UDC 517.983.54; 519.67; 539.122
REGULARIZATION METHODS FOR ILL-POSED PROBLEMS OF QUANTUM OPTICS
Abstract. On the example of a specific physical problem of reduction of noise caused by losses, dark counts, and background radiation, a summary of methods for regularizing ill-posed problems is given in the statistics of photocounts of quantum light. The mathematical formulation of the problem is represented by an operator equation of the first kind. It is shown that the operator is generated by a matrix with elements of a countable set. It is noted that the incorrectness of the Hadamard reconstruction of the statistics of the number of photons of quantum light is due to the compactness of the operator of the mathematical m`odel. It is emphasized that the problem of stable approximation to the exact solution of the operator equation with inaccurate initial data can be solved by one of the most well known regularization methods whose theoretical foundation was laid by A.N. Tikhonov. An important class of regularizers based on a parametric system of functions, called generating functions, is considered. It is confirmed that regularizers of this class allow one to achieve the optimal order of accuracy for equations with source-representable solutions.
Keywords: ill-posed problem, quantum optics, operator, regularization, algorithm, photon.
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