take the $\mathbb{L}_2$ distance on the space of density matrices $\left\|\tau-\rho\right\|^2_2 {\mathrel{\mathop:} = }{\text{tr}}(|\tau-\rho|^2)
= \sum_{j,k=0}^\infty |\tau_{j,k}-\rho_{j,k}|^2$. We consider the mean integrated square error (MISE) and split it into a troncature bias term $b_1^2(n)$, a regularization bias terms $b_2^2(n)$ and a variance term $\sigma^2(n)$. $$\begin{aligned}
    E {{\left(  \sum_{j,k=0}^\infty \left|\hat{\rho}^\eta_{j,k}-
          \rho_{j,k}\right|^2  \right)}} &=& \sum_{j+k \geq N}
    \left|\rho_{j,k}\right|^2+\sum_{j+k=0}^{N-1}
    \left|E[\hat{\rho}^\eta_{j,k}]-
        \rho_{j,k}\right|^2 \\
    &&+ \sum_{j+k=0}^{N-1} E\left|\hat{\rho}^\eta_{j,k} -
        E[\hat{\rho}^\eta_{j,k}]\right|^2 \\
    &{= \mathrel{\mathop:} }& b_1^2(n) + b_2^2(n) + \sigma^2(n).\end{aligned}$$ The following propositions give upper bounds for $b_1^2(n)$, $b_2^2(n)$ and $\sigma^2(n)$ in the different cases $\eta = 1$, $1/2 <
\eta < 1$ or $0 < \eta \leq 1/2$ and $r=2$ or $0<r<2$. Their proofs are defered to Appendix \[sec.proofs\].

\[prop:3\] Let $\hat{\rho}^\eta_{j,k}$ be the estimator defined by (\[estrhojk\]), for $0<\eta<1$, with $\delta\rightarrow 0$ and $N\rightarrow\infty$ as $n\rightarrow \infty$, then for all $B>0$ and $0 < r \leq 2$, $$\label{biais}
        \sup_{\rho\in\mathcal{R}(B,r)} b_1^2(n) \leq
        c_1 N^{2-r/2} e^{-2 B N^{r/2}}$$ where $c_1$ is a positive constant depending on $B$ and $r$.

\[prop:4\] Let $\hat{\rho}^\eta_{j,k}$ be the estimator defined by (\[estrhojk\]), for $0 < \eta \leq 1/2$, with $N \rightarrow
    \infty$ as $n \to \infty$ and $1 / \delta \geq 2\sqrt{N}$. In the case $r = 2$, for $\beta := 