Hermitian Tensor Research Articles

Rank-bmRpositive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors

Rank-bmRpositive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors

Transformed singular values and spectral norms of third-order tensors

The tensor-tensor product (T-product) and tensor singular value decomposition (T-SVD) for third-order tensors, which are based upon the general revertible linear transformation, are widely used in engineering and data science. This paper focuses on studying singular values and spectral norms of third-order tensors under unitary transformation. After presenting the decomposition for the third-order Hermitian tensors and the QR factorization of third-order tensors, with the help of the involved invertible linear transformation, several basic properties of singular values for third-order tensors are studied, which include the Lipschitz continuity that characterizes the singular values perturbation. Based upon these, two kinds of spectral norms for third-order tensors are introduced, one of which is the norm induced by Frobenius norm of tensors in the sense of the T-product.

Convenient Tail Bounds for Sums of Random Tensors

This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Laplace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The proof mechanism can also be applied to tensor-valued martingales and tensor-based Azuma, Hoeffding and McDiarmid inequalities are established.

Separability of Hermitian tensors and PSD decompositions

Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e. it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect the separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors.

Tensor Multivariate Trace Inequalities and Their Applications

In linear algebra, the trace of a square matrix is defined as the sum of elements on the main diagonal. The trace of a matrix is the sum of its eigenvalues (counted with multiplicities), and it is invariant under the change of basis. This characterization can be used to define the trace of a tensor in general. Trace inequalities are mathematical relations between different multivariate trace functionals involving linear operators. These relations are straightforward equalities if the involved linear operators commute, however, they can be difficult to prove when the non-commuting linear operators are involved. Given two Hermitian tensors H₁ and H₂ that do not commute. Does there exist a method to transform one of the two tensors such that they commute without completely destroying the structure of the original tensor? The spectral pinching method is a tool to resolve this problem. In this work, we will apply such spectral pinching method to prove several trace inequalities that extend the Araki–Lieb–Thirring (ALT) inequality, Golden–Thompson(GT) inequality and logarithmic trace inequality to arbitrary many tensors. Our approaches rely on complex interpolation theory as well as asymptotic spectral pinching, providing a transparent mechanism to treat generic tensor multivariate trace inequalities. As an example application of our tensor extension of the Golden–Thompson inequality, we give the tail bound for the independent sum of tensors. Such bound will play a fundamental role in high-dimensional probability and statistical data analysis.

Separability discrimination and decomposition of m -partite quantum mixed states

The separability detecting problem of mixed states is one of the fundamental problems in quantum information theory. In the last 20 years, almost all methods are based on the sufficient or necessary conditions for entanglement. However, in this paper, we only need one algorithm to solve the problem. We propose a tensor optimization method to check whether an $m$-partite quantum mixed state is separable or not and give a decomposition for it if it is. We first convert the separability discrimination problem of mixed states to the positive Hermitian decomposition problem of Hermitian tensors. Then, employing the $E$-truncated $K$-moment method, we obtain an optimization model for discriminating separability. Moreover, applying semidefinite relaxation method, we get a hierarchy of semidefinite relaxation optimization models and propose an $E$-truncated $K$-moment and semidefinite relaxations (ETKM-SDR) algorithm for detecting the separability of mixed states. The algorithm can also be used for symmetric and non-symmetric decomposition of separable mixed states. By numerical examples, we find that not all symmetric separable states have symmetric decompositions. The algorithm can be used for studying properties of mixed states in the future.

Hermitian Tensor Decompositions

Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.

Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region

AbstractLet U be a connected, closed, bounded region in ℝ3 with smooth boundary 𝛛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e−iωt. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛω and magnetic permeability μω, which may both depend upon ω. Both can be arbitrary, time‐independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U.In this paper necessary and sufficient conditions are found for the tangential components, ET or HT on 𝛛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time‐harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified ET, HT, respectively, satisfying the necessary finite‐dimensional conditions are derived by an effective natural algorithm.In the electric ET case, the positivity condition is this: the complex Hermitian matrix μω(p) is to be positive definite while only the real part of ɛω(p), i.e., Re(ɛω(p)), necessarily real symmetric, need be positive definite. In the magnetic‐type HT case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides. © 2019 Wiley Periodicals, Inc.

On a Class of Smooth Complex Finsler Metrics

By associating to a strongly pseudoconvex complex Finsler metric F a Hermitian tensor \({T}\) of type (1, 1), we prove that a weakly Kahler Finsler metric is a Kahler Finsler metric if and only if \({T \equiv 0}\), and a weakly complex Berwald metric is a weakly Kahler Finsler metric if and only if it is a Kahler Finsler metric. We introduce a class of explicitly constructed smooth complex Finsler metrics, called the general complex (\({\alpha, \beta)}\) metric, where \({\alpha}\) is a Hermitian metric and \({\beta}\) is a complex differential form of type (1, 0) on a complex manifold. The holomorphic curvature of the general complex \({(\alpha, \beta)}\) metrics is derived, and a necessary and sufficient condition for these metrics to be weakly Kahler Finsler metrics is obtained. Under the assumption that \({\beta}\) is holomorphic or closed, necessary and sufficient conditions for this class of metrics to be weakly complex Berwald metrics, complex Berwald metrics and Kahler Finsler metrics are obtained, respectively.

Classical and quantum Fisher information in the geometrical formulation of quantum mechanics

The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects together, we show that the Fisher information metric, both classical and quantum, can be described by means of the Hermitian tensor on the manifold of pure states.

CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

CLASSICAL TENSORS FROM QUANTUM STATES

The embedding of a manifold M into a Hilbert-space ℍ induces, via the pull-back, a tensor field on M out of the Hermitian tensor on ℍ. We propose a general procedure to compute these tensors in particular for Manifolds admitting a Lie-Group structure.

Integral representations of separable states

We study a separability problem suggested by mathematical description of bipartite quantum systems. We consider Hermitian 2-forms on the tensor product $H=K\otimes L$, where $K,L$ are finite dimensional complex spaces. Inspired by quantum mechanical terminology we call such a form separable if it is a convex combination of hermitian tensor products $(\sigma_p)^*\odot \sigma_p$ of 1-forms $\sigma_p$ on $H$ that are product forms $\sigma_p=\phi_p\otimes \psi_p$, where $\phi_p\in K^*$, $\psi_p\in L^*$. We introduce an integral representation of separable forms. In particular, we show that the integral of $(D_{z^*}}\Phi)^*\odot D_{z^*}\Phi$ of any square integrable map $\Phi:\C^n\to \C^m$, with square integrable conjugate derivative $D_{z^*}\Phi$, is a separable form. Vice versa, any separable form in the interior of the set of such forms, can be represented in this way. This implies that any separable mixed state (and only such states) can be either explicitly represented in the integral form, or it may be arbitrarily well approximated by such states.

On the eigenvalue and eigentensor problem for a tensor of even rank

In more detail as in [1], we study tensor modules of even orders and the eigenvalue and eigentensor problem for tensors of any even rank. The canonical representation of a tensor in the module ℂ2p(Ω) is given. We present several statements and theorems about the eigentensors for tensors of even rank, and for adjoint, normal, Hermitian, and unitary tensors in a module of even order. We note that the eigenvalue and eigentensor problem for the tensor of elastic moduli was studied by Ya. Rykhlevskii in 1983–1984. Earlier, it was studied for tensors of any even rank by I. N. Vekua.

Hermitian tensor product approximation of complex matrices and separability

The approximation of matrices to the sum of tensor products of Hermitian matrices is studied. A minimum decomposition of matrices on tensor space $H_1\otimes H_2$ in terms of the sum of tensor products of Hermitian matrices on $H_1$ and $H_2$ is presented. From this construction the separability of quantum states is discussed.

General gyrotropic finite element formulation with loss

A four-transverse-field formulation for a lossless Hermitian tensor material was introduced previously for an arbitrary direction of applied bias field. This method is extended here to include loss, which is critical in modelling experimental devices and which causes variational functionals for gyrotropic waveguides to become non-Hermitian. The functional is developed from Maxwell's equations before demonstrating its stationary properties at the boundary value problem solution. Using the finite element method, the functional is implemented and then validated against various waveguide structures. For the first time, the E-t-H-t formulation has been explicitly applied to lossy gyroelectric waveguide cross-sections. This method has important implications for the study and design of future phase shift and control components used in sub-millimeter wave and terahertz systems. Copyright (c) 2006 John Wiley & Sons, Ltd.

Surface Impedance Tensors of Textured Polycrystals

A formula for the surface impedance tensors of orthorhombic aggregates of cubic crystallites is given explicitly in terms of the material constants and the texture coefficients. The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. This tensor plays a fundamental role in Stroh's formalism for anisotropic elasticity. In this paper we account for the effects of crystallographic texture only up to terms linear in the texture coefficients and give an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients.

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang-Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry $U(1,D-1)$ instead of the usual $SO(1,D-1)$. The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces.

Heights and Geometric Invariant Theory

Let $K$ be a number field, $\OK$ be its ring of integers. We introduce the notion of compactified representation of $GL_N(\OK)$ and, we see how to associate to a hermitian vector bundle $\E$ over $\Spec(\OK)$ and a compactified representation $\T$, a hermitian tensor bundle $\E_T$. We can prove then that there exists a lower bound for the heights of points $x\in\P(\E_T)$ with $SL_N(K)$--semistable generic fibre in terms of the degree of $\E$ and some universal constants depending only on the compactified representation. We give then three applications: a universal lower bound for general flag varieties, an application to the adjoint representation of $SL_N(K)$ and a construction of a height on the moduli space of semistable vector bundles over algebraic curves.

Computation of resonant frequencies of cylindrical ferrite resonators using GIBCs

A computational approach for resonant frequencies of cylindrical substrate-mounted resonators is presented in this paper. The resonator is made of ferrite material, which is magnetically anisotropic and characterized by a Hermitian tensor. It is mounted on top of a grounded dielectric substrate, and the entire assembly is covered with a tuning plate, leaving the sidewalls open. The generalized impedance boundary conditions (GIBCs) are derived for the ferrite material and are used to formulate the approximate solution to the boundary-value problem. The resulting transcendental equations are solved numerically, and variations of the resonant frequency with respect to bias field, magnetization, and dimensions of the structure are reported.