Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics

TL;DR

This paper explores how classical and quantum Fisher information metrics can be understood through the geometrical formulation of quantum mechanics, linking statistical measures to the geometry of quantum state space.

Contribution

It unifies the classical and quantum Fisher information metrics within the geometrical framework of quantum mechanics using the Hermitian tensor.

Findings

Fisher information metrics are represented by the Hermitian tensor on pure state manifolds.

The geometrical approach connects classical probability with quantum state geometry.

Provides a unified geometric interpretation of Fisher information in quantum mechanics.

Abstract

The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.