An invitation to quantum tomography (II)
Abstract
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 describe 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2004
 arXiv:
 arXiv:math/0405595
 Bibcode:
 2004math......5595A
 Keywords:

 Mathematics  Statistics;
 Quantum Physics;
 62G07;
 62P35
 EPrint:
 An earlier version of this paper with more mathematical background but less applied statistical content can be found on arXiv as quantph/0303020. An electronic version of the paper with high resolution figures (postscript instead of bitmaps) is available from the authors. v2: added crossvalidation results, references