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arXiv:1112.1687 · Allerton Conference on Communication, Control, and Computing

Non-asymptotic information theoretic bound for some multi-party scenarios

Naresh Kumar Sharma, Naqueeb Ahmad Warsi

Published Oct 1, 2012• 3 citations
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Abstract

In the last few years, there has been a great interest in extending the information-theoretic scenario for the non-asymptotic or one-shot case, i.e., where the channel is used only once. We provide the one-shot rate region for the distributed source-coding (Slepian-Wolf) and the multiple-access channel. Our results are based on defining a novel one-shot typical set based on smooth entropies that yields the one-shot achievable rate regions while leveraging the results from the asymptotic analysis. Our results are asymptotically optimal, i.e., for the distributed source coding they yield the same rate region as the Slepian-Wolf in the limit of unlimited independent and identically distributed (i.i.d.) copies. Similarly for the multiple-access channel the asymptotic analysis of our approach yields the rate region which is equal to the rate region of the memoryless multiple-access channel in the limit of large number of channel uses.

DOI
10.1109/allerton.2012.6483332

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In the last few years, there has been a great interest in extending the information-theoretic scenario for the non-asymptotic or one-shot case, i.e., where the channel is used only once. We provide the one-shot rate region for the distributed source-coding (Slepian-Wolf) and the multiple-access channel. Our results are based on defining a novel one-shot typical set based on smooth entropies that yields the one-shot achievable rate regions while leveraging the results from the asymptotic analysis. Our results are asymptotically optimal, i.e., for the distributed source coding they yield the same rate region as the Slepian-Wolf in the limit of unlimited independent and identically distributed (i.i.d.) copies. Similarly for the multiple-access channel the asymptotic analysis of our approach yields the rate region which is equal to the rate region of the memoryless multiple-access channel in the limit of large number of channel uses.
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One-shot achievable rate region for source coding when coded side information is available at the decoder (source coding with a helper) is proposed. The achievable region proposed is in terms of conditional smooth max Rényi entropy and smooth max Rényi divergence. Asymptotically (in the limit of large block lengths) this region is quantified in terms of spectral-sup conditional entropy rate and spectral-sup mutual information rate. In particular, it coincides with the rate region derived in the limit of unlimited arbitrarily distributed copies of the sources.
We consider the problem of shared randomness-assisted multiple access channel (MAC) simulation for product inputs and characterize the one-shot communication cost region via almost-matching inner and outer bounds in terms of the smooth max-information of the channel, featuring auxiliary random variables of bounded size. The achievability relies on a rejection-sampling algorithm to simulate an auxiliary channel between each sender and the decoder, and producing the final output based on the output of these intermediate channels. The converse follows via information-spectrum based arguments. To bound the cardinality of the auxiliary random variables, we employ the perturbation method from [Anantharamet al., IEEE Trans. Inf. Theory (2019)] in the one-shot setting. For the asymptotic setting and vanishing errors, our result expands to a tight single-letter rate characterization and consequently extends a special case of the simulation results of [Kurriet al., IEEE Trans. Inf. Theory (2022)] for fixed, independent and identically distributed (iid) product inputs to universal simulation for any product inputs. We broaden our discussion into the quantum realm by studying feedback simulation of quantum-to-classical (QC) MACs with product measurements [Atifet al., IEEE Trans. Inf. Theory (2022)]. For fixed product inputs and with shared randomness assistance, we give a quasi tight one-shot communication cost region with corresponding single-letter asymptotic iid expansion.
We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the fully quantum Slepian–Wolf (FQSW) protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in quantum information theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic independent and identically distributed scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previous bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.
This study considers the unconditional smooth Rényi entropy proposed by Renner and Wolf [ASIACRYPT, 2005], the smooth conditional Rényi entropy proposed by Kuzuoka [IEEE Trans. Inf. Th., 66(3), 1674–1690, 2020], and a novel quantity which we term theconditional smooth-⋆entropy.The latter two quantities can be specialized to the first in the absence of side-information. We explore the operational roles of these smooth Rényi entropies by establishing one-shot coding theorems for several information-theoretic problems, including Campbell’s source coding problem, the Arıkan–Massey guessing problem, and the Bunte–Lapidoth task encoding problem. We consider these problems in cases where the errors are non-vanishing and for each problem, we consider two error formalisms: the average and maximum error criteria, where the averaging and maximization are taken with respect to the side-information. Using the one-shot coding theorems, we conclude that Kuzuoka’s smooth conditional Rényi entropy and the conditional smooth-⋆ entropy are the solutions to the problems involving the average and maximum error criteria, respectively. Furthermore, we examine asymptotic expansions of these entropies when the underlying source with its side-information is stationary and memoryless. Applying our asymptotic expansions to the one-shot coding theorems, we derive various fundamental limits for these problems. We show that, under non-degenerate settings, the first-order fundamental limits differ under the average and maximum error criteria. This is in contrast to a different but related setting considered by the present authors [IEEE Trans. Inf. Th., 66(12), 7565–7587, 2020], for variable-length conditional source coding allowing errors, in which the first-order terms are identical but the second-order terms are different under these error criteria.