An invitation to quantum tomography

We describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems. The state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its Wigner function. We illustrate the performance of the estimators on simulated data. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation and studying other estimators; a main purpose of the paper is to bring these to the attention of the statistical community