The Annals of Statistics

Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data

Cristina Butucea, Mădălin Guţă, and Luis Artiles

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Abstract

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.

Article information

Source
Ann. Statist., Volume 35, Number 2 (2007), 465-494.

Dates
First available in Project Euclid: 5 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1183667281

Digital Object Identifier
doi:10.1214/009053606000001488

Mathematical Reviews number (MathSciNet)
MR2336856

Zentralblatt MATH identifier
1117.62027

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties 81V80: Quantum optics

Keywords
Adaptive estimation deconvolution nonparametric estimation infinitely differentiable functions exact constants in nonparametric smoothing minimax risk quantum state quantum homodyne tomography Radon transform Wigner function

Citation

Butucea, Cristina; Guţă, Mădălin; Artiles, Luis. Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data. Ann. Statist. 35 (2007), no. 2, 465--494. doi:10.1214/009053606000001488. https://projecteuclid.org/euclid.aos/1183667281


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