相关论文: F-manifolds and geometry of information
In this article, we describe various aspects of categorification of the structures appearing in information theory. These aspects include probabilistic models both of classical and quantum physics, emergence of F-manifolds, and motivic…
Technology of data collection and information transmission is based on various mathematical models of encoding. The words "Geometry of information" refer to such models, whereas the words "Moufang patterns" refer to various sophisticated…
This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf…
F-manifolds are complex manifolds with a multiplication with unit on the holomorphic tangent bundle with a certain integrability condition. Here the local classification of 3-dimensional F-manifolds with or without Euler fields is pursued.
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
Motivated by the engineering applications of uncertainty quantification, in this work we draw connections between the notions of random quantum states and operations in quantum information with probability distributions commonly encountered…
Manifolds with a commutative and associative multiplication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability condition. They are studied here. They are closely…
New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We…
I review recent works showing that information geometry is a useful framework to characterize quantum coherence and entanglement. Quantum systems exhibit peculiar properties which cannot be justified by classical physics, e.g. quantum…
Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…
The methods of Information geometry have been glowing up to develop various subjects of theoretical physics, including quantum information systems. The present article has two purposes. The first one is to develop general theory of…
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose,…
A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along…
A linear F-manifold is an F-manifold (E, \circ , e) defined on the total space of a vector bundle \pi : E \rightarrow M for which the multiplication and unit field are linear tensor fields. We develop a systematic treatment of linear…
This introductory text arises from a lecture given in G\"oteborg, Sweden, given by the first author and is intended for undergraduate students, as well as for any mathematically inclined reader wishing to explore a synthesis of ideas…
This paper investigates the geometric and algebraic interplay between F-manifolds and a newly defined class of structures termed F$_\text{man}$-algebras. We specialize our study to the category of F-Lie groups, characterized by a Lie group…
Information geometry is a study of statistical manifolds, that is, spaces of probability distributions from a geometric perspective. Its classical information-theoretic applications relate to statistical concepts such as Fisher information,…
In this paper and a companion paper, we show how the framework of information geometry, a geometry of discrete probability distributions, can form the basis of a derivation of the quantum formalism. The derivation rests upon a few…
In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent…
In this chapter, we study Information Geometry from a particular non-parametric or functional point of view. The basic model is a probabilities subset usually specified by regularity conditions. For example, probability measures mutually…