3 research outputs found
Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data
The aim of this work is to estimate a quadratic functional of a unknown
Wigner function from noisy tomographic data. The Wigner function can be seen as
the representation of the quantum state of a light beam. The estimation of a
quadratic functional is done from result of quantum homodyne measurement
performed on identically prepared quantum systems. We start by constructing an
estimator of a quadratic functional of the Wigner function. We show that the
proposed estimator is optimal or nearly optimal in a minimax sense over a class
of infinitely differentiable functions. Parametric rates are also reached for
some values of the smoothness parameters and the asymptotic normality is given.
Then, we construct an adaptive estimator that does not depend on the smoothness
parameters and prove it is minimax over some set-ups
Quadratic functional estimation in inverse problems
We consider in this paper a Gaussian sequence model of observations ,
having mean (or signal) and variance which is
growing polynomially like , . This model describes a large
panel of inverse problems. We estimate the quadratic functional of the unknown
signal when the signal belongs to ellipsoids of both
finite smoothness functions (polynomial weights , ) and
infinite smoothness (exponential weights , , ). We propose a Pinsker type projection estimator in each case and study its
quadratic risk. When the signal is sufficiently smoother than the difficulty of
the inverse problem ( or in the case of exponential
weights), we obtain the parametric rate and the efficiency constant associated
to it. Moreover, we give upper bounds of the second order term in the risk and
conjecture that they are asymptotically sharp minimax. When the signal is
finitely smooth with , we compute non parametric upper
bounds of the risk of and we presume also that the constant is asymptotically
sharp