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Mathematical Methods of Statistics

Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data

Katia Méziani

Published Dec 1, 2007• 7 citations
Publisher

Abstract

The aim of this paper is to answer an important issue in quantum mechanics, namely to estimate the purity of a quantum state of a light beam. Estimation of the purity is based on the results of quantum homodyne measurements performed on independent identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, which can take negative values and must satisfy certain constraints of positivity imposed by quantum physics. We estimate the integrated squared Wigner function by a kernel-based second order U — statistic. This quadratic functional is a physical measure of the purity of the state. We also give an adaptive estimator, which does not depend on the smoothness parameters. We establish upper bounds of the minimax risk over a class of infinitely differentiable functions.

DOI
10.3103/s1066530707040047

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The aim of this paper is to answer an important issue in quantum mechanics, namely to estimate the purity of a quantum state of a light beam. Estimation of the purity is based on the results of quantum homodyne measurements performed on independent identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, which can take negative values and must satisfy certain constraints of positivity imposed by quantum physics. We estimate the integrated squared Wigner function by a kernel-based second order U — statistic. This quadratic functional is a physical measure of the purity of the state. We also give an adaptive estimator, which does not depend on the smoothness parameters. We establish upper bounds of the minimax risk over a class of infinitely differentiable functions.
The aim of this paper is to answer an important issue in quantum me-chanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner func-tion, a density of generalized joint probability which can take negative values and which must respect certain constraints of positivity imposed by quantum physics. We propose to estimate a quadratic functional of the Wigner function by a kernel method as the physical measure of the purity of the state. We give also an adaptive estimator that does not depend on the smoothness parame-ters and we establish upper bound on the minimax risk over a class of infinitely differentiable functions. 1 ha l-0
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.
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.
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R2\mathbb R^2R2 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≤11/2 < η\leq 11/2<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∞\mathbb L_\inftyL∞​-risk over a class of infinitely differentiable. We compute also the lower bound for the L2\mathbb L_2L2​-risk. We construct adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness class functions. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.
We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared pulses. The state is represented through the Wigner function, a “quasi-probability density” on ℝ 2 which may take negative values and must satisfy intrinsic positivity constraints imposed by quantum physics. The data consists of n i.i.d. observations from a probability density equal to the Radon transform of the Wigner function. We construct an estimator for the Wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions. A similar result was previously derived by Cavalier in the context of positron emission tomography. Our work extends this result to the space of smooth Wigner functions, which is the relevant parameter space for quantum homodyne tomography.
In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on R2\mathbb{R}^{2}R2, 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≤11/2<\eta\leq11/2<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∞\mathbb{L}_{\infty}L∞​-risk over a class of infinitely differentiable functions. We also compute the lower bound for the L2\mathbb{L}_{2}L2​-risk. We construct an adaptive estimator, that is, 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.