Model selection for quantum homodyne tomography

Jonas Kahn

ESAIM: Probability and Statistics (2009)

  • Volume: 13, page 363-399
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Kahn, Jonas. "Model selection for quantum homodyne tomography." ESAIM: Probability and Statistics 13 (2009): 363-399. <http://eudml.org/doc/250643>.

@article{Kahn2009,
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. },
author = {Kahn, Jonas},
journal = {ESAIM: Probability and Statistics},
keywords = {Density matrix; model selection; pattern functions estimator; penalized maximum likelihood estimator; penalized projection estimators; quantum calibration; quantum tomography; wavelet estimator; Wigner function.; density matrix; penalized maximum likelihood estimator; quantum calibration; Wigner function},
language = {eng},
month = {9},
pages = {363-399},
publisher = {EDP Sciences},
title = {Model selection for quantum homodyne tomography},
url = {http://eudml.org/doc/250643},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Kahn, Jonas
TI - Model selection for quantum homodyne tomography
JO - ESAIM: Probability and Statistics
DA - 2009/9//
PB - EDP Sciences
VL - 13
SP - 363
EP - 399
AB - 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.
LA - eng
KW - Density matrix; model selection; pattern functions estimator; penalized maximum likelihood estimator; penalized projection estimators; quantum calibration; quantum tomography; wavelet estimator; Wigner function.; density matrix; penalized maximum likelihood estimator; quantum calibration; Wigner function
UR - http://eudml.org/doc/250643
ER -

References

top
  1. L.M. Artiles, R. Gill and M. Guţă, An invitation to quantum tomography. J. Royal Statist. Soc. B67 (2005) 109–134.  Zbl1060.62137
  2. K. Banaszek, G.M. D'Ariano, M.G.A. Paris and M.F. Sacchi, Maximum-likelihood estimation of the density matrix. Phys. Rev. A61 (1999) R010304.  
  3. O.E. Barndorff-Nielsen, R. Gill and P.E. Jupp, On quantum statistical inference (with discussion). J. R. Statist. Soc. B65 (2003) 775–816.  Zbl1059.62128
  4. S.N. Bernstein, On a modification of Chebyshev's inequality and of the error formula of Laplace. In Collected works, Vol. 4 (1964).  
  5. C. Butucea, M. Guţă and L. Artiles, Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist.35 (2007) 465–494.  Zbl1117.62027
  6. L. Cavalier and J.-Y. Koo, Poisson intensity estimation for tomographic data using a wavelet shrinkage approach. IEEE Trans. Inf. Theory48 (2002) 2794–2802.  Zbl1062.92042
  7. G.M. D'Ariano, C. Macchiavello and M.G.A. Paris, Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A50 (1994) 4298–4302.  
  8. G.M. D'Ariano, U. Leonhardt and H. Paul, Homodyne detection of the density matrix of the radiation field. Phys. Rev. A52 (1995) R1801–R1804.  
  9. G.M. D'Ariano, L. Maccone, P. Lo Presti, Quantum calibration of measuring apparatuses. Phys. Rev. Lett.93 (2004) 250407.  
  10. S.R. Deans, The Radon transform and some of its applications. John Wiley & Sons, New York (1983).  Zbl0561.44001
  11. A. Erdélyi, Higher Transcendental Functions, Vol. 2. McGraw-Hill (1953).  Zbl0051.30303
  12. R. Gill, Quantum Asymptotics, volume 36 of IMS Lect. Notes-Monograph Ser. (2001) 255–285.  
  13. C.W. Helstrom, Quantum Detection and Estimation Theory. Academic Press, New York (1976).  Zbl1332.81011
  14. W. Hoeffding, Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc.58 (1964) 13–30.  Zbl0127.10602
  15. A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publishing Company, Amsterdam (1982).  Zbl0497.46053
  16. J. Kahn, Sélection de modèles en tomographie quantique. Master's thesis, École Normale Supérieure, Université Paris-Sud, 2004.  
  17. U. Leonhardt, Measuring the Quantum State of Light. Cambridge University Press (1997).  Zbl0948.81676
  18. U. Leonhardt, H. Paul and G.M. D'Ariano, Tomographic reconstruction of the density matrix via pattern functions. Phys. Rev. A52 (1995) 4899–4907.  
  19. P. Massart, Concentration Inequalities and Model Selection. École d'été de Probabilité de Saint-Flour, 2003. Lect. Notes Math. Springer-Verlag, Berlin (2006).  Zbl1170.60006
  20. D.T. Smithey, M. Beck, M.G. Raymer and A. Faridani, 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.  
  21. K. Vogel and H. Risken, Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A40 (1989) 2847–2849.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.