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New hardware platforms for the study of a free object in a quantum system

Transfer of information has long been studied by scientists and every year there are new developments in the field of information theory. The most relevant in modern conditions are the following questions: whether it is possible to create a device capable of interrupting any form of quantum communication, whether it is possible to maximize the amount of information that can be transmitted with quantum States in a fixed but unknown environment, binary symmetric channels are one of the most useful noise models, because all their theoretical and information properties are easily calculated, whether it is possible to make quantum information theory using classical channels. All of these items depend on a specific free object above the target group.

Relevance of the study

The relevance of solving the problem is displayed in any set of mathematical objects, where free objects are those that satisfy the least number of laws. Since any other monoid over S can be considered as a free monoid together with additional restrictions imposed on its multiplication. Computer scientists call the elements of free monoids “lists”. They are among the most important objects in computing. In particular, a free monoid over a set of one element is a set of positive integers with addition. Elements of a free object are called “channels”. From classical image distortion to quantum communication , it plays an important role in the study of information transmission in a noisy environment.

Purpose of research

The purpose of this paper is a comprehensive, including the study of free object in quantum information theory, consideration of two examples of affine monoids: the first follows from information theory and provides a natural model of image distortion, as well as a multidimensional analog of a binary symmetric channel, the second, from physics, describes the process of teleportation of quantum information with a given entangled state and a free affine monoid over the fourth Klein group.

Research problem

The goal identified the following tasks:

to consider the theoretical and methodological foundations of a free object in quantum information theory;
to study the problem of noise-resistant programming, teleportation, block coding, free affine monoid over a finite group;
prove the isomorphism of two examples of affine monoids;
Scientific novelty of the research

The scientific novelty of the research is as follows:

The object for the implementation of information system security is higher education, the topic of research methods, tools for the construction and implementation of quantum systems. Such systems should be developed in a timely manner and easily adapted to the daily changing conditions on the part of the control object and adapt to modern requirements. The development of a conceptual model is usually divided into several different levels.

The development of quantum systems has created high hopes for reliable processing of quantum information. Although this progress has already led to various experiments with proof of principle on small quantum systems, a large scaling step is required for protocols with many qubits. Fail-safe calculations with protected logical qubits are usually performed at the expense of significant hardware costs. Each of the physical qubits involved still has to satisfy the best properties achieved (coherence times, coupling forces, and tunability).

Two levels will be enough: the upper, research, which covers the entire model of quantum development, modern threat models, etc., lower or consumer, which refers to the individual subsystems of the information system and various services.

Binary symmetric channels in noise-resistant coding. A simple way to present a black and white image on a computer is a set of pixels. A pixel is a tiny rectangular area of the original image. The center of this rectangle is assigned a number that represents its intensity or “gray level.” For example, black may be represented by 0 and white by 255. In General, let’s assume that the intensity is represented by a number whose binary extension can be specified in n bits.

The image is distorted when ambient noise flips some bits in the pixel. This leads to a change of the original intensity of the pixel. For example, if all the bits in a white pixel are inverted, the pixel turns black, causing the image to appear dark where it should be light. To simulate image distortion, we will use a channel whose input is a pixel and whose output is a pixel that has been degraded in some way.

When transmitting digital data over a noisy channel, there is always the possibility that the received data will contain some level of error rate. The receiver generally sets some level of error rate beyond which the received data cannot be used. If the frequency of errors in the received data exceeds the acceptable level, you can use error-correcting coding., which reduces the error rate to acceptable [1].

Coding the detection and correction of errors generally linked to the notion of redundancy of code, which leads in the end to reduce the speed of transmission of information flow on the communication path. Redundancy is that digital messages contain additional characters that ensure the individuality of each code word. The second property associated with error-correcting coding is noise averaging. This effect is that the redundant characters depend on several information characters.

When increasing the length of the code block (i.e. the number of redundant characters), the proportion of erroneous characters in the block tends to the average error rate in the channel. By processing characters in blocks rather than one after the other, you can reduce the overall error rate and, with a fixed probability of block error, the proportion of errors that need to be corrected.

All currently known codes can be divided into two large groups: block and continuous. Block codes are characterized by the fact that the sequence of characters is divided into blocks. Coding and decoding operations in each block are performed separately. Continuous codes are characterized by the fact that the primary sequence of characters carrying information is continuously converted by a certain law into another sequence containing an excessive number of characters. The processes of encoding and decoding does not require dividing the coded symbols in blocks [2].

Types of both block and continuous codes are separable (with the possibility of allocating information and control characters) and inseparable codes. The most numerous class of separable codes are linear codes. Their peculiarity is that control symbols are formed as linear combinations of information symbols. In this Chapter, the following was considered: we studied the case of n = 1, when the intensity is represented by one bit, measured the amount of distortion in the image. The channels of the form of the first affine monoid, which follows from the information theory – binary symmetric channel, were defined.

It is also worth taking into account that after studying binary communication channels, we have learned to simulate image distortion; we will use a channel whose input is a pixel, and whose output is a pixel, which in General has been somehow degraded.

Mixed state in the qubit channel. The fact that the binary channel f :Δ2 → Δ2 operates on Δ2 indicates that only two characters are sent and that a special and fixed way of representing these two characters is chosen. In contrast, in the case of a quantum channel, there are an infinite number of ways to represent bits: each basis of the state space H 2, a two-dimensional complex Hilbert space, offers a different possible representation [3].

The classical binary channel f:Δ2 → Δ2 accepts the distribution of the input signal at the distribution output. In a similar way, a qubit channel will display the input distributions to output distributions. But what is the quantum analog of distribution? Let’s return to the classical case. Each distribution x ∈Δ2 can be written

x= x0 · e0 + x1 · e1

as a convex sum of classical “pure” States. The meaning of this expression is that the system is in the state e 0 with probability x 0 and in the state e 1 with probability x 1. Thus, if the quantum system is in the | Ψ 〉 state with probability x I, the natural way of representing this “distribution” is given by the operator. The density operator in quantum systems is called the mixed state. The set of all density operators on H2 is denoted by Ω2. Thus, by analogy with the classical case, the qubit channel will be a function of the form ε: Ω2 → Ω2[4].

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