\rangle$ and $\| \cdot
\|$ the usual Euclidian scalar product and norm, while $C(\cdot)$ will denote positive constants depending on parameters given in the parentheses.

We suppose that the unknown state belongs to the class $\mathcal{R}(B,r)$ for $B>0$ and $0 < r \leq 2$ defined by $$\label{eq.classcoeff}
    \mathcal{R}(B,r) {\mathrel{\mathop:} = }\{\rho {\rm \ quantum \ state} :
    |\rho_{m,n} |\leq \exp(-B (m+n)^{r/2})\}.$$

For simplicity, we have chosen to express the results relative to a class which is the intersection of the (positive) ball of radius 1 in some Banach space with the hyperplane ${\text{tr}}(\rho) = 1$. Another radius for the class would only change the constant $C$ in front of the asymptotic rates of convergence that we will find.

As it will be made precise in Propositions \[prop:popetown\] and \[prop:popetown2\], quantum states in the class given in (\[eq.classcoeff\]) have fast decreasing and very smooth Wigner functions. From the physical point of view, the choice of such a class of Wigner functions seems to be quite reasonable considering that typical states $\rho$ prepared in the laboratory do satisfy this type of condition.

Statistical model
-----------------

Let us describe the statistical model. Consider $(X_{1}, \Phi_{1}),
\dots ,(X_{n}, \Phi_{n})$ independent identically distributed random variables 