Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography

Abstract : In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R 2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2 < η ≤ 1, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the L∞-risk over a class of infinitely differentiable functions. We also compute the lower bound for the L 2-risk. We construct an adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness of the class of functions up to a logarithmic factor in the sample size. Finite sample behaviour of our adaptive procedure is explored through numerical experiments.
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https://hal.archives-ouvertes.fr/hal-01491197
Contributor : Katia Méziani <>
Submitted on : Thursday, March 16, 2017 - 3:00:00 PM
Last modification on : Tuesday, April 2, 2019 - 2:16:00 PM
Long-term archiving on : Saturday, June 17, 2017 - 2:34:18 PM

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  • HAL Id : hal-01491197, version 1
  • ARXIV : 1506.06941

Citation

Karim Lounici, Katia Meziani, Gabriel Peyré. Adaptive sup-norm estimation of the Wigner function in noisy quantum homodyne tomography. 2017. ⟨hal-01491197⟩

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