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Physica Scripta

State reconstruction via infinite matrix inversion

Jussi Schultz

Published Sep 1, 2010• 1 citation
Publisher

Abstract

We consider the method of infinite matrix inversion in deriving state reconstruction formulae for phase-space distributions. As an example, we demonstrate the method in the case of the Cahill?Glauber s-parametrized quasiprobability distributions. We obtain two formulae by different means, both of which have different conditions for validity.

DOI
10.1088/0031-8949/2010/t140/014060

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We consider the method of infinite matrix inversion in deriving state reconstruction formulae for phase-space distributions. As an example, we demonstrate the method in the case of the Cahill?Glauber s-parametrized quasiprobability distributions. We obtain two formulae by different means, both of which have different conditions for validity.
We propose a method for reconstruction of the density matrix from measurable time-dependent (probability) distributions of physical quantities. The applicability of the method based on least-squares inversion is---compared with other methods---very universal. It can be used to reconstruct quantum states of various systems, such as harmonic and anharmonic oscillators including molecular vibrations in vibronic transitions and damped motion. It also enables one to take into account various specific features of experiments, such as limited sets of data and data smearing owing to limited resolution. To illustrate the method, we consider a Morse oscillator and give a comparison with other state-reconstruction methods suggested recently.
We focus our attention on the problem of reconstruction of density operators of quantum states from propensities, i.e. generalized quasiprobability density distributions obtained by quantum filtering. We consider propensities obtained by filtering with pure Gaussian as well as non-Gaussian states and we present two examples: when the filter is in a squeezed coherent state and in a Fock state, respectively. We also show that even in the case of filtering with statistical mixtures a complete reconstruction of a density operator of the measured quantum state can be performed.