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| Author(s): |
Gabriel dos Reis Trindade
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | São Carlos. |
| Institution: | Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) |
| Defense date: | 2023-09-01 |
| Examining board members: |
Carlos Henrique Grossi Ferreira;
Martha Patrícia Dussan Angulo;
Carlile Campos Lavor;
Ali Tahzibi
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| Advisor: | Carlos Henrique Grossi Ferreira |
| Abstract | |
This paper is a presentation on information geometry, organized as a compilation of fundamental concepts and results of the area, as well as applications in quantum information theory. Nevertheless, it is possible to present the information geometry by regarding it as the area that uses tools from differential geometry, especially from Riemannian geometry, to solve problems arising from statistics. Possessing an interdisciplinary character that permeates its historical development, it interprets statistical models as differentiable manifolds, providing them with a Riemannian metric and a 3-covariant tensor field, called, respectively, Fisher metric and Amari-Chentsov tensor. In the context of finite information geometry, they are the only covariant tensor fields of rank 3 invariant by Markov morphisms induced by congruent Markov kernels. Furthermore, among the families of probability distributions, the one formed by the Gaussian distributions has Fisher metric with a known structure and, consequently, a familiar geometry, which is the two-dimensional hyperbolic geometry. In addition, statistical models can be equipped with a uniparametric family of connections, called α-connections, among which, especially in the finite context, the mixed connection and the exponential connection stand out. Moreover, starting with a manifold equipped with a Riemannian metric g and a 3-symmetric tensor T, it is possible to induce a pair of torsion-free linear connections on it and that are related by a weakening of the notion of compatibility with the metric; that are called dual connections to each other with respect to g. They, in turn, together with the Riemannian metric, induce a 3-symmetric tensor on the manifold. Then, through the study of these connections, the geometry that emerges from the combination of a Riemannian metric g with two flat connections ∇ and ∇* dual to each other with respect to g is equivalent, at least locally, to a single convex function, where this convexity is considered with respect to an affine coordinate system for one of the dual connections. Furthermore, being able to be applied to quantum information theory in order to obtain geometric quantum speed limits, information geometry leads to generalizations of the uncertainty principle for energy and time in quantum systems, where the quantum analog of the metric of classical Fisher produces such limits. (AU) | |
| FAPESP's process: | 21/12253-7 - Information geometry and applications |
| Grantee: | Gabriel dos Reis Trindade |
| Support Opportunities: | Scholarships in Brazil - Master |