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Model selection for quantum homodyne tomography

Published online by Cambridge University Press:  22 September 2009

Jonas Kahn
Jonas Kahn*
Affiliation:
Université Paris-Sud 11, Département de Mathématiques Bât. 425, 91405 Orsay Cedex, France; jonas.kahn@math.u-psud.fr
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Abstract

This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.

Keywords

Density matrixmodel selectionpattern functions estimatorpenalized maximum likelihood estimatorpenalized projection estimatorsquantum calibrationquantum tomographywavelet estimatorWigner function.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 363 - 399
Copyright
© EDP Sciences, SMAI, 2009

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