  We address the use of asymptotic incompatibility (AI) to assess the
quantumness of a multiparameter quantum statistical model. AI is a recently
introduced measure which quantifies the difference between the Holevo and the
SLD scalar bounds, and can be evaluated using only the symmetric logarithmic
derivative (SLD) operators of the model. At first, we evaluate analytically the
AI of the most general quantum statistical models involving two-level (qubit)
and single-mode Gaussian continuous-variable quantum systems, and prove that AI
is a simple monotonous function of the state purity. Then, we numerically
investigate the same problem for qudits ($d$-dimensional quantum systems, with
$2 < d \leq 4$), showing that, while in general AI is not in general a function
of purity, we have enough numerical evidence to conclude that the maximum
amount of AI is attainable only for quantum statistical models characterized by
a purity larger than $\mu_{\sf min} = 1/(d-1)$. In addition, by parametrizing
qudit states as thermal (Gibbs) states, numerical results suggest that, once
the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one
correspondence with the fictitious temperature parameter $\beta$ characterizing
the family of density operators. Finally, by studying in detail the definition
and properties of the AI measure we find that: i) given a quantum statistical
model, one can readily identify the maximum number of asymptotically
compatibile parameters; ii) the AI of a quantum statistical model bounds from
above the AI of any sub-model that can be defined by fixing one or more of the
original unknown parameters (or functions thereof), leading to possibly useful
bounds on the AI of models involving noisy quantum dynamics.
