  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.
