Uncertainty and entropies of classical channels

TL;DR

This thesis develops a mathematical framework to quantify uncertainty in classical channels, extending classical entropy concepts to channels through majorization-based approaches and establishing a solid theoretical foundation.

Contribution

It introduces three distinct approaches to formalize uncertainty in classical channels, unifies them through a common preordering, and extends classical entropy measures to channels.

Findings

Classical channel uncertainty can be characterized by a common preordering.

Classical entropies are extended to channels as additive monotones.

The framework provides a solid foundation for quantifying uncertainty in classical communication.

Abstract

In this thesis, I studied a mathematical development to define and quantify the uncertainty inherent in classical channels. This thesis starts with the introduction and background on how to formally think about uncertainty in the domain of classical states. The concept of probability vector majorization and its variants, relative majorization and conditional majorization, are reviewed. This thesis introduces three conceptually distinct approaches to formalize the notion of uncertainty inherent in classical channels. These three approaches define the same preordering on the domain of classical channels, leading to characterizations from many perspectives. With the solid foundation of uncertainty comparison, classical channel entropy is then defined to be an additive monotone with respect to the majorization relation. The well-known entropies in the domain of classical states are uniquely extended to the domain of channels via the optimal extensions, providing not only a solid foundation but also the quantifiers of uncertainty inherent in classical channels.