# From Classical to Quantum Information Geometry: A Guide for Physicists

## TL;DR

This paper introduces quantum information geometry and connects its key concepts—Berry curvature, fidelity susceptibility, and quantum Fisher information—to various phenomena in quantum many-body physics, serving as an accessible guide for physicists.

## Contribution

It provides a unified conceptual framework linking different facets of quantum information geometry for researchers in quantum many-body and condensed matter physics.

## Key findings

- Connects topological phases with Berry curvature.
- Highlights fidelity susceptibility as a universal criticality probe.
- Shows quantum Fisher information as a resource quantifier.

## Abstract

Recently, there has been considerable interest in the application of information geometry to quantum many body physics. This interest has been driven by three separate lines of research, which can all be understood as different facets of quantum information geometry. First, the study of topological phases of matter characterized by Chern number is rooted in the symplectic structure of the quantum state space, known in the physics literature as Berry curvature. Second, in the study of quantum phase transitions, the fidelity susceptibility has gained prominence as a universal probe of quantum criticality, even for systems that lack an obviously discernible order parameter. Finally, the study of quantum Fisher information (QFI) in many body systems has seen a surge of interest due to its role as a witness of genuine multipartite entanglement and owing to its utility as a quantifier of quantum resources, in particular those useful in quantum sensing. Rather than a thorough review, our aim is to connect key results within a common conceptual framework that may serve as an introductory guide to the extensive breadth of applications, and deep mathematical roots, of quantum information geometry, with an intended audience of researchers in quantum many body and condensed matter physics.