experiments determining the quantum state of a light field, initially with optical pulses in [@Smithey; @Smitheybis; @Leonhardt]. The reconstruction of the density from averages of data has been discussed or studied in for $\eta=1$ (no photon loss). Max-likelihood methods have been studied in and procedure using adaptive tomographic kernels to minimize the variance has been proposed in [@DP]. The estimation of the density matrix of a quantum state of light in case of efficiency parameter $\frac12 < \eta \leq 1$ has been discussed in [@DAriano.1; @DMS; @DAriano.5] and considered in [@Richter] via the pattern functions for the diagonal elements.

Outline of the results
----------------------

The goal of this chapter is to define estimators of both the density matrix and the Wigner function and to compare their performance in $\mathbb{L}_2$ risk. In order to compute estimation risks and to tune the underlying parameters, we define a realistic class of quantum states $\mathcal{R}(B,r)$, depending on parameters $B>0$ and $ 0<
r\leq 2$, in which the elements of the density matrix decrease rapidly.

In Section \[sec.dec.reg\], we prove that the fast decay of the elements of the density matrix implies both rapid decay of the Wigner function and of its Fourier transform, allowing 