Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras

TL;DR

This paper develops a unified geometric framework for classical and quantum parametric estimation theory on finite-dimensional C*-algebras, deriving fundamental bounds like Cramer-Rao and Helstrom bounds.

Contribution

It introduces a geometric formulation that unifies classical and quantum estimation theories within finite-dimensional C*-algebras, enabling derivation of key bounds.

Findings

Unified geometric framework for classical and quantum estimation

Derivation of Cramer-Rao and Helstrom bounds in finite-dimensional setting

Applicable to models with discrete, finite outcome spaces

Abstract

A geometrical formulation of estimation theory for finite-dimensional $C^{\star}$-algebras is presented. This formulation allows to deal with the classical and quantum case in a single, unifying mathematical framework. The derivation of the Cramer-Rao and Helstrom bounds for parametric statistical models with discrete and finite outcome spaces is presented.