Smooth Renyi Entropies and the Quantum Information Spectrum

TL;DR

This paper explores the relationship between the information spectrum method and smooth entropy framework in quantum information theory, showing that spectral entropy rates are the asymptotic limits of smooth entropies, unifying classical and quantum cases.

Contribution

It establishes a formal connection between spectral entropy rates and smooth entropies, extending the relationship to quantum information theory.

Findings

Spectral entropy rate is the asymptotic limit of smooth entropy.

Results apply to both quantum and classical information theory.

Unifies two independent techniques in information spectrum and smooth entropy.

Abstract

Many of the traditional results in information theory, such as the channel coding theorem or the source coding theorem, are restricted to scenarios where the underlying resources are independent and identically distributed (i.i.d.) over a large number of uses. To overcome this limitation, two different techniques, the information spectrum method and the smooth entropy framework, have been developed independently. They are based on new entropy measures, called spectral entropy rates and smooth entropies, respectively, that generalize Shannon entropy (in the classical case) and von Neumann entropy (in the more general quantum case). Here, we show that the two techniques are closely related. More precisely, the spectral entropy rate can be seen as the asymptotic limit of the smooth entropy. Our results apply to the quantum setting and thus include the classical setting as a special case.