Vol 40, No 10 (2019)

We obtain sufficient conditions on kernels of quantum states under which Wigner functions, optical quantum tomograms and linking their formulas are correctly defined. Our approach is based upon the Sobolev Embedding theorem. The transition probability formula and the fractional Fourier transform are discussed in this framework.

Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct the probability representation of the completely positive maps. In this representation, any completely positive map of qubit state density matrix is identified with the set of classical coin probability distributions. Examples of the maps of qubit states are studied in detail. The evolution equation of quantum states is written in the form of the classical-like kinetic equation for probability distributions identified with qubit state.

We consider the transformation of the initial data space for the Schrödinger equation. The transformation is generated by nonlinear Schrödinger operator on the segment [−π, π] satisfying the homogeneous Dirichlet conditions on the boundary of the segment. The potential here has the type \(\xi(u)=\left(1+|u|^{2}\right)^{\frac{p}{2}} u\), where u is an unknown function, p ≥ 0. The Schrödinger operator defined on the Sobolev space \(H_0^2([-\pi, \pi])\) generates a vector field \({\rm{v}}:H_0^2([-\pi, \pi])\rightarrow{H}\equiv{L_2}(-\pi, \pi)\). First, we study the phenomenon of global existence of a solution of the Cauchy problem for p ∈ [0, 4) and, second, the phenomenon of rise of a solution gradient blow up during a finite time for p ∈ [4, +∞). In second case we study qualitative properties of a solution when it approaches to the boundary of its interval of existence. Moreover, we define a solution extension through the moment of a gradient blow up using both the one-parameter family of multifunctions and the one-parameter family of probability measures on the initial data space of the Cauchy problem. We show that this extension describes the destruction of a solution as the destruction of a pure quantum state and the transition from the set of pure quantum states into the set of mixed quantum states.

In the paper we introduce the notion of a local group and define a Fell bundle over a local group and a *-representation of a local group into a C*-algebra. We show that with every partial representation of a group the local group is connected as well as the respective Fell bundle. Also we show that the Cuntz and the Cuntz-Toeplitz algebras are local group graded.

We introduce new supplementary data to the set of the eigenvalues, to determine uniquely potential matrix in the inverse problem for Dirac canonical operator. Besides, we obtain others uniqueness theorems in inverse problems, which are the analogues of well-known Borg, Marchenko and McLaughlin-Rundell theorems in inverse Sturm-Liouville problems.

The coherence of an ensemble of quantum states is considered — a quantity having a similarity with the uncertainty of the quantum observable that arises from the entropic uncertainty relations. We give estimates for the coherence of the ensemble through the uncertainty of the dual observable and the weak uncertainty of the observable through the coherence of the ensemble. It is shown that this numerical characteristic is closely connected with the relative entropy of discord for the corresponding classical-quantum state.

In the paper we study linear operators on complex Hilbert spaces which are strong real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We compare order properties of the orthogonal and of the strong real-orthogonal projections. We prove that the set of all strong real-orthogonal the projections in the complex space is the quantum pseudo logic. We also prove an analogy of Gleason’s theorem.

For a given positive operator G we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with \(\sqrt G \)-bounded and \(\sqrt G \)-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by G (as an energy observable) and the coincidence of the second cone with the completion of the cone of completely positive linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond norm for any energy observable. Some properties of the maps belonging to the introduced cones are described. In particular, the corresponding generalization of the Kretschmann-Schlingemann-Werner theorem is obtained. We also give a nonconstructive description of the completion of the set of all Hermitian-preserving completely bounded linear maps w.r.t. the energy-constrained diamond norm.

An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequalities between the exact and approximate rates.

A covariant definition of the Levy Laplacian on an infinite dimensional manifold is introduced. It is shown that a time-depended connection in a finite dimensional vector bundle is a solution of the Yang—Mills heat equations if and only if the associated flow of the parallel transports is a solution of the heat equation for the covariant Levy Laplacian on the infinite dimensional manifold.

Paper introduces a concept of fully symmetric guided electromagnetic waves. The waves are considered in a plane dielectric slab shielded with infinitely conducting walls. The permittivity of the slab is given by a real diagonal tensor; the permeability is a constant scalar quantity. This concept includes particular cases of transverse-electric and transverse-magnetic guided waves as well as novel (more complicated) types of guided waves. The dispersion equation is found and studied. A discussion of possible ways to develop the suggested concept is also given.

This paper studies the propagation of steady-state oscillations in an irregular rectangular waveguide. The irregularity of the waveguide is caused by the presence inside it of a metallic inclusion in the form of a cylindrical inductive cylinder. To solve the problem in a complete electrodynamic formulation, it is necessary to investigate the boundary problem for the system of Maxwell equations. To study the waveguide system consisting of a waveguide with a well-conducting inclusion, the method of integral equations was applied. The cores of the integral equations are defined through the Green functions of the unfilled waveguide, written in terms of the waveguide modes. Algorithms for their calculation are developed on the basis of the selection of a logarithmic singularity, and algorithms for summing up the series belonging to them are created. The possibilities of the method of integral equations are illustrated with examples of calculating the reflection and transmission coefficients from inductive pins.

The diffraction problem of plane electromagnetic wave by the non-coordinate periodic media interface is reduced to an infinite set of linear algebraic equations for Floquet coefficients of scattered field. Two-dimensional case of diffraction problem is considered in detail. In the three-dimensional case, special attention is paid to problems of program implementation of algorithms of approximate solving of ISLAE.

The problem on normal waves in a radially inhomogeneous dielectric rod is considered. This problem is reduced to the boundary eigenvalue problem for longitudinal components of electromagnetic field in Sobolev spaces. To find the solution, we use the variational formulation of the problem. The variational problem is reduced to study of an operator-function. Discreteness of the spectrum is proved and distribution of the characteristic numbers of the operator-function on the complex plane is found.

The problem of elastic wave diffraction by an isotropic fluid-saturated porous layer is considered. It is assumed that the porosity is constant and elastic parameters are continuously varying deep into the layer. The original problem is reduced to the boundary value problem for ordinary differential equations of the given form. The finite-difference scheme for the boundary value problem is obtained. The theorem is proved that the error of approximation of the solution has a second order of accuracy. Numerical results confirming theoretical conclusions are given.