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arXiv:1301.7644 · Inverse Problems

Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

Pierre Alquier, Katia Méziani, Gabriel Peyré A

Published Jun 24, 2013• 6 citations
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Abstract

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $ρ_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0

DOI
10.1088/0266-5611/29/7/075017

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In the framework of noisy quantum homodyne tomography with efficiency parameter 1/2<η≤11/2 < η\leq 11/2<η≤1, we propose a novel estimator of a quantum state whose density matrix elements ρm,nρ_{m,n}ρm,n​ decrease like Ce−B(m+n)r/2Ce^{-B(m+n)^{r/ 2}}Ce−B(m+n)r/2, for fixed C≥1C\geq 1C≥1, B>0B>0B>0 and $0
In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.
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.
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.
Modern scientiflc studies often involve complex quantum systems, and scientists need to learn the systems from experiment data. As density matrices are usually employed to characterize the quantum states of the systems, this paper investigates estimation of density matrices for the quantum systems. We propose statistical methodologies to construct density matrix estimators and establish an asymptotic theory for the estimation methods. We show that the proposed density matrix estimators are consistent and have high convergence rates. A numerical study is conducted to demonstrate the good flnite sample performances of the proposed estimators.