arXiv:math/0405595v2 [math.ST] 1 Jun 2004 An invitation to quantum tomography L.M. Artiles Eurandom, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, artiles@eurandom.tue.nl, http://euridice.tue.nl/∼lartiles/ R. Gill Mathematical Institute, University of Utrecht, Box 80010, 3508 TA Utrecht, The Netherlands, gill@math.uu.nl, http://www.math.uu.nl/people/gill M.I. Gut ¸˘ a Eurandom, P.O. Box 513, 5600 MB Eindhoven, The Netherlands, guta@eurandom.tue.nl, http://euridice.tue.nl/∼mguta/ Summary. The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We de- scribe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. 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. 1. Introduction It took more than eighty years from its discovery till it was possible to experimentally determine and visualize the most fundamental object in quantum mechanics, the wave function. The forward route from quantum state to probability distribution of measurement results has been the basic stuff of quantum mechanics textbooks for decennia. That the corresponding mathematical inverse problem had a solution, provided (speaking metaphorically) that the quantum state has been probed from a sufficiently rich set of directions, had also been known for many years. However it was only with Smithey et al. (1993), that it became feasible to actually carry out the corresponding measurements on one particular quantum system—in that case, the state of one mode of electromagnetic radiation (a pulse of laser light at a given frequency). Experimentalists have used the technique to establish that they have succeeded in creating non-classical forms of laser light such as squeezed light and Schr¨ odinger cats. The experimental technique we are referring to here is called quantum homodyne tomography: the word homodyne referring to a comparison between the light being measured with a reference light beam at the same frequency. We will explain the word tomography in a moment. The quantum state can be represented mathematically in many different but equivalent ways, all of them linear transformations on one another. One favorite is as the Wigner function W : a real function of two variables, integrating to plus one over the whole plane, but not necessarily nonneg- ative. It can be thought of as a “generalized joint probability density” of the electric and magnetic †Key words and phrases. Quantum tomography, Wigner function, Density matrix, Pattern Functions estima- tion, Sieve Maximum Likelihood estimation, E.M. algorithm.