The Gaussian mode function can be used for describing the output of coherent laser beams, which are the source of light for creating any form of photon-based quantum information processing or information storage applications. The main advantage of using such an approach is that it does not require the use of filters to produce specific characteristics of photons. Gaussian Gaussians can be calculated at the input end of the process but never before in the output. This ensures that there are no unwanted output characteristics in such systems.
Using Gaussian Gaussians, we can analyze information and its properties. In general, we use the following definition:
where N is the number of photons. This is sometimes called “the number of particles” or “the number of quanta”, depending on how far the input signal consists of light (photons), i.e., their frequency or wavelength, and also other properties such as the intensity and power. To calculate a Gaussian, it is necessary to perform several operations on these groups such as normalization, square root, division by constants, etc. These functions can be performed directly on individual input signals or on linear combinations of them. For example, we can define two input signals of the same type, say of the same amplitude with the same intensity, or of different amplitudes with different intensities as a group of similar inputs. Our task is to find the parameters of this group given the input signals.
In our experiment, two linear combinations of N-shaped input signals will be considered. We show them here in figure 1:
The Gaussian can be defined on a set of variables. A set of independent variables in our case is a combination of the output amplitude and the input amplitude. So, for each set of independent inputs, it is possible to calculate the corresponding values and variables. Let us denote the combination given in equation (1) by X1, X2. For each combination of inputs, X1 is the output amplitude and X2 is the input amplitude. But this time we denote them, they are different magnitudes, which means that the values for X1 vary over the input signals but also depend on the amplitude of both and on the intensity of the input signals. The values for X1 vary differently for different combinations depending on the chosen set of independent inputs. Therefore, it should be possible to find a set of parameters for all combination of linear inputs. That is, each set of parameters is comprised of a set of weights and biases which will be able to describe the output and the input of the system.
The parameters for some combinations are unknown until we fit them to the system. However, when we have measured the amplitude using one of the input signals, it is possible to assume that the corresponding values for a combination of linear inputs will be equal to zero. Then, we can use equations (1) to find the parameters for our set of linear input combinations — and then we can derive the characteristics of the output and the input of our set of linear combinations. Finally, we can have a comparison between the parameters for all sets of linear combinations and the one we found for the combined set of linear inputs. When we have a perfect fit, we can have a set of parameters for the system, which describes our system.
In this section, we discuss an application where we will show how the system can be changed through the adjustment of the parameter values of this set of parameters. As an example, this approach will be shown with a hypothetical system consisting of two linear channels. Each channel will be subjected to random noise, which makes only certain combinations of these channels completely blocked. Another example is that in a communication system the information is encoded in the modulated output of the transmitter. We can encode information by showing what kind of signal has been transmitted, say, 10-bit data. It is the transmission over 100 channels. In order to get the correct output for the transmission process, it is necessary to make sure that only those input signals with the corresponding channel will be able to reach the receiver.
In our experiment, the first two channels will have no noise, while the third one will have 50,000 noise channels. The parameters of the third channel will have the same values as in the previous channel. In the second channel, we can see what kind of signal is being transmitted, whether it is a 1 or a 0. When a 1-bit is sent, it goes straight through the channel without experiencing any noise. The noise is the opposite and it is not able to block such signals. Thus, if in a third channel we send 10-bit data we have a combination of input signals with the corresponding channel.
When we want to modify the parameter values of just one channel, then we can use equations (2) so we can calculate the new parameters and compare them with the same group obtained in equation (1). Of course, we could do this for every channel, however, by repeating this procedure many times we can quickly see how complicated the calculation of parameters for various combinations has become. Here we need a large number of parameters that we need to store in memory. If we look at this number, we can see that the computation goes from about 10 (numerical value) to more than 20 million (numeric value). At least, that is one reason why we need to make sure that this method is not too expensive, otherwise it will exceed the limit of practical computation.
So, we can assume that our model is working in the ideal situation where the two channels have no noise. Now let’s consider another scenario, according to which our system is characterized by a lot of parameters.In each channel, we have thousands of parameters and they must be calculated for each possible combination of input signals with all parameters being equal. Therefore, the volume of computations required becomes exponential when we consider a large number of parameters for our model. What we can do is to generate, simulate, and then estimate the values for our parameters by taking into account only the signals of the desired parameters and parameters for all channels that have been generated. So, with all the algorithms and tools available, we can calculate the values and their distributions for the parameters and it will be possible to find the resulting parameters for the set of parameters where the signals have not been corrupted and where all signals having these parameters are available. This can be done for the entire matrix of parameters allowing to find the set that satisfies our criteria. Once these sets of parameters are obtained, these set will be very close to each another. So, it will be possible to make changes to any parameter of our models by changing the values only where these two sets of parameters are changing. They can be solved automatically. Using computer programs like MATLAB, Python, Excel and others, we can create a series of equations that calculates all these parameters for each case. After the calculations, the values were compared to the original set and deviations were calculated. As soon as we know which set of parameters is good for achieving the best results, we can update it and change the values of other parameters by calculating the deviations and correcting them. Thus, we can solve the problem of parameters and their distribution.
We can have different types of modifications to our model, they are usually aimed to add noise to the channels. Or to ensure that we can create filters so that we can filter unwanted signals in some specific regions of the channel. Moreover, we can introduce a number of different signals at once that are used to eliminate noise. Different kinds of noise can be added to channels by modifying the values of our parameters for each channel or by introducing them in the channels by changing some parameters of them. In conclusion, our work shows a technique that allows us to analyze the information and properties of signals in a system. By considering linear combinations of input signals we have been able to demonstrate how easily it is to determine parameters of the system. Two sets of parameters were chosen to represent the parameters of our model which, by adjusting one parameter, can change the output and input of the system. This was an example where we solved two problems simultaneously. The first one, the solution of a linear combination, we found by multiplying two inputs and then adding the result to the right of the equation. The second one, describing the parameters of a multilayer channel, was derived by multiplying a vector of inputs with parameters in different channels and then adding the result to the left. We will continue to follow this approach and apply it in real practice in order to obtain more accurate and efficient information processing methods.