}{C \nu }
        z^{2-\nu} e^{-C z^{\nu}}.$$

First notice that $$\sum_{m+n \geq z} e^{-C (m+n)^{\nu}} = \sum_{t \geq z} (t+1)
        e^{- C t^{\nu}} \leq \int_{z}^\infty (t+1) e^{-C t^{\nu}} dt.
        \nonumber$$ When $t \geq z$ and $z$ is large enough, we have $$\begin{aligned}
        \int_{z}^\infty (t+1) e^{-C t^{\nu}} dt &\leq& \frac{2}{C\nu}
        \int_{z}^{\infty} {{\left(  C \nu t - (2-\nu) t^{1-\nu}  \right)}} e^{-C
          t^{\nu}} dt \\
        &\leq &\frac{2}{C\nu} z^{2-\nu} e^{-Cz^\nu}
    \end{aligned}$$ which is what we needed to prove.

Density matrix estimation {#sec.dens.mat}
=========================

The aim of this part is to estimate the density matrix $\rho$ in the Fock basis directely from the data $(Y_i,\Phi_i)_{i=1,\ldots,n}$. We show that for $0 < \eta \leq 1/2$ it is still possible to estimate the density matrix with an error of estimation tending to $0$ as $n$ tends to infinity (Theorem \[theo:bcpdebruit\]). In both cases ($\eta > \frac12$ and $\eta \leq \frac12$), we construct an estimator of the density matrix $(\rho_{j,k})_{j,k\leq N-1}$ from a sample of QHT data. We give theoretical results for our estimator when the quantum state $\rho$ is in the class of density matrix with decreasing elements defined in (\[eq.classcoeff\]).

Pattern functions
-----------------

The matrix elements $\rho_{j,k}$ of the state $\rho$ in the Fock basis (\[eq:2\]) can 