New foundations of information theory : logical entropy and Shannon entropy / David Ellerman.

Format:

Book

Author/Creator:

Ellerman, David P., author.

Series:

SpringerBriefs in philosophy.

SpringerBriefs in philosophy

Language:

English

Subjects (All):

Entropy (Information theory).

Physical Description:

1 online resource (121 pages)

Edition:

1st ed.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

This monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications.

Contents:

Intro

Acknowledgements

About this book

Contents

About the Author

1 Logical Entropy

1.1 Introduction

1.2 Logical Information as the Measure of Distinctions

1.3 Classical Logical Probability and Logical Entropy

1.4 Information Algebras and Joint Distributions

1.5 Brief History of the Logical Entropy Formula

References

2 The Relationship Between Logical Entropy and Shannon Entropy

2.1 Shannon Entropy

2.2 Logical Entropy, Not Shannon Entropy, Is a (Non-negative) Measure

2.3 The Dit-Bit Transform

3 The Compound Notions for Logical and Shannon Entropies

3.1 Conditional Logical Entropy

3.2 Shannon Conditional Entropy

3.3 Logical Mutual Information

3.4 Shannon Mutual Information

3.5 Independent Joint Distributions

3.6 Cross-Entropies, Divergences, and Hamming Distance

3.6.1 Cross-Entropies

3.6.2 Divergences

3.6.3 Hamming Distance

3.7 Summary of Formulas and Dit-Bit Transforms

4 Further Developments of Logical Entropy

4.1 Entropies for Multivariate Joint Distributions

4.2 An Example of Negative Mutual Information for Shannon Entropy

4.3 Entropies for Countable Probability Distributions

4.4 Metrical Logical Entropy = (Twice) Variance

4.5 Boltzmann and Shannon Entropies: A Conceptual Connection?

4.6 MaxEntropies for Discrete Distributions

4.7 The Transition to Coding Theory

4.8 Logical Entropy on Rooted Trees

5 Quantum Logical Information Theory

5.1 Density Matrix Treatment of Logical Entropy

5.2 Linearizing Logical Entropy to Quantum Logical Entropy

5.3 Theorems About Quantum Logical Entropy

5.4 Quantum Logical Entropies with Density Matrices as Observables

5.5 The Logical Hamming Distance Between Two Partitions

5.6 The Quantum Logical Hamming Distance

6 Conclusion.

6.1 Information Theory Re-founded and Re-envisioned

6.2 Quantum Information Theory Re-envisioned

6.3 What Is to Be Done?

A Basics of Partition Logic

A.1 Subset Logic and Partition Logic

A.2 The Lattice Operations on Partitions

A.3 Implication and Negation in Partition Logic

A.4 Relative Negation in Partition Logic and the Boolean Core

A.5 Partition Tautologies

Index.

Notes:

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

3-030-86552-5

OCLC:

1281766617