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arXiv:physics/0508190 · Optics Letters

Maximum-likelihood estimation of Mueller matrices

Andrea Aiello, Graciana Puentes, D. Voigt, J. P. Woerdman

Published Mar 15, 2006• 53 citations
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Abstract

We show that the method of maximum-likelihood estimation, recently introduced in the context of quantum process tomography, can be applied to the determination of Mueller matrices characterizing the polarization properties of classical optical systems. Contrary to linear reconstruction algorithms, the proposed method yields physically acceptable Mueller matrices even in the presence of uncontrolled experimental errors. We illustrate our method with the case of an unphysical measured Mueller matrix taken from the literature.

DOI
10.1364/ol.31.000817

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When standard methods of process (black-box) estimation are applied straightforwardly then it may happen that some sets of experimental data result in unphysical estimations of the corresponding channels (maps) describing the process. To prevent this problem, one can use the method of maximum likelihood (MML), which provides an efficient scheme for reconstruction of quantum channels. This scheme always results in estimations of channels that are fully physical, e.g. the corresponding maps are linear, positive and completely positive. To show this property, we use the MML for a derivation of physical approximations of truly unphysical operations. In particular, we analyze physical approximations of the universal-NOT gate, the quantum nonlinear polarization rotation and the map ρ → ρ2. Given the result of MML, we examine retrospectively the quality of the experiment. Depending on the resulting value of the MML functional we can determine (physical) consistency of the input data.
We show that the method of maximum likelihood (MML) provides us with an efficient scheme for the reconstruction of quantum channels from incomplete measurement data. By construction this scheme always results in estimations of channels that are completely positive. Using this property we use the MML for a derivation of physical approximations of unphysical operations. In particular, we analyze the optimal approximation of the universal NOT gate as well as the physical approximation of a quantum nonlinear polarization rotation.
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.