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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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

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

First available in Project Euclid: 5 July 2007

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Zentralblatt MATH identifier

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

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


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.

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