This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators,where we may use pattern functions or wavelets, and penalized maximumlikelihood estimators. In all these cases, we get oracle inequalities. In theformer we also have a polynomial rate of convergence for the non-parametricproblem. We finish the paper with applications of similar ideas to thecalibration of a photocounter, a measurement apparatus counting the number ofphotons in a beam. Here the mathematical problem reduces similarly to anon-parametric missingdata problem. We again get oracle inequalities, and better speed if thephotocounter is good.
