# Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems

## TL;DR

This paper introduces a geometric flow framework for classical and quantum mechanical systems using Perelman entropy, exploring relativistic thermodynamics, nonholonomic deformations, and quantum information measures within a unified geometric approach.

## Contribution

It develops a novel geometric flow approach incorporating Perelman entropy for relativistic classical and quantum systems, including nonholonomic deformations and quantum information measures.

## Key findings

- Formulation of relativistic thermodynamic models using geometric flows.
- Development of nonholonomic W-entropy as a complement to Shannon and von Neumann entropy.
- Extension of geometric flow concepts to quantum relative entropy and quantum channels.

## Abstract

This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolution equations of classical mechanical systems are described. There are studied projections of such F- and W-functionals on Lorentz spacetime manifolds and three-dimensional spacelike hypersurfaces. These functionals are used for elaborating relativistic thermodynamic models for Lagrange--Hamilton geometric evolution and respective generalized R. Hamilton geometric flow and nonholonomic Ricci flow equations. The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannon entropy and the quantum von Neumann entropy. There are considered geometric flow generalizations of the approaches based on classical and quantum relative entropy, conditional entropy, mutual information, and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow information theory are elaborated using the formalism of density matrices and measurements with quantum channels for the evolution of quantum mechanical systems.