Hostname: page-component-6898cfcb55-cbk24 Total loading time: 0 Render date: 2026-11-11T11:48:15.390Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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 maximumlikelihood estimators. In all these cases, we get oracle inequalities. In theformer we also have a polynomial rate of convergence for the non-parametricproblem. We finish the paper with applications of similar ideas to thecalibration of a photocounter, a measurement apparatus counting the number ofphotons in a beam. Here the mathematical problem reduces similarly to anon-parametric missingdata problem. We again get oracle inequalities, and better speed if thephotocounter is good.

Keywords

Density matrixmodel selectionpattern functions estimatorpenalized maximum likelihood estimatorpenalized projection estimatorsquantum calibrationquantum tomographywavelet estimatorWigner function.

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009, pp. 363 - 399
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Artiles, L.M., Gill, R. and Guţă, M., An invitation to quantum tomography. J. Royal Statist. Soc. B 67 (2005) 109–134. CrossRef
Banaszek, K., D'Ariano, G.M., Paris, M.G.A. and Sacchi, M.F., Maximum-likelihood estimation of the density matrix. Phys. Rev. A 61 (1999) R010304. CrossRef
Barndorff-Nielsen, O.E., Gill, R. and Jupp, P.E., On quantum statistical inference (with discussion). J. R. Statist. Soc. B 65 (2003) 775–816. CrossRef
S.N. Bernstein, On a modification of Chebyshev's inequality and of the error formula of Laplace. In Collected works, Vol. 4 (1964).
Butucea, C., Guţă, M. and Artiles, L., Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist. 35 (2007) 465–494. CrossRef
Cavalier, L. and Koo, J.-Y., Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inf. Theory 48 (2002) 2794–2802. CrossRef
D'Ariano, G.M., Macchiavello, C. and Paris, M.G.A., Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (1994) 4298–4302. CrossRef
D'Ariano, G.M., Leonhardt, U. and Paul, H., Homodyne detection of the density matrix of the radiation field. Phys. Rev. A 52 (1995) R1801–R1804. CrossRef
D'Ariano, G.M., Maccone, L., Presti, P. Lo, Quantum calibration of measuring apparatuses. Phys. Rev. Lett. 93 (2004) 250407. CrossRef
S.R. Deans, The Radon transform and some of its applications. John Wiley & Sons, New York (1983).
A. Erdélyi, Higher Transcendental Functions, Vol. 2. McGraw-Hill (1953).
R. Gill, Quantum Asymptotics, volume 36 of IMS Lect. Notes-Monograph Ser. (2001) 255–285.
C.W. Helstrom, Quantum Detection and Estimation Theory. Academic Press, New York (1976).
Hoeffding, W., Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58 (1964) 13–30. CrossRef
A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982).
J. Kahn, Sélection de modèles en tomographie quantique. Master's thesis, École Normale Supérieure, Université Paris-Sud, 2004.
U. Leonhardt, Measuring the Quantum State of Light. Cambridge University Press (1997).
Leonhardt, U., Paul, H. and D'Ariano, G.M., Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A 52 (1995) 4899–4907. CrossRef
P. Massart, Concentration Inequalities and Model Selection. École d'été de Probabilité de Saint-Flour, 2003. Lect. Notes Math. Springer-Verlag, Berlin (2006).
Smithey, D.T., Beck, M., Raymer, M.G. and Faridani, A., Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70 (1993) 1244–1247. CrossRef
Vogel, K. and Risken, H., Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40 (1989) 2847–2849. CrossRef