An invitation to quantum tomography

We describe quantum tomography as an inverse statistical problem. We show results of consistency in different norms, for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for the density matrix of the quantum state and its Wigner function. The density matrix and the Wigner function are two different ways of representing quantum state. Results are derived from concentration inequalities and entropy methods. Finally we illustrate via simulated data the performance of the above mentioned estimators. An EM algorithm is proposed for practical implementations. There remain many open problems, ex. rates of convergence, adaptation, studying other estimators, etc, and a main purpose of the paper is to bring these to the attention of the statistical community.