On the philosophy of Cram\'er-Rao-Bhattacharya Inequalities in Quantum Statistics

TL;DR

This paper develops a theoretical framework for quantum Fisher information and Cramér-Rao bounds, extending classical statistical inequalities to quantum systems and providing a basis for quantum parameter estimation.

Contribution

It introduces a new geometric approach to quantum Fisher information and derives an abstract Cramér-Rao-Bhattacharya lower bound applicable to finite and infinite quantum systems.

Findings

Derived a quantum Cramér-Rao-Bhattacharya inequality.

Introduced Fisher maps and tensors for quantum states.

Applicable to both finite and infinite quantum systems.

Abstract

To any parametric family of states of a finite level quantum system we associate a space of Fisher maps and introduce the natural notions of Cram\'er-Rao-Bhattacharya tensor and Fisher information form. This leads us to an abstract Cram\'er-Rao-Bhattacharya lower bound for the covariance matrix of any finite number of unbiased estimators of parameteric functions. A number of illustrative examples is included. Modulo technical assumptions of various kinds our methods can be applied to infinite level quantum systems as well as parametric families of classical probability distributions on Borel spaces.