Open Access
April 2007 Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data
Cristina Butucea, Mădălin Guţă, Luis Artiles
Ann. Statist. 35(2): 465-494 (April 2007). DOI: 10.1214/009053606000001488


We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on ℝ2 which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise.

We construct a kernel estimator for the Wigner function, prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions, and implement it for numerical results. We construct adaptive estimators, that is, which do not depend on the smoothness parameters, and prove that in some setups they attain the minimax rates for the corresponding smoothness class.


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Cristina Butucea. Mădălin Guţă. Luis Artiles. "Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data." Ann. Statist. 35 (2) 465 - 494, April 2007.


Published: April 2007
First available in Project Euclid: 5 July 2007

zbMATH: 1117.62027
MathSciNet: MR2336856
Digital Object Identifier: 10.1214/009053606000001488

Primary: 62G05 , 62G20 , 81V80

Keywords: adaptive estimation , Deconvolution , exact constants in nonparametric smoothing , Infinitely differentiable functions , minimax risk , nonparametric estimation , quantum homodyne tomography , quantum state , Radon transform , Wigner function

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 2 • April 2007
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