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arXiv:quant-ph/9804050

Reconstruction of photon distribution with positivity constraints

Konrad Banaszek

Published Apr 22, 1998• 2 citations
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Abstract

An iterative algorithm for reconstructing the photon distribution from the random phase homodyne statistics is discussed. This method, derived from the maximum-likelihood approach, yields a positive definite estimate for the photon distribution with bounded statistical errors even if numerical compensation for the detector imperfection is applied.

DOI
10.48550/arxiv.quant-ph/9804050

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An iterative algorithm for reconstructing the photon distribution from the random phase homodyne statistics is discussed. This method, derived from the maximum-likelihood approach, yields a positive definite estimate for the photon distribution with bounded statistical errors even if numerical compensation for the detector imperfection is applied.
We present a numerical algorithm for the maximum-likelihood estimation of the density matrix, and apply it to the homodyne tomography of a single-mode radiation field. The algorithm is based on a specific form of the Gauss decomposition for positive definite Hermitian matrices. Results from Monte Carlo simulated experiments are presented. PACS: 42.50.Dv, 03.65.Bz Over past several years a great deal of interest has been paid to the reconstruction of the complete quantum state from experimental data [1]. Theoretical research in this field, apart from designing new measurement schemes, explores also the problem of extracting the optimal amount of information from data available in a realistic experimental setup. In this communication, we present a numerical algorithm for reconstructing the density matrix using the maximum-likelihood method [2, 3]. We discuss this algorithm in the context of quantum homodyne tomography [4, 5], which is actually the most successful method for quantum state reconstruction, and, in fact, the unique technique that has been so far experimentally implemented to measure nonclassical states of light [4, 6]. The realistic homodyne measurement of the quadrature is described by the positive operator-valued measure
The photon statistics of a strongly attenuated continuous-wave laser (1550 nm), characterized by an Avalanche Photodiode operating in Geiger mode, was reconstructed with the aid of an iterative Maximum Likelihood Estimation model and without photon counting. Statistical analysis of the reconstructed distribution was performed to determine the algorithm convergence criteria. Parameters such as reconstruction error, Fischer Dispersion Index, chi-square, and p-value were evaluated. The photon number distribution for attenuated coherent sources was successfully reconstructed according to the defined convergence criteria.