The Homological Nature of Entropy
We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dyna...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2015-05-01
|
| Series: | Entropy |
| Subjects: | |
| Online Access: | http://www.mdpi.com/1099-4300/17/5/3253 |
| _version_ | 1811307512337530880 |
|---|---|
| author | Pierre Baudot Daniel Bennequin |
| author_facet | Pierre Baudot Daniel Bennequin |
| author_sort | Pierre Baudot |
| collection | DOAJ |
| description | We propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system. |
| first_indexed | 2024-04-13T09:05:36Z |
| format | Article |
| id | doaj.art-8caba1e587b3439bb32d35873a4a07c7 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-13T09:05:36Z |
| publishDate | 2015-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-8caba1e587b3439bb32d35873a4a07c72022-12-22T02:53:00ZengMDPI AGEntropy1099-43002015-05-011753253331810.3390/e17053253e17053253The Homological Nature of EntropyPierre Baudot0Daniel Bennequin1Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, GermanyUniversite Paris Diderot-Paris 7, UFR de Mathematiques, Equipe Geometrie et Dynamique, Batiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris Cedex 13, FranceWe propose that entropy is a universal co-homological class in a theory associated to a family of observable quantities and a family of probability distributions. Three cases are presented: (1) classical probabilities and random variables; (2) quantum probabilities and observable operators; (3) dynamic probabilities and observation trees. This gives rise to a new kind of topology for information processes, that accounts for the main information functions: entropy, mutual-informations at all orders, and Kullback–Leibler divergence and generalizes them in several ways. The article is divided into two parts, that can be read independently. In the first part, the introduction, we provide an overview of the results, some open questions, future results and lines of research, and discuss briefly the application to complex data. In the second part we give the complete definitions and proofs of the theorems A, C and E in the introduction, which show why entropy is the first homological invariant of a structure of information in four contexts: static classical or quantum probability, dynamics of classical or quantum strategies of observation of a finite system.http://www.mdpi.com/1099-4300/17/5/3253Shannon informationhomology theoryentropyquantum informationhomotopy of linksmutual informationsKullback–Leiber divergencetreesmonadspartitions |
| spellingShingle | Pierre Baudot Daniel Bennequin The Homological Nature of Entropy Entropy Shannon information homology theory entropy quantum information homotopy of links mutual informations Kullback–Leiber divergence trees monads partitions |
| title | The Homological Nature of Entropy |
| title_full | The Homological Nature of Entropy |
| title_fullStr | The Homological Nature of Entropy |
| title_full_unstemmed | The Homological Nature of Entropy |
| title_short | The Homological Nature of Entropy |
| title_sort | homological nature of entropy |
| topic | Shannon information homology theory entropy quantum information homotopy of links mutual informations Kullback–Leiber divergence trees monads partitions |
| url | http://www.mdpi.com/1099-4300/17/5/3253 |
| work_keys_str_mv | AT pierrebaudot thehomologicalnatureofentropy AT danielbennequin thehomologicalnatureofentropy AT pierrebaudot homologicalnatureofentropy AT danielbennequin homologicalnatureofentropy |