with values in $\mathbb{R}\times [0,\pi]$ and distribution $P_{\rho}$ having density $p_{\rho}(x,\phi)$ (given by (\[eq:prhophi\]) with respect to $\frac{1}{\pi}\lambda$, $\lambda$ being the Lebesgue measure on $\mathbb{R}\times [0,\pi]$. The aim is to recover the density matrix $\rho$ and the Wigner function $W_\rho $ from the observations.

However, there is a slight complication. What we observe are not the variables $(X_{\ell}, \Phi_{\ell})$ but the noisy ones $(Y_{\ell},
\Phi_{\ell})$, where $$\label{noisy.data}
    Y_\ell:=\sqrt{\eta}X_\ell+\sqrt{(1-\eta)/2}~\xi_\ell,$$ with $\xi_\ell$ a sequence of independent identically distributed standard Gaussians which are independent of all $(X_j, \Phi_{j})$. The detection efficiency parameter $0 < \eta \leq 1$ is known from the calibration of the apparatus and we denote by $N^\eta$ the centered Gaussian density of variance $(1-\eta)/2$, and $\widetilde N^\eta$ its Fourier transform. Then the density $p_{\rho}^{\eta}$ of $(Y_{\ell},
\Phi_{\ell})$ is given by the convolution of the density $p_\rho(\cdot/\sqrt{\eta}, \phi)/\sqrt{\eta}$ with $N^\eta$ $$\begin{aligned}
    p^\eta_\rho(y,\phi) &=& \int_{-\infty}^\infty
    \frac{1}{\sqrt{\eta}} p_\rho\left(\frac{y-x}{\sqrt{\eta}}, \phi
    \right) N^\eta (x)dx\\
    &~ {= \mathrel{\mathop:} }& \left(\frac{1}{\sqrt\eta}
        p_\rho\left(\frac{\cdot}{\sqrt\eta},\phi\right) \ast N^\eta
    \right)(y).\end{aligned}$$ In the Fourier domain this relation becomes $$\begin{aligned}
    \label{fourierproun}
    \mathcal{F}_1[p^\eta_\rho(\cdot,\phi)](t)
    &=& \mathcal{F}_1[p_\rho(\cdot,\phi)](t \sqrt{\eta})
    \widetilde{N}^\eta(t),\end{aligned}$$ where $\mathcal{F}_{1} $ denotes the Fourier transform with respect to the first variable.

The theoretical foundation of quantum homodyne tomography was outlined in and has inspired the first 