Vol 40, No 10 (2019)
- Year: 2019
- Articles: 29
- URL: https://journals.rcsi.science/1995-0802/issue/view/12771
Article
On Definition of Quantum Tomography via the Sobolev Embedding Theorem
Abstract
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.
On Linear Structure of Non-commutative Operator Graphs
Abstract
We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.
Probability Representation of Quantum Channels
Abstract
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.
Rearrangements of Tripotents and Differences of Isometries in Semifinite von Neumann Algebras
Abstract
Let τ be a faithful normal semifinite trace on a von Neumann algebra ℳ, and ℳu be a unitary part of ℳ. We prove a new property of rearrangements of some tripotents in ℳ. If V ∈ ℳ is an isometry (or a coisometry) and U − V is τ-compact for some U ∈ ℳu then V ∈ ℳu. Let ℳ be a factor with a faithful normal trace τ on it. If V ∈ ℳ is an isometry (or a coisometry) and U − V is compact relative to ℳ for some U ∈ ℳu then V ∈ ℳu. We also obtain some corollaries.
Phase Flows Generated by Cauchy Problem for Nonlinear Schrödinger Equation and Dynamical Mappings of Quantum States
Abstract
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.
On Quantum Operations of Photon Subtraction and Photon Addition
Abstract
The conventional photon subtraction and photon addition transformations, ϱ → taϱa† and ϱ → ta†ϱa, are not valid quantum operations for any constant t > 0 since these transformations are not trace nonincreasing. For a fixed density operator ϱ there exist fair quantum operations, \(\mathcal{N}_{-}\) and \(\mathcal{N}_{+}\), whose conditional output states approximate the normalized outputs of former transformations with an arbitrary accuracy. However, the uniform convergence for some classes of density operators ϱ has remained essentially unknown. Here we show that, in the case of photon addition operation, the uniform convergence takes place for the energy-second-moment-constrained states such that tr[ϱH2] ≤ E2 < ∞, H = a†a. In the case of photon subtraction, the uniform convergence takes place for the energy-second-moment-constrained states with nonvanishing energy, i.e., the states ϱ such that tr[ϱH] ≥ E1 > 0 and tr[ϱH2] ≥ E2 < ∞. We prove that these conditions cannot be relaxed and generalize the results to the cases of multiple photon subtraction and addition.
On a Grading of the Cuntz Algebra
Abstract
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.
Additivity of Quadratic Maps on JB Algebras
Abstract
The problem of automatic additivity of Jordan type homomorphisms has received a great deal of attention in theory of operator algebras as well in ring theory. We study additivity of quadratic maps, that is the maps between Jordan Banach algebras that preserve the quadratic product (a, b) → aba. The main result shows that any continuous quadratic bijective map between unital Jordan Banach algebras that is linear on associative subalgebras is automatically additive. This contributes to the Borification program in quantum theory and Mackey-Gleason problem on linearity of quasi-linear maps.
The Uniqueness Theorems in the Inverse Problems for Dirac Operators
Abstract
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.
Coherence of Quantum Ensemble as a Dual to Uncertainty for a Single Observable
Abstract
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.
H-Theorem for Systems with an Interaction Invariant Distribution Function
Abstract
H-theorem gives necessary conditions for a system to evolve in time with a non-diminishing entropy. In a quantum case the role of H-theorem plays the unitality criteria of a quantum channel transformation describing the evolution of the system’s density matrix under the presence of the interaction with an environment. Here, we show that if diagonal elements of the system’s density matrix are robust to the presence of interaction the corresponding quantum channel is unital.
Strong Projections in Hilbert Space and Quantum Logic
Abstract
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.
Maximization of the Overlap between Density Matrices for a Two-Level Open Quantum System Driven by Coherent and Incoherent Controls
Abstract
The article considers a two-level open quantum system evolving under the action of coherent and incoherent controls using the general control method proposed in Phys. Rev. A. 73, 062102 (2006). Coherent control determines the Hamiltonian aspects of the dynamics whereas incoherent control determines the dissipative aspects. The goal is to find controls which steer the initial density matrix into a state which maximizes overlap with a predefined target density matrix. The controlled dynamics is represented as evolution in the Bloch ball and is analyzed analytically using Pontryagin maximum principle and numerically using optimization either in the functional space of controls (with the conditional and projected gradient methods) or through reduction to a finite-dimensional control space.
On Completion of the Cone of Completely Positive Linear Maps with Respect to the Energy-Constrained Diamond Norm
Abstract
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.
Energy-Сonstrained Diamond Norms and Quantum Dynamical Semigroups
Abstract
In the developing theory of infinite-dimensional quantum channels the relevance of the energy-constrained diamond norms was recently corroborated both from physical and information-theoretic points of view. In this paper we study necessary and sufficient conditions for differentiability with respect to these norms of the strongly continuous semigroups of quantum channels (quantum dynamical semigroups). We show that these conditions can be expressed in terms of the generator of the semigroup. We also analyze conditions for representation of a strongly continuous semigroup of quantum channels as an exponential series converging w.r.t. the energy-constrained diamond norm. Examples of semigroups having such a representation are presented.
Non-Markovian Evolution of Multi-level System Interacting with Several Reservoirs. Exact and Approximate
Abstract
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.
Higher-order Corrections to the Redfield Equation with Respect to the System-bath Coupling Based on the Hierarchical Equations of Motion
Abstract
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 required. Here we propose a general method of derivation of higher-order corrections to the Redfield quantum master equation based on the hierarchical equations of motion (HEOM). Also we derive conditions of validity of the Redfield equation as well as the additional secular approximation for it.
Levy Laplacian on Manifold and Yang—Mills Heat Flow
Abstract
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.
On the Eigen Frequencies of Rectangular Resonator with a Hole in the Wall
Abstract
The rectangular waveguide is attached to a hole in a wall of rectangular resonator and the corresponding to the hole part of the waveguide boundary is cross-section of the waveguide. The problem of excitation of the resonator by an eigen wave of the waveguide is investigated. The condition on a hole is obtained from the condition defining the wave in the waveguide, outgoing from cross-section. The initial diffraction problem is reduced to some infinite set of linear algebraic equations. Numerical experiment shows that the dependence of expansion coefficients of the field in the resonator on frequency of exciting wave has resonant nature. We propose to use the real values of the resonance frequencies of the running on the hole wave as the initial approximations for eigen frequencies of the resonator with a hole in the wall.
Fully Symmetric Guided Electromagnetic Waves in a Shielded Plane Dielectric Slab
Abstract
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.
Reconstruction of Inhomogeneities in a Hemisphere from the Field Measurements
Abstract
In this paper a scalar inverse problem on a hemisphere is considered. The incident field is radiated by point source located outside the body. We are looking for a solution of the inverse problem using the measurements of the field outside the body. We suggest an original approach to solve the problem. The various numerical results of solving the problem are presented.
The Method of Integral Equations in Problems of Wave Diffraction in Waveguides
Abstract
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.
Problem of Coupled Electromagnetic TE-TE Wave Propagation in a Layer Filled with Nonlinear Medium with Saturation
Abstract
Propagation of coupled electromagnetic TE-TE wave in a nonlinear plane layer is considered. The layer is located between two half-spaces with constant permittivities. The permittivity in the layer is described by nonlinearity with saturation. The physical problem is reduced to a nonlinear two-parameter eigenvalue problem for a system of (nonlinear) ordinary differential equations. Existence and uniqueness of solution to the two-parameter eigenvalue problem is proved. Iteration method for solving given problem is presented. Convergence theorem of the iteration method is proved.
Infinite Sets of Linear Algebraic Equations in the Problems of Diffraction of Electromagnetic Waves by the Non-Coordinate Periodic Media Interfaces
Abstract
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.
Qualitative Analysis of Quadratic Polynomial Dynamical Systems Associated with the Modeling and Monitoring of Oil Fields
Abstract
The behavior and properties of solutions of two-dimensional quadratic polynomial dynamical system on the phase plane of variables and time are considered. A complete qualitative theory is constructed which includes the analysis of all singular points and the features of solutions depending on all parameters of the problem. A main result is that, with the discriminant criteria created on the basis of the growth model constructed in the present study, it is possible to formulate practical recommendations for regulating and monitoring the process of waterflooding and the development of an oil field.
Mathematical Theory of Normal Waves in Radially Inhomogenous Dielectric Rod
Abstract
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 Faster Methods for Computing Bessel Functions of the First Kind of an Integer Order with Application to Graphic Processors
Abstract
Algorithms for fast computations of the Bessel functions of an integer order with required accuracy are considered. The domain of functions is split into two intervals: 0 ≤ x ≤ 8 and x > 8. For the finite interval, expansion in the Chebyshev polynomials is applied. An optimal algorithm for computing functions J0(x) and J1(x) is presented. It is shown that the sufficient number of mathematical operations equals 15 for computing the function J0(x) and 16 for computing the function J1(x) in the interval x ≤ 8 with the approximation error O(10−6). Several algorithms for approximation of the functions Jn(x) at n > 1 are presented. The increase in speed of computations of the Bessel functions obtained through using our in-house methods in place of the Toolkit library is evaluated. Graphs showing the improvement of performance are presented.
Solving the Problem of Elastic Waves Diffraction by a Fluid-Saturated Porous Gradient Layer Using a Second-Order Finite-Difference Scheme
Abstract
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.