Diagonal Form Research Articles

Using linear and nonlinear entanglement witnesses to generate and detect bound entangled states on an IBM quantum processor

Abstract We investigate bound entanglement in three-qubit mixed states which are diagonal in the Greenberger-Horne-Zeilinger (GHZ) basis. Entanglement in these states is detected using entanglement witnesses and the analysis focuses on states exhibiting positive partial transpose (PPT). We then compare the detection capabilities of optimal linear and nonlinear entanglement witnesses. In theory, both linear and nonlinear witnesses produce non-negative values for separable states and negative values for some entangled GHZ diagonal states with PPT, indicating the presence of entanglement. Our experimental results reveal that in cases where linear entanglement witnesses fail to detect entanglement, nonlinear witnesses are consistently able to identify its presence. Optimal linear and nonlinear witnesses were generated on an IBM quantum computer and their performance was evaluated using two bound entangled states (Kay and Kye states) from the literature, and randomly generated entangled states in the GHZ diagonal form. Additionally, we propose a general quantum circuit for generating a three-qubit GHZ diagonal mixed state using a six-qubit pure state on the IBM quantum processor. We experimentally implemented the circuit to obtain expectation values for three-qubit mixed states and compute the corresponding entanglement witnesses.

Pulsar timing array harmonic analysis and source angular correlations

Gravitational waves (GWs) influence the arrival times of radio signals coming from pulsars. Here, we investigate the harmonic space approach to describing a pulsar’s response to GWs. We derive and discuss the “diagonalized form” of the response, which is a sum of spin-2-weighted spherical harmonics of the GW direction multiplied by normal (spin-weight 0) spherical harmonics of the pulsar direction. We show how this allows many useful objects, for example, the Hellings and Downs two-point function, to be easily calculated. The approach also provides a clear description of the gauge dependence. We then employ this harmonic approach to model the effects of angular correlations in the sky locations of GW sources (sometimes called “statistical isotropy”). To do this, we construct ensembles made up of many Gaussian subensembles. While each of the individual subsensembles breaks rotational invariance, the full ensemble is rotationally invariant. Using harmonic techniques, we compute the cosmic covariance and the total covariance of the Hellings and Downs correlation in these models. The results may be used to assess the impact of angular source correlations on the Hellings and Downs correlation, and for optimal reconstruction of the Hellings and Downs curve in models where GW sources have correlated sky locations. Published by the American Physical Society 2024

Special cubic zeros and the dual variety

AbstractLet be a diagonal cubic form over in six variables. From the dual variety in the delta method of Duke–Friedlander–Iwaniec and Heath‐Brown, we unconditionally extract a weighted count of certain special integral zeros of in regions of diameter . Heath‐Brown did the same in four variables, but our analysis differs and captures some novel features. We also put forth an axiomatic framework for more general .

Self-similar collapse in Painlevé–Gullstrand coordinates

We report a family of self-similar exact solutions in General Relativity. The solutions are found in a Painleve-Gullstrand coordinate system but can also be transformed smoothly into a diagonal form. The solutions represent a gravitational collapse leading to three possible outcomes, depending on the parameter space: (i) a collapse followed by a bounce and dispersal of the clustered matter distribution, (ii) a rapid collapse followed by a bounce and an eventual re-collapse, and (iii) a standard collapse leading to zero proper volume. Profiles of the energy conditions are studied for all of the scenarios, and it is noted that a bounce is usually associated with a violation of the Null Energy Condition. It is found that more than one null surfaces (apparent horizons) can develop during the collapse. We also discuss that for a general metric tensor having a conformal symmetry, some regions of the parameter space allows a formation of null throat, much like a wormhole. Matching the metric with a Schwarzschild metric in Painleve–Gullstrand form leads to the geodesic equation for a zero energy falling particle in the exterior.

Weak magnetohydrodynamic turbulence theory revisited

Two recent papers, P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021) and Yoon et al., Phys. Plasmas 29, 112303 (2022), utilized in the derivation of the kinetic equation for the intensity of turbulent fluctuations the assumption that the wave spectra are isotropic, that is, the ensemble-averaged magnetic field tensorial fluctuation intensity is given by the isotropic diagonal form, ⟨δBiδBj⟩k=⟨δB2⟩kδij. However, it is more appropriate to describe the incompressible magnetohydrodynamic turbulence involving shear Alfvénic waves by modeling the turbulence spectrum as being anisotropic. That is, the tensorial fluctuation intensity should be different in diagonal elements across and along the direction of the wave vector, ⟨δBiδBj⟩k=12 ⟨δB⊥2⟩k(δij−kikj/k2)+⟨δB∥2⟩k(kikj/k2). In the present paper, we thus reformulate the weak magnetohydrodynamic turbulence theory under the assumption of anisotropy and work out the form of nonlinear wave kinetic equation.

Efficient simulation of potential energy operators on quantum hardware: a study on sodium iodide (NaI)

This study introduces a conceptually novel polynomial encoding algorithm for simulating potential energy operators encoded in diagonal unitary forms in a quantum computing machine. The current trend in quantum computational chemistry is effective experimentation to achieve high-precision quantum computational advantage. However, high computational gate complexity and fidelity loss are some of the impediments to the realization of this advantage in a real quantum hardware. In this study, we address the challenges of building a diagonal Hamiltonian operator having exponential functional form, and its implementation in the context of the time evolution problem (Hamiltonian simulation and encoding). Potential energy operators when represented in the first quantization form is an example of such types of operators. Through systematic decomposition and construction, we demonstrate the efficacy of the proposed polynomial encoding method in reducing gate complexity from O(2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(2^n)$$\\end{document} to O∑i=1rnCr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( \\sum _{i=1}^{r} {} ^nC_r \\right)$$\\end{document} (for some r≪n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ll n$$\\end{document}). This offers a solution with lower complexity in comparison to the conventional Hadamard basis encoding approach. The effectiveness of the proposed algorithm was validated with its implementation in the IBM quantum simulator and IBM quantum hardware. This study demonstrates the proposed approach by taking the example of the potential energy operator of the sodium iodide molecule (NaI) in the first quantization form. The numerical results demonstrate the potential applicability of the proposed method in quantum chemistry problems, while the analytical bound for error analysis and computational gate complexity discussed, throw light on issues regarding its implementation.

The Simultaneous Reduction of a Pair of Unitoid Matrices to Diagonal Form Revisited

The Simultaneous Reduction of a Pair of Unitoid Matrices to Diagonal Form Revisited

From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications

In this paper, we present the connection of two concepts as induced representation and partially reduced irreducible representations (PRIR) appear in the context of port-based teleportation protocols. Namely, for a given finite group G with arbitrary subgroup H, we consider a particular case of matrix irreducible representations, whose restriction to the subgroup H, as a matrix representation of H, is completely reduced to diagonal block form with an irreducible representation of H in the blocks. The basic properties of such representations are given. Then as an application of this concept, we show that the spectrum of the port-based teleportation operator acting on n systems is connected in a very simple way with the spectrum of the corresponding Jucys–Murphy operator for the symmetric group S(n-1)⊂S(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S(n-1)\\subset S(n)$$\\end{document}. This shows on the technical level relation between teleporation and one of the basic objects from the point of view of the representation theory of the symmetric group. This shows a deep connection between the central object describing properties of deterministic PBT schemes and objects appearing naturally in the abstract representation theory of the symmetric group. In particular, we present a new expression for the eigenvalues of the Jucys–Murphy operators based on the irreducible characters of the symmetric group. As an additional but not trivial result, we give also purely matrix proof of the Frobenius reciprocity theorem for characters with explicit construction of the unitary matrix that realizes the reduction in the natural basis of induced representation to the reduced one.

Low-complexity GFDM transmission scheme for low earth orbit satellite communications

As a promising multi-carrier modulation scheme, Generalized Frequency Division Multiplexing (GFDM) has certain advantages in low earth orbit satellite communications. However, the cost of GFDM to obtain these advantages is the self-interference caused by non-orthogonality and the increase of computational complexity. This paper proposes two low-complexity transceiver schemes for GFDM based on matrix sparsity. Firstly, the paper summarizes two GFDM modulation structures, referred to as structure types I and II, based on modulation data. Next, the modulation matrix in both structures is divided into sub-matrices, and the modulation sub-matrix is made sparse by leveraging the properties of the Fourier transform, so as to reduce the complexity of the transmitter. Similarly, by performing sparse processing on the received matrix and converting it into a block diagonal matrix form, the complexity of the receiver can be effectively reduced. The computational costs of our proposed schemes are analyzed in detail and compared with the existing solutions that are known to have the lowest complexity. The simulation results demonstrate that the two proposed schemes can provide satisfactory low-complexity performance depending on the time–frequency structure.

Dual symplectic classical circuits: An exactly solvable model of many-body chaos

We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for a choice of observables.

ESTIMATION OF FATIGUE CURVES USING ORTHOGONAL POLYNOMIALS

The article proposes a methodology for processing fatigue tests carried out to assess the fatigue curve of materials and structural elements, the most important design characteristic of the endurance, service life and reliability of aircraft products, transport engineering and other structures operating under conditions of both regular and irregular loading. The task of estimating the parameters of the fatigue curve is often complicated by the limited scope of fatigue tests, significant scattering of cyclic durability, uneven duplication of experiments at a given level of alternating stress amplitudes, the presence of censoring, especially at low levels, and other factors. which necessitates the use of the weighted least squares method. These circumstances lead to the violation of the conditions for using standard methods of regression analysis and the least squares method. This is especially true for the confidence assessment of the fatigue curve, the accuracy of which significantly depends on these factors. For this purpose, the article proposes to apply the procedure of orthogonalization of factor characteristics to estimate the parameters of the fatigue curve, which reduces the errors arising from standard approaches and makes it possible to obtain accurate confidence intervals for durability. A special feature of the technique is the use of orthogonal polynomials in statistical procedures, which makes it possible to modify the covariance matrix of estimates by converting it to a diagonal form, regardless of the spread of the weight parameters of the original data. These transformations make it possible to perform statistical procedures for point and confidence estimation when processing the results of fatigue tests. To test the methodology, a statistical analysis of a large volume (about 200 samples) of fatigue tests of samples with varying degrees of stress concentration from titanium alloy VT3-1 and aluminum alloy V-95 was performed, algorithms and computer programs were developed in the public domain.

A Universal Test on Spikes in a High-Dimensional Generalized Spiked Model and Its Applications

This paper aims to test the number of spikes in a generalized spiked covariance matrix, the spiked eigenvalues of which may be extremely larger or smaller than the non-spiked ones. For a high-dimensional problem, we first propose a general test statistic and derive its central limit theorem by random matrix theory without a Gaussian population constraint. We then apply the result to estimate the noise variance and test the equality of the smallest roots in generalized spiked models. Simulation studies showed that the proposed test method was correctly sized, and the power outcomes showed the robustness of our statistic to deviations from a Gaussian population. Moreover, our estimator of the noise variance resulted in much smaller mean absolute errors and mean squared errors than existing methods. In contrast to previously developed methods, we eliminated the strict conditions of diagonal or block-wise diagonal form of the population covariance matrix and extend the work to a wider range without the assumption of normality. Thus, the proposed method is more suitable for real problems.

Spectral Decomposition of Gramians of Continuous Linear Systems in the Form of Hadamard Products

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given.

Experimental and numerical study on seismic behavior of special shaped steel truss and concrete composite shear wall

A steel truss-concrete composite (STCC) shear wall is fabricated by replacing the reinforcement of the shear wall by steel truss, which has high bearing capacity, good fire resistance, good seismic performance, and an easy construction. In order to investigate the seismic performance of special shaped composite shear wall, a quasi-static test was carried out on four T-shaped and two L-shaped composite shear walls. The results showed that the change of diagonal bracing form from strip steel to channel steel can improve the ductility and energy dissipation capacity of the shear wall and delay the strength degradation. Increasing the thickness of steel vertical supports can effectively improve the initial stiffness, while decreasing the thickness of diagonal braces will reduce the bearing capacity of the wall. The strength and stiffness of shear walls loaded along the web in the positive direction (flange in tension) is high, but the ductility is relatively poor when compared with walls loaded in the negative direction (flange in compression). Numerical models were established by ABAQUS, in which the steel hysteretic constitutive model was adjusted to simulate the influence of bond-slip effect. The influence of shear span ratio, axial load ratio, steel ratio of vertical support, volumetric ratio of diagonal brace, tilt angle of diagonal brace and horizontal force direction on the seismic performance of T & L shaped composite shear walls were analyzed. Finally, a design method was proposed, and the calculated results by the design method were in good agreement with the test.

An efficient reconfigurable code rate cooperative low-density parity check codes for gigabits wide code encoder/decoder operations

In recent days, extensive digital communication process has been performed. Due to this phenomenon, a proper maintenance of authentication, communication without any overhead such as signal attenuation code rate fluctuations during digital communication process can be minimized and optimized by adopting parallel encoder and decoder operations. To overcome the above-mentioned drawbacks by using proposed reconfigurable code rate cooperative (RCRC) and low-density parity check (LDPC) method. The proposed RCRC-LDPC is capable to operate over gigabits/sec data and it effectively performs linear encoding, dual diagonal form, widens the range of code rate and optimal degree distribution of LDPC mother code. The proposed method optimize the transmission rate and it is capable to operate on 0.98 code rate. It is the highest upper bounded code rate as compared to the existing methods. The proposed method optimizes the transmission rate and is capable to operate on a 0.98 code rate. It is the highest upper bounded code rate as compared to the existing methods. the proposed method's implementation has been carried out using MATLAB and as per the simulation result, the proposed method is capable of reaching a throughput efficiency greater than 8.2 (1.9) gigabits per second with a clock frequency of 160 MHz.

On the local Gevrey integrability

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm.

Rethinking accountability in developing countries: an institutional pillars perspective

PurposeThis paper aims to fill gap in the literature and explore policy options for resolving the problems of accountability by framing three research questions. The research questions are (i) whether certain elements of Scott’s (2014) institutional pillars attenuate (accentuate) corporate and public accountability; (ii) whether the presence of ruling party-affiliated enterprises (RPAEs) create an increase (decrease) in the degree of corporate (public) accountability; and (iii) whether there is a particular form of ownership change that transforms RPAEs into public investment companies.Design/methodology/approachUsing a qualitative research methodology that involves term frequency and thematic analysis of publicly available textual information, the paper examines Mechkova et al.’s (2019 forms of government accountability. The paper analyzes the gaps between the de jure and de facto accountability using the institutional pillars framework.FindingsThe findings of the paper are three. First, there are gaps between de jure and de facto in all three (vertical, horizontal and diagonal) forms of government (public) accountability. Second, the study finds that more than three fourth of the parties that contested the June 2021 election did have regional focus. They did not advocate for accountability. Third, Ethiopia’s RPAEs are unique. They have regional focus and are characterized by severe forms of agency and information asymmetry problems.Research limitations/implicationsThe main limitation of the paper is its exploratory nature. Extending this research by using cross-country data could provide a more complete picture of the link between corporate (public) accountability and a country’s institutional pillars.Practical implicationsAcademic research documents that instilling modern corporate (public) governance standards in the Sub Sahara Africa (SSA) region has shown mixed results. The analysis made in this paper is likely to inform researchers and policymakers about the type of change that leads to better corporate (and public) accountability outcomes.Social implicationsThe institutional change proposed in the paper is likely to advance the public interest by mitigating agency and information asymmetry problems and enhancing government accountability. The changes make the enterprises investable, save scarce jobs, enhance diversity and put the assets in RPAEs to better use.Originality/valueTo the best of the authors’ knowledge, this is the first paper that uses the institutional pillars analytical framework to examine an SSA country's corporate (public) accountability problem. It demonstrates that accountability is a domestic and a (novel) traveling theory. The paper identifies the complexity of resolving the interlock between political institutions and business enterprises. It theorizes that it is impossible to instill modern corporate (public) accountability standards without changing regulatory, normative and cultural cognitive pillars of institutions. The paper contributes to the change management and public interest literature.

Quantum Dynamics of Oblique Vibrational States in the Hénon-Heiles System.

In this paper, we study the quantum time evolution of oblique nonstationary vibrational states in a Hénon-Heiles oscillator system with two dissociation channels, which models the stretching vibrational motions of triatomic molecules. The oblique nonstationary states we are interested in are the eigenfunctions of the anharmonic zero-order Hamiltonian operator resulting from the transformation to oblique coordinates, which are defined as those coming from nonorthogonal coordinate rotations that express the matrix representation of the second-order Hamiltonian in a block diagonal form characterized by the polyadic quantum number n = n1 + n2. The survival probabilities calculated show that the oblique nonstationary states evolve within their polyadic group with a high degree of coherence up to the dissociation limits on the short time scale. The degree of coherence is certainly much higher than that exhibited by the local nonstationary states extracted from the conventional orthogonal rotation of the original normal coordinates. We also show that energy exchange between the oblique vibrational modes occurs in a much more regular way than the exchange between the local modes.

Fixed/preassigned-time output synchronization for T–S fuzzy complex networks via quantized control

This article is devoted to the analysis of fixed-time (FXT) and preassigned-time (PAT) output synchronization for T–S fuzzy complex networks (CNs). Firstly, to improve transmission efficiency of information, a kind of pure power-law control strategy with logarithmic quantization is designed and the power-law terms have only one free parameter. Then, by applying FXT stability theorem as well as inequality skills, some compact criteria are obtained to achieve FXT output synchronization of T–S fuzzy CNs, and the restriction on the output matrix with positive definite and diagonal form is removed. Furthermore, the PAT output synchronization of T–S fuzzy CNs is further explored by developing a quantized control scheme with power-law form. Three numerical examples in the end are given to attest the proposed control design and criteria.

Совершенствование проточной части мультифазных ступеней с использованием мультифазного коэффициента относительной скорости движения дискретных частиц

Equations are derived to design and develop the multiphase stages flow path in a diagonal form using the dimensionless multiphase coefficient of the discrete particles relative speed. An analysis was made of the formation fluid flow containing free gas bubbles in the serial multiphase stages flow path of axial and axis-diagonal types in comparison with the developed diagonal-type stages and with inclined-cylindrical and helical blades. It is shown that, compared to the axial and axis-diagonal type stages, multiphase stages of the diagonal type with the inclined-cylindrical and helical blades have higher pressure when operating on the gas-liquid mixtures, especially in the multi-stage design. When operating on water without free gas, pressure and efficiency of the multiphase stages exceed characteristics of the best analogues, which makes it possible to manufacture the full-size pumps operating efficiently on liquid without gas and on the gas-liquid mixture.