## 6 Conclusions  
In summary, we developed a novel method for the modeling of quantum light scattering and quantum antennas. It is formulated in terms of a linear super-operator defined as an operator in the Hilbert space, which acts in the subspace of quantum state operators, satisfying the physically appropriate commutation relations. It is shown that in the general case, the naive quantization via the formal exchange of the classical values by the corresponding operators leads to an ill-posed super-operator problem. The reason is that the action of super-operator \(\overline{\mathbf{A}}\) may throw the result outside of the required subspace, which violates the correct commutation relations. In order to satisfy them, one needs to complete the regularization of the super-operator equation.  
The proposed regularization technique is based on the introduction of ancillary channels, which are associated with the novel type of quantum noise, whose origin is not related to any dissipation of energy. The regularized super-operator equation has the form \(\overline{\mathbf{A}}\hat{E}=\hat{E}_{0}+\hat{Q}\), where \(\hat{Q}\) is a noise operator. Its form and commutation relations are determined in this paper. In general case, the super-operator equations may be solved numerically. Exact analytical solutions exist for special types of configurations. Sometimes, the approximate analytical solution is promised (for example, the perturbation technique). The solution of the regularized super-operator equation may be expressed in terms of classical resolvent, which may be found by one of the methods of classical computational electromagnetics [48].  
Figure 4: The contour \(C\) (colored brown) is the integration contour in Equation (56). The cut is placed along the negative semi-axis and shown by the jagged line.  
Simple applications of the theory are considered using several examples (two-element dipole quantum antenna, reflection-transmission in a 1D dielectric layer, and scattering by a dielectric cylinder). It is shown that a special noise is produced via the exchange of energy between the body and the external space. From the \"point of view\" of the dielectric body, the process of multimode scattering looks like losses (or equivalent gain, if the energy enters from outside) in spite of the absence of any heating or real physical amplification. As a result, the exchange of energy is accompanied by a noise similar to that in systems with real physical dissipation. Thus, it becomes possible to speak about the general \"fluctuation-scattering theorem\" similar to the well-known \"fluctuation-dissipating theorem\": the scattering of the quantum field is accompanied by fluctuations. In partial cases, a similar mechanism manifested itself in a parallel-plane beam splitter [49] and time lenses [50]. In [49], the initial photon state as a product of number states in both modes is considered. It was shown that the lossless feature due to beam-splitter coupling created a noise in the photon number of each mode in spite of the total photon number being invariant and noise-free.  
The theory developed in this paper may be generalized for many other scattering problems (different types of quantum antennas, scattering from 3D dielectric bodies, or dielectric bodies with different types of environments, etc.). It suggests a new method for using scattering analysis as a tool for controlling the statistical properties of quantum light. It may be applied to various problems posed by quantum technologies, metamaterials [51], radars and lidars [14; 16], quantum circuits, and nanoantennas [52; 53; 54; 55].  
Conceptualization, G.S. and A.B.; Formal analysis, G.S. and A.B.; Writing--original draft, G.S. and A.B.; Writing--review & editing, G.S. and A.B. All authors have read and agreed to the published version of the manuscript.  
This research was funded by NATO grant number NATO SPS-G5860 and by H2020, project TERASSE 823878.  
Not applicable.  
The authors declare no conflict of interest.