We construct explicit codes for transmitting classical or quantum information over the thermal-noise Bosonic Gaussian channel. For classical information transmission, our codes achieve the classical capacity, recently characterized by Giovannetti et al. [Nature Photonics 8, 796 (2014)], while in the quantum case they transmit at a rate given by the Gaussian coherent information. The construction is based on concatenating polar codes with a suitable discretization of the channel input into a finite constellation of coherent states. The codes have efficient encoders, but the complexity of decoding is unknown. Our results are based on a quantum version of Wu and Verd\'u's technique of determining the achievable rates of finite constellations for the classical Gaussian-noise channel [Allerton 2010, pp. 620].
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.