# Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

## TL;DR

This paper explores the intersection of geometric flows, entanglement entropy, and quantum information theory within relativistic mechanical systems, developing covariant methods on curved phase spaces.

## Contribution

It introduces a covariant framework linking geometric flows with quantum entanglement entropy, integrating thermodynamic and information-theoretic concepts in relativistic phase spaces.

## Key findings

- Formulated entanglement entropy for quantum geometric flows.
- Derived properties and inequalities for quantum and thermodynamic entropies.
- Connected geometric flow entropy with quantum information measures.

## Abstract

This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigory Perelman's entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy, mutual information etc. We formulate the concept of the entanglement entropy of quantum geometric information flows and study properties and inequalities for quantum, thermodynamic and geometric entropies characterising such systems.