# Anton TrushechkinRussian Academy of Sciences | RAS · Steklov Mathematical Institute, Department of Mathematical Physics

Anton Trushechkin

Ph.D. (Phys&Math)

## About

64

Publications

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## Publications

Publications (64)

Detection-efficiency mismatch is a common problem in practical quantum key distribution (QKD) systems. Current security proofs of QKD with detection-efficiency mismatch rely either on the assumption of the single-photon light source on the sender side or on the assumption of the single-photon input of the receiver side. These assumptions impose res...

The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here, we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the h...

The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the hi...

The dynamical convergence of a system to the thermal distribution, or Gibbs state, is a standard assumption across all of the physical sciences. The Gibbs state is determined just by temperature and energies of the system. However, at decreasing system sizes, i.e., for nanoscale and quantum systems, the interaction with their environments is not ne...

The dynamical convergence of a system to the thermal distribution, or Gibbs state, is a standard assumption across all of the physical sciences. The Gibbs state is determined just by temperature and the system's energies alone. But at decreasing system sizes, i.e. for nanoscale and quantum systems, the interaction with their environments is not neg...

In the framework of theory of open quantum systems, we derive quantum master equations for the ultrastrong system-bath coupling regime and, more generally, the strong-decoherence regime. In this regime, the strong decoherence is complemented by slow relaxation processes. We use a generalization of the Foerster and modified Redfield peturbation theo...

Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we give an alternative derivation of the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems, as well as higher-order corrections to it. This derivation naturally considers initially correlated s...

Derivation of a quantum master equation for a system weakly coupled to a bath which takes into account nonsecular effects, but nevertheless has the mathematically correct Gorini-Kossakowski-Lindblad-Sudarshan form (in particular, it preserves positivity of the density operator) and also satisfies the standard thermodynamic properties is a known lon...

Дается вывод квантового линейного кинетического уравнения Редфилда, широко используемого в теории открытых квантовых систем, с помощью идей, использованных Н.Н. Боголюбовым при выводе классических и квантовых нелинейных кинетических уравнений. Данный метод позволяет вычислять совместное состояние системы и резервуара в каждый момент времени, а такж...

Derivation of a quantum master equation for a system weakly coupled to a bath which take into account non-secular effects, but, nevertheless, conserves positivity, is a known long-standing problem in theory of open quantum systems. Though there are several heuristic approaches, here we provide a fully rigorous derivation based on a formalization of...

Quantum cryptography or, more precisely, quantum key distribution (QKD), is one of the advanced areas in the field of quantum technologies. The confidentiality of keys distributed with the use of QKD protocols is guaranteed by the fundamental laws of quantum mechanics. This paper is devoted to the decoy state method, a countermeasure against vulner...

Stream ciphers form one of two large classes of ciphers with private keys in classical cryptography. In this paper, we introduce the concept of a quantum stream cipher. Special types of quantum stream ciphers were proposed earlier by numerous researchers. We prove a general result on the nonexistence of an unconditionally strong quantum stream ciph...

Quantum cryptography or, more precisely, quantum key distribution (QKD), is one of the advanced areas in the field of quantum technologies. The confidentiality of keys distributed with the use of QKD protocols is guaranteed by the fundamental laws of quantum mechanics. This paper is devoted to the decoy state method, a countermeasure against vulner...

We discuss the operational meaning of a commonly accepted security parameter in quantum key distribution, which is based on the trace distance. We separately consider the cases of using a key in a one-time pad and in a computationally secure cipher. Some practical aspects of using the security parameter are also elucidated, which are usually not pa...

The detection-efficiency mismatch is a common problem in practical quantum key distribution (QKD) systems. The security of quantum key distribution in this case is proved only under the assumption that either the output of the sender side or the input to the receiver side are single-photon signals, which impose a restriction over the class of possi...

The properties of solutions of the Gorini–Kossakowski-Sudarshan–Lindblad (GKSL) equation for the density operator (matrix) of a system that has nondegenerate energy spectrum and weakly interacts with a reservoir are considered. Conditions for the existence of solutions for which the density matrix has off-diagonal entries (“coherences”) not tending...

The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is require...

Одним из основных методов описания динамики открытых квантовых систем являются квантовые кинетические уравнения. Эти уравнения описывают динамику приведeнного оператора плотности системы, взаимодействующей с резервуаром. При этом по степеням свободы резервуара проводится усреднение, что не позволяет описывать динамику наблюдаемых резервуара. В данн...

One of the main methods for describing the dynamics of open quantum systems is the method of quantum master equations. These equations describe the dynamics of the reduced density operator of a system interacting with a reservoir. In this case, averaging is performed over the degrees of freedom of the reservoir, which does not allow one to describe...

Förster and modified Redfield theories play one of the central roles in the description of excitation energy transfer in molecular systems. However, in the present state, these theories describe only the dynamics of populations of local electronic excitations or delocalized exciton eigenstates, respectively, i.e., the diagonal elements of the densi...

We discuss the question of the general definition of the production of entropy per unit time for a quantum system governed by the Lindblad equation. The difficulty is as follows: in order to determine the total production of entropy, one must know the entropy flow from the system into the environment. This requires additional information on the env...

We propose a quantum branch-and-bound algorithm based on the general scheme of the branch-and-bound method and the quantum nested searching algorithm and examine its computational efficiency. We also compare this algorithm with a similar classical algorithm on the example of the travelling salesman problem. We show that in the vast majority of prob...

The Redfield equation describes the dynamics of a quantum system weakly coupled to one or more reservoirs and is widely used in theory of open quantum system. However, the assumption of weak system-reservoir coupling is often not fully adequate and higher-order corrections to the Redfield equation with respect to the system-bath coupling is require...

Quantum key distribution (QKD), ensuring the unconditional security of information, attracts a significant deal of interest. An important task is to design QKD systems as a platform for education as well as for research and development applications and fast prototyping new QKD protocols. Here, we present a modular QKD setup driven by National Instr...

One of the challenges in practical quantum key distribution is dealing with efficiency mismatch between different threshold single-photon detectors. There are known bounds for the secret key rate for the BB84 protocol with detection-efficiency mismatch provided that the eavesdropper sends no more than one photon to the legitimate receiver. Here we...

Foerster and modified Redfield theories play one of the central roles in the description of excitation energy transfer in molecular systems. However, in the present state, these theories describe only the dynamics of populations of local electronic excitations or delocalized exciton eigenstates, respectively, i.e., the diagonal elements of the dens...

One of the practical challenges in practical quantum key distribution is dealing with the efficiency mismatch between different threshold detectors. There are known bounds for the secret key rate for the BB84 protocol with the detection efficiency mismatch provided that the eavesdropper sends exactly one photon to the receiver. Here we improve thes...

A necessary and sufficient condition is derived for a density operator to be a stationary solution for a certain class of Lindblad equations in the theory of open quantum systems. This condition is based on the properties of a functional that in some cases corresponds to entropy production. Examples are given where this condition is used to find st...

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equat...

We report the results of the implementation of a quantum key distribution (QKD) network using standard fibre communication lines in Moscow. The developed QKD network is based on the paradigm of trusted repeaters and allows a common secret key to be generated between users via an intermediate trusted node. The main feature of the network is the inte...

Quantum key distribution (QKD) offers a way for establishing information-theoretically secure communications. An important part of QKD technology is a high-quality random number generator (RNG) for quantum states preparation and for post-processing procedures. In the present work, we consider a novel class of prepare-and-measure QKD protocols, util...

Blockchain is a distributed database which is cryptographically protected against malicious modifications. While promising for a wide range of applications, current blockchain platforms rely on digital signatures, which are vulnerable to attacks by means of quantum computers. The same, albeit to a lesser extent, applies to cryptographic hash functi...

We present the results of realizing a quantum key distribution (QKD) network with the use the standard fiber channels in Moscow. The developed QKD network is based on the trusted repeater paradigm. The QKD network allows establishing a common key between users over an intermediate trustworthy node. In our experiment, the developed QKD network conne...

We consider an information reconciliation protocol for quantum key distribution (QKD). In order to correct down the error rate, we suggest a method, which is based on symmetric blind information reconciliation for the low-density parity-check (LDPC) codes. We develop a subsequent verification protocol with the use of $\epsilon$-universal hash funct...

Decoy-state quantum key distribution is a standard tool for long-distance quantum communications. An important issue in this field is the processing the decoy-state statistics taking into account statistical fluctuations (or "finite-key effects"). In this work, we propose and analyse an option for decoy statistics processing, which is based on the...

Quantum key distribution (QKD) is a quantum-proof key exchange scheme which is fast approaching the communication industry. An essential component in QKD is the information reconciliation step, which is used for correcting the quantum channel noise errors. The recently suggested blind reconciliation technique, based on low-density parity-check (LDP...

The semiclassical long-time limit of free evolution of quantum wave packets on the torus is under consideration. Despite of simplicity of this system, there are still open questions concerning the detailed description of the evolution on time scales beyond the Ehrenfest time. One of the approaches is based on the limiting Wigner or Husimi distribut...

We present algorithmic solutions aimed on post-processing for industrial quantum key distribution systems with hardware sifting. The main steps of the procedure are error correction, parameter estimation, and privacy amplification. Authentication of a classical public communication channel is also considered.

The problem of construction of a quantum master equation for a system of
sites weakly coupled to each other and to one or more reservoirs (open quantum
network) is considered. Microscopic derivation of a quantum master equation
requires a diagonalization of the Hamiltonian of the network, which can be a
difficult task. When the inter-site couplings...

Nonselective quantum measurements, i.e., measurements without reading the results, are often considered as a resource for manipulating quantum systems. In this work, we investigate optimal acceleration of the Landau-Zener (LZ) transitions by nonselective quantum measurements. We use the measurements of a population of a diabatic state of the LZ sys...

N. N. Bogolyubov discovered that the Boltzmann-Enskog kinetic equation has microscopic solutions. They have the form of sums of deltafunctions and correspond to trajectories of individual hard spheres. But the rigorous sense of the product of the delta-functions in the collision integral was not discussed. Here we give a rigorous sense to these sol...

As established by N.N. Bogolyubov, the Boltzmann-Enskog kinetic equation admits the so-called microscopic solutions. These solutions are generalized functions (have the form of sums of delta functions); they correspond to the trajectories of a system of a finite number of balls. However, the existence of these solutions has been established at the...

We find explicit soliton-like solutions for the nonlinear
integro-differential Boltzmann-Enskog kinetic equation (for both elastic and
inelastic hard spheres). They are analogues of multisoliton solutions of the
Korteweg-de Vries equation. To our awareness, these are the first classical
explicit solutions of the Boltzmann-Enskog equation. The const...

We construct families of squeezed quantum states on an interval (depending on
boundary conditions, we interpret the interval as a circle or as the infinite
square potential well) and obtain estimates of position and momentum
dispersions for these states. A particular attention is paid to the possibility
of proper localization of a particle in nanos...

The Boltzmann–Enskog kinetic equation is known to describe the irreversible dynamics of a dilute hard sphere gas. From the other side, the Boltzmann–Enskog equation is known to have reversible microscopic solutions. We show that the reversibility or irreversibility property of the Boltzmann–Enskog equation depends on the considered class of solutio...

We study a peculiar semiclassical limit of the dynamics of quantum states on a circle and in a box (infinitely deep potential well with rigid walls) as the Planck constant tends to zero and time tends to infinity. Our results describe the dynamics of coherent states on the circle and in the box at all time scales in semiclassical approximation. The...

We consider the microscopic solutions of the Boltzmann-Enskog equation
discovered by Bogolyubov. The fact that the time-irreversible kinetic equation
has time-reversible microscopic solutions is rather surprising. We analyze this
paradox and show that the reversibility or irreversibility property of the
Boltzmann-Enskog equation depends on the cons...

Quantum cryptography is used to find practical and useful applications. Recently, some first quantum cryptographic solutions became available on the market. For clients, it is important to be able to compare the quality and properties of the proposed products. To this end, one needs to elaborate on specifications and standards of solutions in quant...

A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume
is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation
of classical mechanics. According to the functional approach to mechanics, a state of a s...

In this report we discuss the organization of different levels of nature and
the corresponding space-time structures by the consideration of a particular
problem of time irreversibility. The fundamental time irreversibility problem
consists in the following: how to reconcile the time-reversible microscopic
dynamics and the irreversible macroscopic...

We analyze the role of an instrument in the recently proposed functional formulation of classical mechanics, whose basic equation
is the Liouville equation. Its solution has the delocalization (spreading) property, which is interpreted as irreversibility
on the microlevel. We show that the reversible and recurrent dynamics for a particle can be obs...

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion
of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary
real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility p...

We construct families of squeezed quantum states on an interval and analyze their asymptotic behavior. We study the localization
properties of a kind of such states constructed on the basis of the theta function. For the coordinate and momentum dispersions
of a quantum particle on an interval, we obtain estimates that apply, in particular, to nanos...

Relations between number theory and cryptography are discussed. A general mathematical framework for quantum key distribution based on the concepts of quantum channel and Turing machine is suggested. The security for its special case is proved. The assumption is that the adversary can perform only individual (in essence, classical) attacks. For thi...

A rather general mathematical model of quantum key distribution is constructed and cryptographic stability for its particular case is proved. Cryptographic stability is proved in assumption that the opponent runs only individual (essentially, classical) attacks. For this case, the advantage of quantum cryptography over the classical one is demonstr...

A general mathematical framework for quantum key distribution based on the
concepts of quantum channel and Turing machine is suggested. The security for
its special case is proved. The assumption is that the adversary can perform
only individual (in essence, classical) attacks. For this case an advantage of
quantum key distribution over classical o...

## Projects

Project (1)