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IEICE Technical Report; IEICE Tech. Rep.

Variations of the Strong Converse Theorem on the Intrinsic Randomness Problem for General Sources

Shota Saito, Matsushima Toshiyasu

Published Jul 6, 2015• 0 citations
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The purpose of this paper is to establish a new unified method for random number generation from general sources. Specifically, we introduce an alternative definition of the smooth Rényi entropy of order infinity, and show a unified approach to represent the intrinsic randomness in terms of this information quantity. Our definition of the smooth Rényi entropy is easy to calculate for finite block lengths. We also represent δ-intrinsic randomness and the strong converse property in terms of the smooth Rényi entropy.

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Compression of sources of probability distributions and density operators

Andreas Winter

We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.

arXiv:quant-ph/0208131•Aug 21, 2002•62 citations
The purpose of this paper is to establish a new unified method for random number generation from general sources. Specifically, we introduce an alternative definition of the smooth Rényi entropy of order infinity, and show a unified approach to represent the intrinsic randomness in terms of this information quantity. Our definition of the smooth Rényi entropy is easy to calculate for finite block lengths. We also represent δ-intrinsic randomness and the strong converse property in terms of the smooth Rényi entropy.
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.
We study the problem of efficient compression of a stochastic source of probability distributions. It can be viewed as a generalization of Shannon's source coding problem. It has relation to the theory of common randomness, as well as to channel coding and rate--distortion theory: in the first two subjects ``inverses'' to established coding theorems can be derived, yielding a new approach to proving converse theorems, in the third we find a new proof of Shannon's rate--distortion theorem. After reviewing the known lower bound for the optimal compression rate, we present a number of approaches to achieve it by code constructions. Our main results are: a better understanding of the known lower bounds on the compression rate by means of a strong version of this statement, a review of a construction achieving the lower bound by using common randomness which we complement by showing the optimal use of the latter within a class of protocols. Then we review another approach, not dependent on common randomness, to minimizing the compression rate, providing some insight into its combinatorial structure, and suggesting an algorithm to optimize it. The second part of the paper is concerned with the generalization of the problem to quantum information theory: the compression of mixed quantum states. Here, after reviewing the known lower bound we contribute a strong version of it, and discuss the relation of the problem to other issues in quantum information theory.