Model selection for quantum homodyne tomography

Jonas Kahn (1)

1 UP11 - Université Paris-Sud - Paris 11

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 maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.

Mots clés: Mathematics

Jonas Kahn. Model selection for quantum homodyne tomography. ESAIM: Probability and Statistics, 2009, 13, pp.363-399. ⟨10.1051/ps:2008017⟩. ⟨hal-00520008⟩