State estimation in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ≤ 1, we propose two estimators of a quantum state whose density matrix elements ρ_m,n decrease as , for fixed known B > 0 and 0 < r ≤ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ≤ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.