Regularization Methods for Ill-Posed Problems in Quantum Optics

On the example of the specific physical problem of noise reduction associated with losses, dark counts, and background radiation, in the statistics of photocounts of quantum light, the summary of methods for regularizing ill-posed problems is given. The mathematical formulation of the problem is presented by an operator equation of the first kind. It is shown that the operator is generated by a matrix with countable elements. It is pointed out that in the sense of Hadamard, the problem of reconstructing the number of photons of quantum light is due to the compactness of the operator of the mathematical model. It is emphasized that the problem of stable approximation to the exact solution of the operator equation with inexact initial data can be overcome by one of the most well-known regularization methods, whose theoretical foundations were laid in the works of A. M. Tikhonov. An important class of regularizers based on the construction of a parametric family of functions called generator is considered. It is confirmed that regularizers of this class allow achieving the optimal order of accuracy for equations with source-representable solutions.