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arXiv:1301.6339 · IEEE International Symposium on Information Theory (ISIT)

Lovász's theta function, Rényi's divergence and the sphere-packing bound

Marco Dalai

Published Jul 1, 2013• 6 citations
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Abstract

Lovász's bound to the capacity of a graph and the the sphere-packing bound to the probability of error in channel coding are given a unified presentation as information radii of the Csiszár type using the Rényi divergence in the classical-quantum setting. This brings together two results in coding theory that are usually considered as being of a very different nature, one being a “combinatorial” result and the other being “probabilistic”. In the context of quantum information theory, this difference disappears.

DOI
10.1109/isit.2013.6620222

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Lovász's bound to the capacity of a graph and the the sphere-packing bound to the probability of error in channel coding are given a unified presentation as information radii of the Csiszár type using the Rényi divergence in the classical-quantum setting. This brings together two results in coding theory that are usually considered as being of a very different nature, one being a “combinatorial” result and the other being “probabilistic”. In the context of quantum information theory, this difference disappears.
In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: 1) extending classical bounds of Shannon to classical-quantum channels, and 2) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lovász' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lovász' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lovász theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel, and the sphere packing bound for classical-quantum channels are established.
In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: 1) extending classical bounds of Shannon to classical-quantum channels, and 2) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lovász' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lovász' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lovász theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel, and the sphere packing bound for classical-quantum channels are established.
The sphere packing bound, in the form given by Shannon, Gallager and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lovász theta function. In this paper, we extend the study to the case of constant composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lovász theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is then applied to auxiliary channels to deduce a bound which can be interpreted as an extension of the Elias bound.
The sphere packing bound, in the form given by Shannon, Gallager, and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lovász theta function. In this paper, we extend the study to the case of constant-composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lovász theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is finally applied to auxiliary channels to deduce a bound, which is useful in the low rate region and which can be interpreted as an extension of the Elias bound.
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.
There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution. In [IEEE Transactions on Information Theory, 41(1):26-34, 1995], Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of constant composition classical-quantum channel coding.