Topological forms of information.
Pierre Baudot and Daniel Bennequin
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Submission date: 14. Nov. 2014
published in: Bayesian inference and maximum entropy methods in science and engineering : (MaxEnt 2014) : Clos Lucé, Amboise, France, 21-26 September 2014 / A. Mohammad-Djafari (ed.)
Melville, NY : AIP Publising, 2015. - P. 213 - 221
(AIP conference proceedings ; 1641)
DOI number (of the published article): 10.1063/1.4905981
MSC-Numbers: 55U99, 55P48, 55N99, 92B05, 94A17, 94A15
PACS-Numbers: 02.40.Re, 03.67.-a, 05.20.-y
Keywords and phrases: Shannon information, Homology Theory, entropy, Quantum Information, Homotopy of Links, Mutual Informations, Trees, Monads
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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. We discuss briefly its application to complex data, in particular to the structures of information flows in biological systems. This short note summarizes results obtained during the last years by the authors. The proofs are not included, but the definitions and theorems are stated with precision.