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Project Number17-11-01388

Project titleMathematical methods for problems of quantum technologies and dynamics of open quantum systems

Project LeadPechen Alexander

AffiliationSteklov Mathematical Institute of Russian Academy of Sciences,

Implementation period 2017 - 2019  extension for 2020 - 2021


Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-113 - Mathematical physics

Keywordscontrol of open quantum systems, quantum technologies, quantum decoherence, many-body systems, thermalization, quantum information theory, quantum entanglement, superadditivity of quantum information



Quantum technologies - technologies, which are based on the use of individual quantum systems of atomic, molecular and nanoscopic scale. This area represents a high level of interest associated with a wide range of applications, from synthetic materials with predetermined properties and control of chemical reactions to information processing, cryptography, metrology. Further progress in quantum technologies requires the development of mathematical methods for the analysis of dynamics and control of open, that is, interacting with the environment, quantum systems. One of the major problems required for the development of quantum technologies is to develop methods for derivation of dynamic equations for controlled open quantum systems and methods for control of quantum systems using modulated laser pulses or other external influences. The project aims at solving the actual mathematical problem in the field of quantum technology, namely, the development of new methods to obtain dynamic equations for control of open quantum systems, with applications to control of systems with similar spectral properties, the study of decoherence in the many-body quantum systems, establishing the connection between the quantitative characteristics of various quantum phenomena. The following tasks will be solved during the project. The first task is to derive a master equation for open quantum systems which evolve under the simultaneous action of coherent and incoherent controls, and to develop a new mathematical formalism that allows to derive dynamical models for open quantum systems directly from the measured data. This task aims to ensure physical validity of the obtained equations of motion and high fidelity of describing the observed dynamics. For simulations a wide range of models will be used which appear in quantum control, quantum optics, metrology, statistical dynamics, quantum transport. The second task is to develop new mathematical tools to study the decoherence of many-particle quantum systems. A distinctive feature of the dynamics of quantum many-particle systems is the separation of time scales of decoherence and thermalization. It is planned to develop a formalism for the perturbative description of decoherence in many-particle systems based on the separation of scales, followed by the the analysis of the performance of quantum devices. The study of quantum decoherence is fundamental for explaining the quantum-classical transition, and also plays an important role in understanding performance and limitations for complex quantum devices, such as quantum simulations and, in the future, quantum computers. The third task is to establihs a relation between different quantum phenomena such as superadditivity of capacity for c-q channels, multipartite quantum entanglement, quantum uncertainty, the possibility of quantum key distribution. This task is important for the development of cryptographic protocols and proving their privacy, and also for the development of encoding methods to achieve superadditivite capacity. Importance of these unsolved problems will require the development of new methods for the theory of open quantum systems and control of quantum systems, which allow to derive the correct dynamic equations in the presence of coherent and incoherent controls, analyze the decoherence in complex many-body systems, and provide universal way to describe the quantitative characteristics of various quantum phenomena. Wide applications in problems of quantum technologies provide the actuality of the project. Despite the high level of interest, these issues are unsolved and form the basis of the proposed project, implementation of which will make a significant progress in the development of mathematical methods for quantum technology.

Expected results
During the project it is planned to obtain the following results. 1) Mathematical and statistical tools for a new approach to the conclusion of dynamic equations for open quantum systems, which provide a correct description of the dynamics of the density matrix of the system, while the impact of coherent and incoherent controls. The solution to this problem is of great fundamental scientific importance for the development of theory of control of open quantum systems. The new generation of numerical methods for efficient, scalable and sustainable solutions derived the equations of motion. The high significance of this result is determined by the possibility of its application in problems of quantum technologies, control quantum systems in quantum optics, metrology, data processing and description of transport phenomena. 2) Perturbative description of decoherence in many-particle systems based on the separation of time scales for thermalizationand and decoherence. Analysis of decoherence in the experimentally realizable of many-body systems and devices, in particular, in quantum simulations and quantum memory. The high significance of the results is determined by the fact that the description of quantum decoherence is fundamental to explaining the quantum-classical transition, and also plays an important role in understanding performance and limitations of complex quantum devices, such as quantum simulations and, in the future, quantum computers. 3) Introduction of the numerical characteristics of quantum uncertainty, obtaining estimates for the relationship of quantum uncertainty and quantum entanglement, superadditivity of quantum information and possibility to generate a secret key. The high significance of this result is determined by the possibility of applying to the development of secure cryptography protocols and proving their security, the development of coding techniques to achieve superadditive capacity. The high importance of the expected results is provided by the fact that in most situations the physical systems used in problems of quantum technologies, are open, that is, interacting with the environment. This fact requires the development of methods for the correct description of the dynamics of controlled open quantum systems. The results will be at the high international scientific level in the field of mathematical problems of quantum technologies and are planned for publication in world-leading scientific journals. Methods which are planned to be developed can be used to study various aspects of atomic and molecular dynamics induced by external laser fields, control of quantum systems, decoherence in nano-mechanical, microwave and superconducting quantum networks. The proposed mathematical research will have a significant impact on the emerging field of quantum technologies.



Annotation of the results obtained in 2019
In 2019, the research was carried out for solving problems of controlling quantum systems, quantum systems with similar spectra, derivation and investigation of master equations for open quantum systems, analysis of decoherence and compensation of dissipative dynamics, analysis of communication protocols for public information transmission and secret key distribution for one or several recipients. Nine papers were published or at press, including three papers which are at press, several manuscripts are submitted or at the final stage of preparation and submission to journals. The results were presented on 31 talks of the project's participants at conferences and seminars; videos of 10 talks were recorded and uploaded for free access on the All-Russian Mathematical Portal MathNet.Ru. On November 25, 2019, Third Youth Workshop «Mathematical Methods in the Problems of Quantum Technologies» was organised at Steklov Mathematical Institute, were also results performed within the project were presented and video of talks were uploaded for free access on the portal MathNet.Ru. The website of this workshop: The research on the derivation of master equations for open quantum systems starting from Ehrenfest-type relations for averages was performed. A consideration was performed for quantum systems with infinite-dimensional Hilbert space using differential equations that relate the average values ​​of the energy, coordinate and momentum of a quantum oscillator evolving under the simultaneous influence of coherent and incoherent controls and interacting with the reservoir of other quantum oscillators. In this problem, using the method of operational dynamic modeling, a general class of master equations was derived that leads to given relations for means with operators in the dissipator which are linear in the creation and annihilation operators. Dynamical master equation for the density matrix was derived with the generator of Gorini-Kossakowski-Sudarshan-Lindblad form and theory of low density limit was generalized to the case of gas particles with internal degrees of freedom. Simplified expressions for the Lamb shift and the dissipator were derived in the first Born approximation for the case of fast particles. The obtained results were compared with the semiclassical collision model in the stroboscopic limit. The two models were shown to coincide in the limit of infinite temperature for an arbitrary interaction potential. In the case of finite temperature, the solutions do not differ much if the gas concentration is small (nd^3<<1, where n is concentration and d is the characteristic radius of potential) and the thermal energy is much greater than the interaction energy and the characteristic quantum energy in scattering potential (kT>>U, kT >>h^2/md^2). The low density limit in the Born approximation for fast particles was shown to be equivalent to the semiclassical collision model in the stroboscopic approximation. The difference between two solutions was estimated for a finite temperature using the second order perturbation theory. The analysis of mathematical optimal control problems and development of numerical methods for two-level open quantum systems which are determined by the dynamical master equation for density matrix with simultaneous coherent and incoherent controls was continued. Two approaches for solving the problem of time-optimal maximization of the overlap between a certain target density matrix and the final density matrix, which is obtained for the controlled dynamics, were investigated. Reachability and controllability sets, reachable tubes in the Bloch ball for an open two-level quantum system, which is driven simultaneously by coherent and incoherent controls, were numerically constructed. For constructing approximations of reachable and controllability sets, the section method, where solving an optimal control problem is as elementary operation, was adapted. Numerical optimization methods of various types were adopted and used: (a) reduction to finite-dimensional stochastic global optimization without using gradients of the objective functions; (b) GRAPE (GRadient Ascent Pulse Engineering) in combination with the gradient projection method and the projection version of the heavy-ball method in a finite-dimensional search space; (c) the first order Krotov method with regularization and other methods operating in the functional space of controls. The analysis of comparative effectiveness of the optimization methods was performed. Approximations of reachable sets were constructed for different final times, degree of purity of athe initial state, constraints for the values of the coherent and incoherent controls. Application of mashine learning to the time-optimal generation of a target density matrix for a two-level open quantum system whose dynamics is driven by piecewise constant coherent and incoherent controls was analyzed. The problem of controlling quantum systems with similar spectra was analyzed for the example of laser assisted zirconium isotopes separation. Parameters of the Lennard-Jones potential and infrared absorption spectra of the (C_5H_5)_2Zr(OCN)_2 molecule, proposed for the separation of zirconium isotopes by the method of selective retardation of condensation in overcooled gas flow (SILARC), were calculated using Hartree-Fock approximation. A new formula was derived which relates photo-absorption cross sections corresponding to the fundamental mode and its first overtone vibrations via the first and second derivatives of the fundamental mode transition dipole momen. Constraints on the laser intensity, associated with applicability of the SILARC method, were derived. Optimization criterion was formulated for zirconium isotope separation policy taking into account constraints on the control variables. The consideration of the control problem for open quantum systems was peformed, where the control is modeled by a finite set of completely positive, trace-preserving mappings. Based on the established correspondence between the Diophantine equations and such control problems for open quantum systems, the algorithmic unsolvability of the control problem was shown in the case when it is enough to achieve maximum value of the target functional not exactly, an approximate, but with sufficiently high fidelity. In particular, the problem of controlling the state of a system of two quantum oscillators (i.e., a two-mode coherent field) controlled by displacement operators was considered. In connection to the analysis of decoherence and compensation of dissipative dynamics the investigation of subspace of errors generated by the interaction of quantum systems with an infinite-dimensional reservoir was performed and the use of entangled states for eliminating errors was analysed. The search was performed for exactly solvable models of the dynamics describing the system of two interacting quantum oscillators. The possibility of using entangled states for error correction in this system is analysed. For this system with infinite-dimensional Hilbert space we studied the dynamics driven by the Schrodinger equation with Hamiltonian describing the interaction between the system and the reservoir in which the initial separable quantum states become entangled. The action of the unitary group defined by the Schrodinger equation for the composite quantum system is explicitly obtained. The orbits of the *-automorphism group produced by this unitary group were studied. A family of infinite-dimensional projectors was constructed for which the action of the group of automorphisms generates an operator subspace containing the identical operator, so that the obtained operator space is an operator system. It was shown how the application of coherent states and squeezed coherent states allows to construct examples of operator systems in infinite-dimensional spaces and corresponding orthogonal projectors, the action of which maps these systems into a one-dimensional subspace. It was proved that the found projector is maximal in the sense that there is no projector having the same property and which exceeds it in the sense of spectral order. For the constructed operator system a set of vectors in the Hilbert space is found allowing to performing coding for erroc correction. Communication protocols for transmitting public data and distributing a secret key to one or several recipients, where entanglement is considered as a resource, were considered. Estimates were obtained about the data rate for approximate cloning, broadcasting, and entanglement dilution. An estimate was obtained for the coherence of a dual ensemble with a chaotic mean state through the uncertainty of the rank-one observable. The concept of weak uncertainty of the observable was introduced, which considers the uncertainty of the measurement results with respect to the orthonormal basis. An estimate was obtained for the weak uncertainty of the dual observable through the coherence of an ensemble of pure states with a chaotic mean state. It was shown that the coherence of the ensemble used and the weak uncertainty of the observable are necessary for such an important application of quantum technologies as the quantum key distribution, while weak uncertainty makes sense as taking into account attenuation in the communication line. The security analysis of the BB84 quantum key distribution protocol with detection-efficiency mismatch was completed. In 2018, the security of the BB84 quantum key distribution protocol detection-efficiency mismatch was proved and the formula for the maximal achievable secret key generation rate was derived. But this was done under the assumption of a single-photon source. However, in practice, quantum cryptography usually uses not single-photon sources, but sources of coherent states. The number of photons in a coherent state is distributed according to the Poisson law, i.e., multi-photon pulses may occur. This allows the enemy to carry out various attacks (for example, the photon number splitting attack), which make multi-photon pulses unreliable. This must be taken into account in the calculation of the achievable secret key generation rates. In 2019, the case of using sources of coherent states instead of single-photon sources was considered. For this case, the decoy state method for the quantum cryptography protocol BB84 for the case of mismatched efficiencies of single-photon detectors: estimations for the number of multi-photon pulses and the number of errors in single-photon pulses were developed. The assumptions of the method were analysed and justified in detail.



1. Bochkov M.K., Trushechkin A.S. Security of quantum key distribution with detection-efficiency mismatch in the single-photon case: Tight bounds Physical Review A, Vol. 99, P. 032308 (year - 2019).

2. Filippov S.N., Kuzhamuratova K.V. Quantum informational properties of the Landau–Streater channel Journal of Mathematical Physics, Vol. 60, P. 042202 (year - 2019).

3. Morzhin O.V., Pechen A.N. Krotov method for optimal control of closed quantum systems Успехи математических наук, 74:5(449), 83–144 (year - 2019).

4. Morzhin O.V., Pechen A.N. Minimal time generation of density matrices for a two-level quantum system driven by coherent and incoherent controls International Journal of Theoretical Physics, (year - 2019).

5. Morzhin O.V., Pechen A.N. Maximization of the overlap between density matrices for a two-level open quantum system driven by coherent and incoherent controls Lobachevskii Journal of Mathematics, 40:10, 1532–1548 (year - 2019).

6. Kronberg D.A. Coherence of quantum ensemble as a dual to uncertainty for a single observable Lobachevskii Journal of Mathematics, 40:10, 1507–1515 (year - 2019).

7. Filippov S.N., Semin G.N., Pechen A.N. Quantum master equations for a system interacting with a quantum gas in the low-density limit and for the semiclassical collision model Physical Review A, - (year - 2019).

8. Bondar D.I., Pechen A.N. Uncomputability and complexity of quantum control Scientific Reports, - (year - 2019).

9. G.G. Amosov, A.S. Mokeev, A.N. Pechen Non-commutative graphs and quantum error correction for a two-mode quantum oscillator Quantum Information Processing, - (year - 2019).

Annotation of the results obtained in 2017
In the course of the project in 2017, important results were achieved to solve the fundamental scientific problem of developing mathematical methods for problems of quantum technologies and for theory of control of open quantum systems, including the development of methods for deriving dynamic equations for controlled open quantum systems, the study of decoherence of many-particle quantum systems, the establishment of a connection between quantum uncertainty, quantum coupling, and the development of methods for controlling quantum systems with similar spectra. The dynamics of an open quantum system with a time-dependent free Hamiltonian was considered. For the weak interaction with the reservoir, master equations describing the dynamics of an n-level atom interacting with incoherent and, in general, non-thermal radiation, and a quantum oscillator interacting with phonons, in both cases at the same time under the influence of coherent control, were obtained. An algorithm for the synthesis of control strategies for open quantum systems is developed, explicitly taking into account the given constraints on the form of the control action. A coherent control of the dynamics of quantum systems by electromagnetic radiation and incoherent control by a thermostat are considered under strong constraints on controls. In applications it is often very difficult to achieve high management efficiency because of the limitations on the shape of the control. At the same time, the presence of restrictions on available classes of control actions can lead to their cheaper experimental implementation. For example, relatively cheap and reliable telecommunications equipment provides generation of a sequence of arbitrary length, consisting of simple laser pulses. In connection with this problem, mathematical methods for manipulating a quantum system using highly limited controls have been developed, when a controller can only take a finite number of values, as for example, "zero" and "one" in telecommunication lasers. The applications of control strategy synthesis algorithms to controlling quantum systems with similar spectra are demonstrated on the example of the laser separation of boron isotopes by using profiled laser pulses in the method of selective inhibition of condensation in supercooled gas jets. The objective function for the optimal isotope separation problem within the framework of this method was constructed. The isotope separation process in an external laser field was modeled using the solution of the transport equations for the relative mole fractions of different types of particles in the flow. The influence of the shape of the laser pulse on the efficiency of separation was studied. It was shown that not only the spectrum of a single pulse plays an important role, but also the their periodicity. A squaring parametrization of sets of quantum states obeying symmetries was developed. This parametrization automatically ensures positivity of the density matrix. Moreover, it explicitly preserves the tensor product structure of a compound system, which is important for investigating decoherence in many-particle systems. On the basis of this parametrization, an equation for the boundary of the set of quantum states was derived. The merits of the developed technique were illustrated with several examples. In particular, Werner states of three qubits constrained by additional symmetries, such as T-invariance or permutation invariance, were constructed. An analysis of decoherence in quantum integrable and nonintegrable systems of various types was performed. An extended eigenstate decoherence hypothesis (EDH) was formulated. Previously known version of EDH stated that a density matrix of a subsystem obtained by partial tracing from the projector to the eigenstate of the closed system is a classical-like state. This form of EDH, however, did not address the fact that a typical decoherence time scale is small – orders of magnitude smaller than thermalization time scales. Extended EDH fills this gap. It concerns nondiagonal matrix elements of special operators acting in a subsystem, quantumness witnesses, which have highly nonclassical eigenstates. The extended EDH states that nondiagonal matrix elements of these operators in an eigenstate basis are small as long as the energy difference of corresponding eigenstates is smaller than the inverse decoherence time. We demonstrated that the EDH and the extended EDH are satisfied in model systems of spin 1/2, both integrable and nonintegrable. A numerical characteristic of the quantum uncertainty for an ensemble of two pure nonorthogonal quantum states of dimension two was introduced. The problem of extracting information from a set of non-orthogonal quantum states was studied. This problem is related to applications in quantum cryptography, where legitimate users evaluate interceptor's information using states that have reached the receiving side. Estimates of the interceptor's information were obtained in the case when its action is limited by varying the intensity of the coherent states at the receiving side. Information constraints were studied in the situation when users receive parts of two pure non-orthogonal quantum states and extract information from them.



Annotation of the results obtained in 2018
In 2018, the work was carried out on solving problems of controlling open quantum systems, quantum systems with similar spectra on the example of laser separation of boron isotopes, study of compensation of dissipative dynamics and preservation of coherence of quantum systems, applications of the quadratic parametrisation of density matrices to solve problems of quantum information theory and quantum theory of condensed matter, solving problems of quantum cryptography. Seven articles were published, several articles were submitted to journals and now under review, others are in preparation. The results were presented on about 25 talks of the participants of the project. The problem of transferring a given initial state of an open quantum system to a given final state under the simultaneous influence of coherent (which enters in the Hamiltonian) and incoherent (which enters in the dissipator) controls was considered. This problem was solved for two-level quantum systems by reducing it to a series of optimal control problems, each of which is considered with its own fixed final time and with terminal objective functional taking into account the terminal constraint with the target density matrix. A decreasing sequence of terminal times was considered. For solving each such a problem with a fixed terminal time, the Pontryagin maximum principle was applied along with a two-parameter gradient projection method which allows for a search in the functional space of controls. The structure of optimal controls, minimal time, non-uniqueness of optimal control depending on the initial and final states were investigated. The problem of controlling quantum systems with similar spectra was considered and solved for the example of finding optimal thermodynamic (pressure and temperature) and quantum (laser pulse shape) conditions for boron isotopes separation. Laser pulse shape is the coherent control, while pressure and temperature, as well as geometry and operational characteristics of the experimental setup are incoherent controls. Constraints on the laser pulse spectrum, photo-absorption spectrum, resonance and non-resonance absorption in the separation cell, as well as relations between geometry of the separation cell, boundary layer model, and condensation rate were considered. Optimization problem, where the objective function expresses requirement of maximal production rate with the lowest energy consumption, under these constraints was solved. The maximum principle for this problem was obtained. A new formula for the excitation rate of target isotopologues by a laser pulse with an arbitrary spectrum was derived. It was shown that, provided that positions of the resonance peaks, corresponding to different combinations of chlorine isotopes, coincide with the related peaks of the CO2 laser pulse spectrum, continuous excitation is more optimal than multi-mode pulsed irradiation with excitation and absorption modes mismatch at least 0.02cm-1. Ranges of constraints on the laser beam intensity in the case of multimode irradiation, corresponding to measurements of BCl3 photo-absorption spectrum in overcooled supersonic gas flow were found. The parameters of the vacuum chamber of the experimental setup were obtained by solving the system of matter, momentum and energy balance equations. As part of the study of compensating the dissipative dynamics and preserving the coherence of a quantum system, the subspaces of errors generated by interaction with the reservoir were investigated, and the use of entangled states to eliminate errors was analysed. The reducible unitary representations of the commutative group of rotations and the non-commutative Heisenberg-Weyl group in the tensor product of two Hilbert spaces of any finite dimension, modelling the dynamics of an open quantum system such that it entangles the system and the reservoir, were determined. Orthogonal projectors were constructed which are the sums of one-dimensional separable states such that the linear span of the orbits of an automorphic action of the unitary group contains an identical operator and is thus an operator system. The obtained operator system is a set of admissible errors determined by the quantum dynamics. For this operator system sets of vectors in the Hilbert space were found which allow to perform an encoding that eliminates errors. It was shown that such a coding requires using entangled states. Quantum control of open quantum systems was studied when the controller is modelled by a finite set of completely positive trace-preserving maps (incoherent control). A correspondence was established between such control problems and Diophantine equations. This correspondence was applied to determine the degree of complexity of control tasks for open quantum systems. It was shown that some control problems involving as little as two quantum particles are NP-complete. The low density limit and the collisional model, which describe Markovian dynamics of a quantum system interacting with a diluted gas of particles were compared. The equivalence of dissipators in both theories up to a factor in the case of fast particles was proved. The applicability of stroboscopic limit was shown to be equivalent to the applicability of Born approximation for fast particles. This allows one to simplify the derivation of the master equation since the motion of particles in the collisional model is classical. The quadratic parametrization of density matrices of quantum systems was analysed and applied to problems of quantum information theory and the condensed matter, namely, to diagnose the separability of density matrices of quantum systems with symmetries. The diagnostics of separability refers to the calculation of the distance between the density matrix of the composite system and the space of separable density matrices. The entanglement of special states of a system of several qubits, divided into two subsystems, was analysed. As part of the study the quantum entanglement, states with a given value of total spin, with global SU (2) symmetry and with symmetry with respect to permutation of the qubits (Werner states), were considered. A formula was obtained for one of the characteristics of entanglement, purity of the quantum state, i.e., trace of the square of the reduced density matrix. The resulting formula allows to effectively obtain the numerical values ​​of the degree of purity for sufficiently large systems. An explicit attack on the B92 protocol with strong reference state was constructed and a critical error was obtained, for which this attack is possible. The upper bounds on the secret key rate were provided for various channel lengths, an arbitrary inner product of the initial states, and various restrictions on the intensity of the states of the receiver. A modification of the active beam splitting attack for the differential phase shift quantum key distribution protocol was constructed. For this attack, the dependence of the secret key rate on the channel length with arbitrary parameters of the protocol was obtained. The optimal value of the intensity for coherent states of the legitimate users and the maximum key rate were obtained if the eavesdropper uses this attack. The values of the mutual information of the transmitter and receiver for the ensemble of two non-orthogonal states were obtained using a repetition code with arbitrary block length, both for individual and collective measurements. The estimates were obtained for the degree of mixture of the states in the picture dual to the case of individual measurements. The security of the BB84 quantum key distribution protocol with detectors with different efficiencies was proved and accurate estimates for the achievable secret key generation rate were derived. To obtain this result, a new method was developed, based on an analytical solution of the problem of minimizing the quantum relative entropy of coherence.



1. Lyakhov K.A., Pechen A.N., Lee H.-J. The efficiency of one-line versus multi-line excitation of boron isotopes within the method of selective laser assisted retardation of condensation AIP Advances, Vol. 8, Iss. 9, P. 095325 (2018) (year - 2018).

2. Il'in N., Shpagina E., Uskov F., Lychkovskiy O. Squaring parametrization of constrained and unconstrained sets of quantum states Journal of Physics A: Mathematical and Theoretical, Vol. 51, Iss. 8, P. 085301 (year - 2018).

3. Kronberg D.A., Kurochkin Y.V. Role of intensity fluctuation in quantum cryptograph based on coherent states Quantum Electronics, Vol. 48, Iss. 9, P. 843–848 (2018) (year - 2018).

4. Avanesov A.S., Kronberg D.A., Pechen A.N. Active beam splitting attack applied to differential phase shift quantum key distribution protocol P-Adic Numbers, Ultrametric Analysis and Applications, Vol. 10, Iss. 3, P. 222–232 (2018) (year - 2018).

5. Amosov G.G., Mokeev A.S. On non-commutative operator graphs generated by covariant resolutions of identity Quantum Information Processing, Vol. 17, Iss. 12, P. 325 (2018) (year - 2018).

6. Lyakhov K.A., Pechen A.N. Целевая функция в задаче оптимального лазерного разделения изотопов методом селективного торможения Квантовая вероятность, Итоги науки и техн. Сер. Соврем. мат. и ее прил. Темат. обз., Т. 151. М.: ВИНИТИ РАН, Т. 151. М.: ВИНИТИ РАН, 2018. С. 62-66 (year - 2018).

7. Amosov G.G., Mokeev A.S. On Non-Commutative Operator Graphs Generatedby Reducible Unitary Representation of the Heisenberg-Weyl Group International Journal of Theoretical Physics, - (year - 2018).