**5**, 1

Research Article

## Introduction to logical entropy and its relationship to Shannon entropy

Faculty of Social Sciences, University of Ljubljana, Ljubljana 1000, Slovenia

^{*} Corresponding author: david@ellerman.org

Received:
23
August
2021

Accepted:
5
October
2021

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.

Key words: Logical entropy / Shannon entropy / Partitions / MaxEntropy / Quantum logical entropy / Von Neumann entropy

*© D. Ellerman et al., Published by EDP Sciences, 2022*

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.