Unitary Eigenvalue Research Articles

On a problem of E. Meckes for the unitary eigenvalue process on an arc

On a problem of E. Meckes for the unitary eigenvalue process on an arc

Multi-scale transition matrix approach to time series

Statistical and structural characteristics provide us with a multi-dimensional picture of time series. Rooting the properties in a unified scheme is the preliminary step to develop a model that can reproduce most part or even the whole of the picture. In this paper, we proposed a concept called multi-scale transition matrix, which is a series of transition matrices describing the jumping probabilities between states after specified numbers of jumps each. The eigenvectors corresponding to the unitary eigenvalues are identical with the probability distribution function. The second largest eigenvalues depict the upper-boundary curve of the persistence. The change speed of the eigenvector corresponding to the second largest eigenvalue along with time scale displays the relaxation behavior of the time series. The multi-scale matrix maintains the structure of autocorrelation in the original time series and its evolution, which are merged by the averaging procedure in the statistical properties. These predictions are confirmed by using the series generated with the Auto-Regressive Conditional Heteroskedasticity Model, the Auto-Regression Model and the fractional Brownian motion, and the empirical records for Shenzhen Component index in Mainland China and the word length series of the novel entitled Remembrance of Things Past written by Marcel Proust. Hence, the concept is a good candidate of bridges between the multi-dimensional picture and the dynamical models of the time series.

A descent cautious BFGS method for computing US-eigenvalues of symmetric complex tensors

Unitary symmetric eigenvalues (US-eigenvalues) of symmetric complex tensors and unitary eigenvalues (U-eigenvalues) for non-symmetric complex tensors are very important because of their background of quantum entanglement. US-eigenvalue is a generalization of Z-eigenvalue from the real case to the complex case, which is closely related to the best complex rank-one approximations to higher-order tensors. The problem of finding US-eigenpairs can be converted to an unconstrained nonlinear optimization problem with complex variables, their complex conjugate variables and real variables. However, optimization methods often need a first- or second-order derivative of the objective function, and cannot be applied to real valued functions of complex variables because they are not necessarily analytic in their argument. In this paper, we first establish the first-order complex Taylor series and Wirtinger calculus of complex gradient of real-valued functions with complex variables, their complex conjugate variables and real variables. Based on this theory, we propose a norm descent cautious BFGS method for computing US-eigenpairs of a symmetric complex tensor. Under appropriate conditions, global convergence and superlinear convergence of the proposed method are established. As an application, we give a method to compute U-eigenpairs for a non-symmetric complex tensor by finding the US-eigenpairs of its symmetric embedding. The numerical examples are presented to support the theoretical findings.

Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement

The purpose of this paper is to study the problem of computing unitary eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of symmetric embedding of complex tensors, the relationship between U-eigenpairs of a non-symmetric complex tensor and unitary symmetric eigenpairs (US-eigenpairs) of its symmetric embedding tensor is established. An algorithm (Algorithm \ref{algo:1}) is given to compute the U-eigenvalues of non-symmetric complex tensors by means of symmetric embedding. Another algorithm, Algorithm \ref{algo:2}, is proposed to directly compute the U-eigenvalues of non-symmetric complex tensors, without the aid of symmetric embedding. Finally, a tensor version of the well-known Gauss-Seidel method is developed. Efficiency of these three algorithms are compared by means of various numerical examples. These algorithms are applied to compute the geometric measure of entanglement of quantum multipartite non-symmetric pure states.

Calculating Entanglement Eigenvalues for Nonsymmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint, and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.

Precoding for Uplink Distributed Antenna Systems With Transmit Correlation in Rician Fading Channels

In this paper, we develop an optimal precoder and a suboptimal precoder for uplink distributed antenna systems with transmit correlation over Rician channels. The objective is to minimize a pairwise error probability (PEP) with constrained transmit power. The optimal scheme uses an efficient projection-based steepest descent algorithm to obtain the Gram matrix of the precoder, while the suboptimal scheme exploits the eigen-structure of the Gram matrix to obtain its unitary matrix (beamforming) and eigenvalues (power control). For a given beamforming (BF), an efficient power adaptation based on a fuzzy signal-to-noise ratio (SNR) is developed. The PEP analysis at low SNR is used to obtain a suboptimal BF. The derived BF and the fuzzy power adaptation have closed-form expressions that make the suboptimal scheme computationally efficient in comparison with existing methods. The BF can be re-optimized via the Riemannian steepest descent method to further reduce the PEP. Simulation results show that the proposed optimal scheme not only gives the performance the same as the existing optimal scheme when the latter can converge, but it also has lower computational complexity and faster convergence. Moreover, the suboptimal scheme can give a nearly optimal error rate.

Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique

The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented.

Compression of unitary rank-structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos-type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank-structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank-structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration.

Geometric Measure of Entanglement and U-Eigenvalues of Tensors

We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rank-one approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.

A unitary Hessenberg QR-based algorithm via semiseparable matrices

In this paper, we present a novel method for solving the unitary Hessenberg eigenvalue problem. In the first phase, an algorithm is designed to transform the unitary matrix into a diagonal-plus-semiseparable form. Then we rely on our earlier adaptation of the QR algorithm to solve the dpss eigenvalue problem in a fast and robust way. Exploiting the structure of the problem enables us to yield a quadratic time using a linear memory space. Nonetheless the algorithm remains robust and converges as fast as the customary QR algorithm. Numerical experiments confirm the effectiveness and the robustness of our approach.

New tests of the correspondence between unitary eigenvalues and the zeros of Riemann s zeta function

This paper presents some new statistical tests and new conjectures regarding the correspondence between the eigenvalues of random unitary matrices and the zeros of Riemann's zeta function. Global features such as the trace and number of eigenvalues in intervals are compared. Our results show satisfying match-ups between the two domains. They give examples of large natural datasets that follow classical distributions to high accuracy.

On the Perturbation Theory for Unitary Eigenvalue Problems

Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant--Fischer-type theorem for unitary matrices is derived and an inclusion theorem analogous to the Kahan theorem for Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices are discussed.

Inverse unitary eigenproblems and related orthogonal functions

This paper explores the relationship between certain inverse unitary eigenvalue problems and orthogonal functions. In particular, the inverse eigenvalue problems for unitary Hessenberg matrices and for Schur parameter pencils are considered. The Szego recursion is known to be identical to the Arnoldi process and can be seen as an algorithm for solving an inverse unitary Hessenberg eigenvalue problem. Reformulation of this inverse unitary Hessenberg eigenvalue problem yields an inverse eigenvalue problem for Schur parameter pencils. It is shown that solving this inverse eigenvalue problem is equivalent to computing Laurent polynomials orthogonal on the unit circle. Efficient and reliable algorithms for solving the inverse unitary eigenvalue problems are given which require only O(\(mn\)) arithmetic operations as compared with O(\(mn^2\)) operations needed for algorithms that ignore the structure of the problem.

Perturbation and interlace theorems for the unitary eigenvalue problem

Two aspects of the perturbation problem for the eigenvalues of a unitary matrix U are treated. Firstly, analogues of the Hoffman-Wielandt theorem and a weyl-type theorem proved by Bhatia and Davis are derived, which are based on a different measure of the distance of spectra. Using a suitable parametrization of the unit circle by an angle, the new results are called tangent theorems, in contrast to the first- mentioned well-known results, which are sine theorems. Moreover, we illuminate the unknown minimizing permutations in the above Weyl-type theorems. With respect to their angles the eigenvalues of U and Ũ (the perturbed matrix) are naturally ordered on the unit circle counterclockwise, after a point is cut on the unit circle. We prove a well-known open conjecture; there exists a cutting point such that the Weyl-type theorems, both sine and tangent, are true when the ordered eigenvalues of U and Ũ are paired with each other. Secondly, the Cauchy interlacing theorem for Hermitian matrices is generalized. It is shown that certain modified principal submatrices of U, called the modified kth leading principal submatrices, have the property that their eigenvalues interlace those of U. Finally we discuss block reflectors, appearing in the description of the modified principal submatrices, and generalize a result of Schreiber and Parlett.

A case against a divide and conquer approach to the nonsymmetric eigenvalue problem

Divide and conquer techniques based on rank-one updating have proven fast, accurate, and efficient in parallel for the real symmetric tridiagonal and unitary eigenvalue problems and for the bidiagonal singular value problem. Although the divide and conquer mechanism can also be adapted to the real nonsymmetric eigenproblem in a straightforward way, most of the desirable characteristics of the other algorithms are lost. In this paper, we examine the problems of accuracy and efficiency that can stand in the way of nonsymmetric divide and conquer eigensolver based on low-rank updating.

Schur parameter pencils for the solution of the unitary eigenproblem

Let U− λV be an n× n pencil with unitary matrices U and V. An algorithm is presented which reduces U and V simultaneously to unitary block diagonal matrices G o = Q H UP and G e = Q H VP with block size at most two. It is an O( n 3) process using Householder eliminations, and it is backward stable. In the special case V= I the block diagonal matrices G o , G H e can be normalized so that their entries are just the Schur parameters of the Hessenberg condensed form of U. We call G o − λG e a Schur parameter pencil. It can also be derived from U,V by a Lanczos-like process. For the solution of the eigenvalue problem for G o − λG e a QR-type algorithm can be developed based on this unitary reduction of a pencil U− λV to a Schur parameter pencil. The condensed form is preserved throughout the process. Each iteration step needs only O( n) operations. This method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the QR method for the Hermitian eigenvalue problem.