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Entropy 2017, 19(7), 355; doi:10.3390/e19070355

Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback–Leibler

Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy
This paper is an extended version of our paper published in the Proceedings of the 2016 International Zurich Seminar on Communications, Zurich, Switzerland, 2–4 March 2016.
Received: 20 May 2017 / Revised: 3 July 2017 / Accepted: 10 July 2017 / Published: 12 July 2017
(This article belongs to the Section Information Theory)
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Abstract

We review the use of binary hypothesis testing for the derivation of the sphere packing bound in channel coding, pointing out a key difference between the classical and the classical-quantum setting. In the first case, two ways of using the binary hypothesis testing are known, which lead to the same bound written in different analytical expressions. The first method historically compares output distributions induced by the codewords with an auxiliary fixed output distribution, and naturally leads to an expression using the Renyi divergence. The second method compares the given channel with an auxiliary one and leads to an expression using the Kullback–Leibler divergence. In the classical-quantum case, due to a fundamental difference in the quantum binary hypothesis testing, these two approaches lead to two different bounds, the first being the “right” one. We discuss the details of this phenomenon, which suggests the question of whether auxiliary channels are used in the optimal way in the second approach and whether recent results on the exact strong-converse exponent in classical-quantum channel coding might play a role in the considered problem. View Full-Text
Keywords: channel coding; sphere packing bound; classical-quantum channels; hypothesis testing channel coding; sphere packing bound; classical-quantum channels; hypothesis testing
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Dalai, M. Some Remarks on Classical and Classical-Quantum Sphere Packing Bounds: Rényi vs. Kullback–Leibler. Entropy 2017, 19, 355.

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