Classification and characterization of quantum parametric models in quantum estimation theory

TL;DR

This paper classifies quantum parametric models based on the Holevo bound, exploring their relationships and geometric structures to deepen understanding of quantum estimation theory.

Contribution

It introduces a detailed classification of quantum models into four classes and analyzes their interrelations and geometric properties.

Findings

Classical models characterized by diagonal elements are intersection of D-invariant and asymptotically classical models.

A gap exists between classical and quasi-classical models where logarithmic derivatives commute.

Each class is characterized by equivalent conditions, revealing geometric insights into quantum statistical models.

Abstract

In this paper, we characterize quantum parametric models into different classes based on the estimation error bound, known as the Holevo bound. These classes are given by the classical, quasi-classical, D-invariant, and asymptotically classical models. We first explore the relationships among these four models and show that: i) The classical model having the diagonal elements only is characterized by the intersection of the D-invariant and asymptotically classical models. ii) There exists a gap between the classical model and the quasi-classical model, where all logarithmic derivative operators commute with each other. Further, we characterize each class with several equivalent conditions. This result then reveals the geometrical understanding of quantum statistical models.