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arXiv:math/0405595

An invitation to quantum tomography (II)

L.M. Artiles, Richard David Gill, M.I. Guţǎ

Published May 31, 2004• 0 citations
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Abstract

The quantum state of a light beam can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We describe quantum tomography as an inverse statistical problem in which the state is the unknown parameter and the data is given by results of measurements performed on identical quantum systems. We present consistency results for Pattern Function Projection Estimators as well as for Sieve Maximum Likelihood Estimators for both the density matrix of the quantum state and its Wigner function. Finally we illustrate via simulated data the performance of the estimators. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation, studying other estimators, etc., and a main purpose of the paper is to bring these to the attention of the statistical community.

DOI
10.48550/arxiv.math/0405595

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Summary We describe quantum tomography as an inverse statistical problem in which the quantum state of a light beam is the unknown parameter and the data are given by results of measurements performed on identical quantum systems. The state can be represented as an infinite dimensional density matrix or equivalently as a density on the plane called the Wigner function. We present consistency results for pattern function projection estimators and for sieve maximum likelihood estimators for both the density matrix of the quantum state and its Wigner function. We illustrate the performance of the estimators on simulated data. An EM algorithm is proposed for practical implementation. There remain many open problems, e.g. rates of convergence, adaptation and studying other estimators; a main purpose of the paper is to bring these to the attention of the statistical community.
Summary form only given. Experimental techniques for measurement of the quantum state of light have been the subject of intensive investigation for some time. Tomographic techniques have been applied to experiments such as the homodyne measurement of the Wigner function of a single mode of light and of the density matrix of the polarization degrees of freedom of a pair of entangled photons. In this technique, the density matrix (or Wigner function) which characterizes the quantum state of the system being measured is found from a linear transformation of experimental data. There are a number of drawbacks to the method, principally in that the recovered state might not, because of experimental noise, correspond to a physical state. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite. To avoid this problem, the maximum likelihood approach to the estimation of quantum states has been developed. We evaluate two different approaches to the maximum likelihood technique, namely defining the likelihood function f as a weighted sum of squared variances from measured data, and defining it in terms of information content. These are compared with results of standard tomographical schemes.
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.
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.
Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.