us to translate the classes $\mathcal{R}(B,r)$ in terms of Wigner functions.

In Section \[sec.dens.mat\], we give estimators of the density matrix $\rho$. The legend was somehow forged that no estimation of the matrix is possible when $0 < \eta \leq 1/2$. The physicists argue that their machines actually have high detection efficiency, around 0.8; it is nevertheless satisfying to be able to solve this problem in any noise condition. We give here the so-called *pattern functions* to use for estimating the density matrix in the noisy case with *any* value of $\eta$ between 0 and 1. These pattern functions allow us to solve an inverse problem which becomes (severly) ill-posed when $0 < \eta \leq 1/2$. In this case, we regularize the inverse problem and this introduces a smoothing parameter which we will choose in an optimal way. We compute the upper bounds for the rates achieved by our methods, with $\mathbb{L}_2$ risk measure.

In Section \[sec.Wigner\], we study a kernel estimator of the Wigner function in $\mathbb{L}_2$ risk, over the same class of Wigner functions. It is a truncated version of the estimator in and tuned accordingly. We compute upper bounds for the rates of convergence of this estimator 