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SPHERE PACKING AND ZERO-RATE BOUNDS TO THE RELIABILITY OF CLASSICAL-QUANTUM CHANNELS

Marco Dalai

Published Jan 25, 2012• 1 citation
OpenAlex

Abstract

In this paper, the sphere packing bound of Fano, Shannon, Gal- lager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability func- tion obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo (21). Thus, for pure state channels, the reliability function at high rates is now ex- actly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.

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In this paper, the sphere packing bound of Fano, Shannon, Gal- lager and Berlekamp and the zero-rate bound of Berlekamp are extended to general classical-quantum channels. The upper bound for the reliability func- tion obtained from the sphere packing coincides at high rates, for the case of pure state channels, with a lower bound derived by Burnashev and Holevo (21). Thus, for pure state channels, the reliability function at high rates is now ex- actly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet. Finally, the obtained zero-rate upper bound to the reliability function of a general classical-quantum channel with no zero-error capacity coincides with a lower bound obtained by Holevo, thus determining the exact expression.
In this paper, the Sphere-Packing-Bound of Fano, Shannon, Gallager and Berlekamp is extended to general classical-quantum channels. The obtained upper bound for the reliability function, for the case of pure-state channels, coincides at high rates with a lower bound derived by Burnashev and Holevo [1]. Thus, for pure state channels, the reliability function at high rates is now exactly determined. For the general case, the obtained upper bound expression at high rates was conjectured to represent also a lower bound to the reliability function, but a complete proof has not been obtained yet.
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 aim of this work is to study the zero-error capacity of pure-state classical-quantum channels in the setting of list decoding. We provide an achievability bound for list-size two and a converse bound holding for every fixed list size. The two bounds coincide for channels whose pairwise absolute state overlaps form a positive semi-definite matrix. Finally, we discuss a remarkable peculiarity of the classical-quantum case: differently from the fully classical setting, the rate at which the sphere-packing bound diverges might not be achievable by zero-error list codes, even when we take the limit of fixed but arbitrarily large list size.
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.
We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information.
We study lower bounds on the optimal error probability in channel coding at rates below capacity, commonly termed sphere-packing bounds. In this work, we establish a sphere-packing bound for classical-quantum channels, which significantly improves previous prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of o(log n/n), indicating our sphere-packing bound is almost exact in the high rate regime. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.