in $\mathbb{L}_2$ risk.

To conclude, we may infer that the performances of both estimators are comparable. We obtain nearly polynomial rates for the case $r = 2$ and intermediate rates for $0 < r < 2$ (faster than any logarithm, but slower than any polynomial). It is convenient to have methods to estimate directly both representations of a quantum state. The estimator of the matrix $\rho$ can be more easily projected on the space of proper quantum states. On the other hand, we may capture some features of the quantum states more easily on the Wigner function, for instance when this function has significant negative parts, the fact that the quantum state is non classical.

Decrease and smoothness of the Wigner distribution {#sec.dec.reg}
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We recall that the Wigner distribution $W_{\rho}$ was defined in the introduction. In the Fock basis, we can write $W_{\rho}$ in terms of the density matrix $[\rho_{m,n}]$ as follows (see Leonhardt [@Leonhardt] for the details). $$W_{\rho}(q,p) = \sum_{m,n} \rho_{m,n} W_{m,n}(q,p) \nonumber$$ where $$\label{eq:1}
    W_{m,n}(q,p) = \frac1\pi \int e^{2ipx} h_m(q-x) h_n(q+x) dx.$$ It can be seen that $W_{m,n}(q,p) = W_{n,m}(q,-p)$ and if $m \geq n$, $$\begin{aligned}
   W_{m,n}(q,p)&=&\frac{(-1)^m}{\pi} {{\left( \frac{n!}{m!} \right)}}^\frac12
    e^{-{{\left( q^2+p^2 \right)}}} \nonumber\\
    && \times {{\left( 