An invitation to quantum tomography
L.M. Artiles, R. Gill and M.I. Guta
Journal of the Royal Statistical Society B: Methodological
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.