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Physical Review A

Direct sampling of density matrices in field-strength bases

A. Zucchetti, W. Vogel, Manfred Tasche, D.‐G. Welsch

Published Aug 1, 1996• 27 citations
Publisher

Abstract

A simple and efficient method for direct sampling of the density matrix of a signal mode in optical homodyne tomography is developed. The method is based on approximate, quasianalytical techniques of Fourier transform and yields the density matrix in an arbitrary field-strength basis. It is also suitable for direct sampling of density matrices of matter systems in generalized position or momentum bases. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.54.1678

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A simple and efficient method for direct sampling of the density matrix of a signal mode in optical homodyne tomography is developed. The method is based on approximate, quasianalytical techniques of Fourier transform and yields the density matrix in an arbitrary field-strength basis. It is also suitable for direct sampling of density matrices of matter systems in generalized position or momentum bases. \textcopyright{} 1996 The American Physical Society.
It is shown that the density matrix elements of a radiation mode in a chosen field strength representation can be obtained from the field strength distributions (with the phase being varied in a π interval) by a twofold Fourier transformation, and from the quasiprobability distributions P(α; s) (s < 0) by a threefold one. The method also applies, in principle, to the smoothed distributions observed in balanced homodyne (four-port and eight-port) detections with detector efficiencies less than unity.
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We propose a general method for reconstructing directly the density matrix of a single light mode in optical homodyne tomography. In our scheme the density matrix 〈a\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}${\mathit{a}}^{\ensuremath{'}}$〉 is obtained by averaging a set of pattern functions ${\mathit{F}}_{\mathit{aa}}^{\ensuremath{'}}$(${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$,\ensuremath{\theta}) with respect to the homodyne data ${\mathit{x}}_{\mathrm{\ensuremath{\theta}}}$. The functions show the typical features of the quadrature distributions for the corresponding density-matrix elements. It is also possible to compensate the effect of detection losses which requires, however, extra effort in both experimental and numerical precision. We calculate the pattern functions for the coherent-state and Fock representations and study their properties. We believe that our method is the most efficient way for reconstructing the density matrix from homodyne measurements.
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.