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DESCRIPTION OF BASIC MATHEMATICAL MODELS IN QUANTUM CRYPTOGRAPHY

E. V. Borisova, postgraduate student, RSREU, Ryazan, Russia;
orcid.org/0000-0002-0198-377X, e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Quantum cryptography is one of the three parts that form interdisciplinary field of research-quantum computer science. In addition to quantum cryptography, quantum information science includes research on quantum computers and quantum communication theory. Since quantum informatics originated at the intersection of various disciplines, its understanding requires, first of all, the knowledge of languages and methods of quantum theory as well as classical information theory [1].
This article discusses quantum cryptography, on the one hand, as quantum communication, in particular, quantum key distribution, and on the other hand, quantum computing, which poses a threat to classical cryptography and to public key data transfer protocols. Both are essentially different branches of quantum computer science – the science of how information can be transmitted and transformed using quantum systems. The concept of a qubit is defined as a simple two-dimensional quantum state, the basis of which is a logical zero and a logical unit. In this paper, we will define the concept of qubit as a two-dimensional quantum system, i.e., a superposition of two basis vectors: 0 and 1, with coefficients C0 and C1 being complex.
The aim of this paper is to present a mathematical description of qubit, qudit, N-qubit system, and ensemble density matrix. In this paper, we define the concept of qubit as a quantum system with two welldistinguishable orthogonal quantum states. Qubits are shown to move on to qudits, i.e., to d-dimensional system, which, respectively, will be described by 2d-2 number of parameters, the possibility to move to N qubit system, the dimension of which, in general, growing exponentially is also proved. In case when the system is factorized, it grows linearly.

Key words:  qubit, qudit, quantum cryptography, ensemble density matrix.

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