# Quantum Information Geometry and its classical aspect

## TL;DR

This thesis investigates quantum information geometry, highlighting differences from classical cases, and demonstrates that classical analogs can simplify calculations for Gaussian states, with applications to entanglement and classical measures.

## Contribution

It introduces classical analogs of quantum geometric concepts, providing new tools for analyzing Gaussian states and classical systems in quantum information geometry.

## Key findings

- Classical analogs of quantum geometric tensor and covariance matrix are effective for Gaussian states.
- Quantum entanglement measures can be derived from classical analogs for certain systems.
- Calculations for systems like harmonic oscillators and qubits validate the approach.

## Abstract

This thesis explores important concepts in the area of quantum information geometry and their relationships. We highlight the unique characteristics of these concepts that arise from their quantum mechanical foundations and emphasize the differences from their classical counterparts. However, we also demonstrate that for Gaussian states, classical analogs can be used to obtain the same mathematical results, providing a valuable tool for simplifying calculations. To establish the groundwork for the subsequent analysis, we introduce some fundamental ideas from quantum field theory. We then explore the structure of parameter space using the fidelity and the Quantum Geometric Tensor (QGT), which is composed of the Quantum Metric Tensor and the Berry curvature. We also introduce the Quantum Covariance Matrix (QCM) and show its relationship to the QGT. We present how the QCM can be used to study entanglement between quantum systems by obtaining the purity, linear entropy, and von Neumann entropy. To illustrate these concepts, we calculate all these quantities for several systems, including the Stern-Gerlach, a two qubits system, two symmetrically coupled harmonic oscillators, and N coupled harmonic oscillators. In the final section of this thesis, we examine how the aforementioned quantum concepts can be applied in a classical sense, following the approach taken by Hannay with the Berry phase. We examine classical analogs of the QGT and QCM and since for Gaussian states, all the necessary information to calculate purity, linear entropy, and von Neumann entropy is contained within the QCM, we also generate classical analogs for them. These results in turn can be used to derive measures of separability for classical systems.