Introduction to Logical Entropy and its Relationship to Shannon Entropy

TL;DR

This paper introduces logical entropy as a foundational measure of information based on distinctions, relates it to Shannon entropy through a transformation, and extends it to quantum systems, providing a new perspective on information theory.

Contribution

It formally defines logical entropy, establishes its relationship with Shannon entropy via a dit-bit transform, and extends the concept to quantum logical entropy.

Findings

Logical entropy measures distinctions in a set.

Shannon entropy can be derived from logical entropy through a nonlinear transform.

Quantum logical entropy relates to measurement probabilities in quantum states.

Abstract

We live in the information age. Claude Shannon, as the father of the information age, gave us a theory of communications that quantified an "amount of information," but, as he pointed out, "no concept of information itself was defined." Logical entropy provides that definition. Logical entropy is the natural measure of the notion of information based on distinctions, differences, distinguishability, and diversity. It is the (normalized) quantitative measure of the distinctions of a partition on a set--just as the Boole-Laplace logical probability is the normalized quantitative measure of the elements of a subset of a set. And partitions and subsets are mathematically dual concepts--so the logic of partitions is dual in that sense to the usual Boolean logic of subsets, and hence the name "logical entropy." The logical entropy of a partition has a simple interpretation as the probability that a distinction or dit (elements in different blocks) is obtained in two independent draws from the underlying set. The Shannon entropy is shown to also be based on this notion of information-as-distinctions; it is the average minimum number of binary partitions (bits) that need to be joined to make all the same distinctions of the given partition. Hence all the concepts of simple, joint, conditional, and mutual logical entropy can be transformed into the corresponding concepts of Shannon entropy by a uniform non-linear dit-bit transform. And finally logical entropy linearizes naturally to the corresponding quantum concept. The quantum logical entropy of an observable applied to a state is the probability that two different eigenvalues are obtained in two independent projective measurements of that observable on that state. Keywords: logical entropy, Shannon entropy, partitions, MaxEntropy, quantum logical entropy, von Neumann entropy