Classification and characterization of quantum parametric models in quantum estimation theory

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.