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New foundations for information theo...

Ellerman, David.

New foundations for information theorylogical entropy and Shannon entropy /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	New foundations for information theoryby David Ellerman.
其他題名:	logical entropy and Shannon entropy /
作者:	Ellerman, David.
出版者:	Cham :Springer International Publishing :2021.
面頁冊數:	xiii, 113 p. :ill., digital ;24 cm.
Contained By:	Springer Nature eBook
標題:	Entropy (Information theory)
電子資源:	https://doi.org/10.1007/978-3-030-86552-8
ISBN:	9783030865528$q(electronic bk.)

New foundations for information theorylogical entropy and Shannon entropy /
Ellerman, David.

New foundations for information theorylogical entropy and Shannon entropy /[electronic resource] :by David Ellerman. - Cham :Springer International Publishing :2021. - xiii, 113 p. :ill., digital ;24 cm. - SpringerBriefs in philosophy,2211-4556. - SpringerBriefs in philosophy..

- Logical entropy -- The relationship between logical entropy and Shannon entropy -- The compound notions for logical and Shannon entropies -- Further developments of logical entropy -- Logical Quantum Information Theory -- Conclusion -- Appendix: Introduction to the logic of partitions.

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. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or "dit" of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory) and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits--so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general--and to Hilbert spaces in particular--for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.

ISBN: 9783030865528$q(electronic bk.)

Standard No.: 10.1007/978-3-030-86552-8doiSubjects--Topical Terms:

199046
Entropy (Information theory)

LC Class. No.: Q370 / .E45 2021

Dewey Class. No.: 003.54

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520 $a 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. Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or "dit" of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory) and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits--so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general--and to Hilbert spaces in particular--for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement. Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.
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