Nonparametric goodness-of fit testing in quantum homodyne tomography with noisy data

Abstract

In the framework of quantum optics, we study the problem of goodness-of-fit testing in a severely ill-posed inverse problem. A novel testing procedure is introduced and its rates of convergence are investigated under various smoothness assumptions. The procedure is derived from a projection-type estimator, where the projection is done in $\mathbb{L}_{2}$ distance on some suitably chosen pattern functions. The proposed methodology is illustrated with simulated data sets.