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International Symposium on Information Theory and its Applications

A Study on the Problem of Channel Resolvability for Channels with Countable Input Alphabet

Shigeaki Kuzuoka

Published Oct 1, 2018• 1 citation
Publisher

Abstract

The problem of channel resolvability for channels with countably infinite input alphabet is considered. In particular, Hayashi's lower bound (i.e., converse part) on the optimal rate of channel resolvability, which was proved under the assumption of the finiteness of the input alphabet of the channel, is revisited. Our motivation is to prove the bound without the finiteness of the input alphabet. Although we cannot accomplish the final goal, we introduce a slight modification into the problem of channel resolvability and establish a coding theorem (particularly converse coding theorem) corresponding to Hayashi's result without the assumption of the finiteness of the input alphabet.

DOI
10.23919/isita.2018.8664277

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The problem of channel resolvability for channels with countably infinite input alphabet is considered. In particular, Hayashi's lower bound (i.e., converse part) on the optimal rate of channel resolvability, which was proved under the assumption of the finiteness of the input alphabet of the channel, is revisited. Our motivation is to prove the bound without the finiteness of the input alphabet. Although we cannot accomplish the final goal, we introduce a slight modification into the problem of channel resolvability and establish a coding theorem (particularly converse coding theorem) corresponding to Hayashi's result without the assumption of the finiteness of the input alphabet.
The authors have introduced the new notion of resolvability of a channel, as the dual of the capacity, which is defined as the minimum complexity per input letter needed to generate an input process whose output distribution via the channel arbitrarily accurately approximates any prescribed achievable output distribution. The resolvability thus introduced has revealed a deep relationship between the minimum achievable rate for source coding, the channel capacity, the identification capacity and the problem of random number generation. However, the validity of the proof of the converse for the resolvability formula established by Han and Verdu (see IEEE Trans. on Inform. Theory, vol.39, no.3, p.379, 1993) hinged heavily on the assumption that the input alphabet of the channel is finite. Our main purpose in this paper is to show that we can relax this restriction and to show that the resolvability formula of Han and Verdu continues to hold also for a wide class of channels with continuous input alphabet, including as a special case, additive white Gaussian noise (AWGN) channels with power constraint.>
Given a channel and an input process we study the minimum randomness of those input processes whose output statistics approximate the original output statistics with arbitrary accuracy. We introduce the notion of resolvability of a channel, defined as the number of random bits required per channel use in order to generate an input that achieves arbitrarily accurate approximation of the output statistics for any given input process. We obtain a general formula for resolvability which holds regardless of the channel memory structure. We show that, for most channels, resolvability is equal to Shannon capacity. By-products of our analysis are a general formula for the minimum achievable (fixed-length) source coding rate of any finite-alphabet source, and a strong converse of the identification coding theorem, which holds for any channel that satisfies the strong converse of the channel coding theorem.
Given a general channel, we first formulate the idetification capacity problem as well as the resolvability problem with input cost constraint in as the general form as possible, along with relevant fundamental theorems. Next, we establish some mild sufficient condition for the key lemma linking the identification capacity with the resolvability to hold for the continuous input alphabet case with input cost constraint. Under this mild condition, it is shown that we can reach the {\em continuous}-input fundamental theorem of the same form as that for the fundamental theorem with {\em finite} input alphabet. Finally, as important examples of this continuous-input fundamental theorem, we show that the identification capacity as well as the resolvability coincides with the channel capacity for stationary additive white (and also non-white) Gaussian noise channels.
Channel resolvability concerns the minimum resolution for approximating the channel output. We study the resolvability of classical-quantum channels in two settings, for the channel output generated from the worst input, and from the fixed independent and identically distributed (i.i.d.) input. The direct part of the worst-input setting is derived from sequential hypothesis testing as it involves non-i.i.d. inputs. The strong converse of the worst-input setting is obtained via the connection to identification codes. For the fixed-input setting, while the direct part follows from the known quantum soft covering result, we exploit the recent alternative quantum Sanov theorem to prove the strong converse.