Weyl Inequalities Research Articles

Simple and deterministic spectral concentration bound for local Hamiltonians

We give a simple proof of a Chernoff bound for the spectrum of a $k$-local Hamiltonian based on Weyl's inequalities. The complexity of estimating the spectrum's $\epsilon(n)$-th quantile up to constant relative error thus exhibits the following dichotomy: For $\epsilon(n)=d^{-n}$ the problem is NP-hard and maybe even QMA-hard, yet there exists constant $a>1$ such that the problem is trivial for $\epsilon(n)=a^{-n}$.

Nodal domain count for the generalized graph p-Laplacian

Inspired by the linear Schrödinger operator, we consider a generalized p-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous p-Laplacian, we prove that the variational spectrum of the discrete generalized p-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized p-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case p=2, in addition to recovering well-known features, the new results provide novel properties of the linear Schrödinger operator.

Of shadows and gaps in spatial search

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty \etal ({\em Physical Review A}, {\bf 102}:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

Quantum walks do not like bridges

We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing theorem, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.

The dominant eigenvector of a noisy quantum state

Although near-term quantum devices have no comprehensive solution for correcting errors, numerous techniques have been proposed for achieving practical value. Two works have recently introduced the very promising error suppression by derangements (ESD) and virtual distillation (VD) techniques. The approach exponentially suppresses errors and ultimately allows one to measure expectation values in the pure state as the dominant eigenvector of the noisy quantum state. Interestingly this dominant eigenvector is, however, different than the ideal computational state and it is the aim of the present work to comprehensively explore the following fundamental question: how significantly different are these two pure states? The motivation for this work is two-fold. First, comprehensively understanding the effect of this coherent mismatch is of fundamental importance for the successful exploitation of noisy quantum devices. As such, the present work rigorously establishes that in practically relevant scenarios the coherent mismatch is exponentially less severe than the incoherent decay of the fidelity—where the latter can be suppressed exponentially via the ESD/VD technique. Second, the above question is closely related to central problems in mathematics, such as bounding eigenvalues of a sum of two matrices (Weyl inequalities)—solving of which was a major breakthrough. The present work can be viewed as a first step towards extending the Weyl inequalities to eigenvectors of a sum of two matrices—and completely resolves this problem for the special case of the considered density matrices.

General neutrino mass spectrum and mixing properties in seesaw mechanisms * *Supported partly by the Polish National Science Centre (NCN) under the Grant Agreement 2017/25/B/ST2/01987 and the COST (European Cooperation in Science and Technology) Action CA16201 PARTICLEFACE

Neutrinos stand out among the elementary particles because of their unusually small masses. Various seesaw mechanisms attempt to explain this fact. In this work, applying insights from matrix theory, we are in a position to treat variants of seesaw mechanisms in a general manner. Specifically, using Weyl's inequalities, we discuss and rigorously prove under which conditions the seesaw framework leads to a mass spectrum with exactly three light neutrinos. We find an estimate of the mass of heavy neutrinos to be the mass obtained by neglecting light neutrinos, shifted at most by the maximal strength of the coupling to the light neutrino sector. We provide analytical conditions allowing one to prescribe that precisely two out of five neutrinos are heavy. For higher-dimensional cases the inverse eigenvalue methods are used. In particular, for the CP-invariant scenarios we show that if the neutrino sector has a valid mass matrix after neglecting the light ones, i.e. if the respective mass submatrix is positive definite, then large masses are provided by matrices with large elements accumulated on the diagonal. Finally, the Davis-Kahan theorem is used to show how masses affect the rotation of light neutrino eigenvectors from the standard Euclidean basis. This general observation concerning neutrino mixing, together with results on the mass spectrum properties, opens directions for further neutrino physics studies using matrix analysis.

Advances in the Sequential Design of Computer Experiments Based on Active Learning

We present some advances in the design of computer experiments. A Gaussian Process (GP) model is fitted to the computer experiment data as a surrogate model. We investigate using the Active Learning (AL) strategy of finding design points that maximize reduction on predictive variance. Using a series of approximations based on standard results from linear algebra (Weyl's inequalities), we establish a score that approximates the AL utility. Our method is illustrated with a simulated example as well as with an intermediate climate computer model.

On s-numbers, quasi s-numbers, s-moduli and Weyl inequalities of operators in Banach spaces

In the paper we continue the investigations of the papers (Carl in Proc. Am. Math. Soc. 137:155–159, 2009, and Carl and Hinrichs in Bull. Lond. Math. Soc. 41:332–340, 2009) between s-numbers and eigenvalues of operators in Banach spaces. We introduce new s-numbers as well as quasi s-numbers generated by the classical s-numbers. An elementary mixing multiplicativity observed in (Carl in Proc. Am. Math. Soc. 137:155–159, 2009) will be very useful for establishing Weyl inequalities for operators in Banach spaces.

On s -numbers and Weyl inequalities of operators in Banach spaces

Let s = (sn) be an injective s-number sequence in the sense of Pietsch. We show the following Weyl inequality between geometric means of eigenvalues and s-numbers for a Riesz-operator T: X → X acting on a (complex) Banach space of weak type 2: for any 0 < δ ⩽ 1 and all n ∈ ℕ, we have ( Π i = 1 n | λ i ( T ) | ) 1 / n ⩽ c ( δ ) ω C 2 ( X ) ( 1 + ln ω C 2 ( X ) ) ( Π i = 1 n δ S i ( T ) ) 1 / n δ , where wC2(X) is the weak cotype 2 constant of X, nδ ≔ [n/(1+δ)] and c(δ) ⩽ c0 (1+1/δ ln (1/δ)) with an absolute constant c0 ⩾ 1.

Low rank update of singular values

The notion of a low rank update arises in many important applications. This paper deals with the inverse problem of updating a rectangular matrix by additive low rank matrices so as to reposition the associated singular values. The setting is analogous to the classical pole assignment problem where eigenvalues of a square matrix are relocated. Precise and easy-to-check necessary and sufficient conditions under which the problem is solvable are completely characterized, generalizing some traditional Weyl inequalities for singular values. The constructive proof makes it possible to compute such a solution numerically. A pseudo algorithm is outlined.

The strengthened versions of the additive and multiplicative Weyl inequalities

The strengthened versions of the classical additive and multiplicative Weyl inequalities for the singular values of A + B and AB*, where A and B are rectangular matrices, and for the eigenvalues of A + B and AB, where A and B are Hermitian matrices, are established under certain assumptions on the subspaces spanned by some singular vectors or eigenvectors, respectively, of A and B. Bibliography: 6 titles.

Canonical expansion of -symmetric operators and perturbation theory

Let $H$ be any $\PT$ symmetric Schr\"odinger operator of the type $ -\hbar^2\Delta+(x_1^2+...+x_d^2)+igW(x_1,...,x_d)$ on $L^2(\R^d)$, where $W$ is any odd homogeneous polynomial and $g\in\R$. It is proved that $\P H$ is self-adjoint and that its eigenvalues coincide (up to a sign) with the singular values of $H$, i.e. the eigenvalues of $\sqrt{H^\ast H}$. Moreover we explicitly construct the canonical expansion of $H$ and determine the singular values $\mu_j$ of $H$ through the Borel summability of their divergent perturbation theory. The singular values yield estimates of the location of the eigenvalues $\l_j$ of $H$ by Weyl's inequalities.

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles.

Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation

We give necessary conditions for a sequence of complex numbers closed under complex conjugation to be the spectrum of the weakly damped wave operator. These restrictions are a consequence of some new results on spectral sequences for Hermitian quadratic pencils which are based on Weyl's and Mirsky's classical eigenvalue inequalities. In the case of finite-dimensional weakly damped pencils our conditions are both necessary and sufficient. We also obtain some conditions for overdamped pencils of degree m, and show that some of the inequalities that have to be satisfied in the weakly damped case are now reversed.

Eigenvalue inequalities and equalities

We consider cases of equality in three basic inequalities for eigenvalues of Hermitian matrices: Cauchy's interlacing inequalities for principal submatrices, Weyl's inequalities for sums, and the residual theorem. Several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal Hermitian matrices.

Eigenvalue inequalities and equalities

We consider cases of equality in three basic inequalities for eigenvalues of Hermitian matrices: Cauchy's interlacing inequalities for principal submatrices, Weyl's inequalities for sums, and the residual theorem. Several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal Hermitian matrices.

Commutativity and spectra of Hermitian matrices

If two Hermitian matrices commute, then the eigenvalues of their sum are just the sums of the eigenvalues of the two matrices in a suitable order. Examples show that the converse is not true in general. In this paper, partial converses are obtained. The technique involves a characterization of the equality cases for Weyl's inequalities. Moreover, a new proof on the commutativity of two Hermitian matrices with property L and analogous results for the product of two positive definite Hermitian matrices are included.

Multilinear methods in linear algebra

Several classical and a few new results are presented in which multilinear algebra has proven to be an effective tool. Some of the topics covered are: Weyl's inequalities; the Hadamard product; mappings on tensor spaces; Gram matrices in tensor spaces; strong nonsingularity and the LDU theorem.

Radically simplified proofs of the Weyl and Stenger inequalities concerning eigenvalues of self-adjoint operators

Radically simplified proofs of the generalized Weyl inequality and the Stenger inequality for eigenvalues of self-adjoint operators is presented. The main application of these results lie in atomic and molecular physics. The generalized inequality has application in obtaining bounds on molecular energy levels in cases where the molecular hamiltonian can be written as a sum of several atomic ones and the Stenger inequality has application in putting bounds on energy levels of atomic and molecular resonances to which the Feshbach formalism applies.

Harnack's and Weyl's Inequalities

Harnack's and Weyl's Inequalities