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arXiv:quant-ph/9904063

Maximum-likelihood algorithm for quantum tomography

Konrad Banaszek

Published Apr 15, 1999• 1 citation
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Abstract

Optical homodyne tomography is discussed in the context of classical image processing. Analogies between these two fields are traced and used to formulate an iterative numerical algorithm for reconstructing the Wigner function from homodyne statistics.
DOI
10.48550/arxiv.quant-ph/9904063

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The problem of optical homodyne tomography is considered in the context of a Bayesian model-fitting or inverse problem approach. An algorithm is formulated, based on matrix computation rather than the numerical approximation of an analytic inverse transform. This automatically takes into account the effects of noise, detector inefficiencies and incomplete sampling of the data. The relationships with conventional reconstruction schemes, methods for including various forms of prior information and for calculating error estimates are discussed. The process of reconstructing the photon number distribution and the density matrix are illustrated using both simulated and experimental data.
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The intrinsic relationship between the maximum-likelihood quantum-state estimation and the representation of the signal is elaborated. A quantum analogy of the transfer function determines the space where a successful reconstruction can be achieved. This provides a tool for reducing the number of dimensions of the observed system based on physical characteristics of the reconstruction scheme rather than some ad hoc truncations. The method is illustrated with two examples of practical importance: an optical quantum homodyne tomography and a simple and robust tomography of an optical signal recorded by realistic binary detectors.