Positive Semidefinite Hermitian Research Articles

On symmetry adapted bases in trigonometric optimization

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).

On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: A Riemannian optimization interpretation

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top p eigenvalues, the simplest orthogonalization-free method is to find the best rank-p approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer–Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer–Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures–Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest k eigenvalues for large-scale matrices if the largest k eigenvalues are nearly distributed uniformly.

The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the norm of the matrix exponential for the associated conservative-dissipative ODE system.

Rényi Entropy, Signed Probabilities, and the Qubit.

The states of the qubit, the basic unit of quantum information, are 2 × 2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

On Non-Hermitian Positive (Semi)Definite Linear Algebraic Systems Arising from Dissipative Hamiltonian DAEs

We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based on efficient three-term recurrences. We illustrate the performance of these iterative methods on several examples. The semidefinite case can be challenging and requires additional techniques to deal with the “singular part,” while the “positive definite part” can still be treated with the three-term recurrence methods.

Matrix Pencils with Coefficients that have Positive Semidefinite Hermitian Parts

We analyze when an arbitrary matrix pencil is strictly equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with coefficients that have positive semidefinite Hermitian parts. We relate the Kronecker structure of these pencils to that of an underlying skew-Hermitian pencil and discuss their regularity, index, numerical range, and location of eigenvalues. Further, we study matrix polynomials with positive semidefinite Hermitian coefficients and use linearizations with positive semidefinite Hermitian parts to derive sufficient conditions for having a spectrum in the left half plane and to derive bounds on the index.

Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes

For [Formula: see text], consider the sample covariance matrix [Formula: see text] from a data set [Formula: see text], where [Formula: see text] is a [Formula: see text] matrix having i.i.d. entries with mean zero and variance one, and [Formula: see text] are deterministic positive semi-definite Hermitian matrices of dimension [Formula: see text] and [Formula: see text], respectively. We assume that [Formula: see text] is bounded in spectral norm, and [Formula: see text] is a Toeplitz matrix with its largest eigenvalues diverging to infinity. The matrix [Formula: see text] can be viewed as a data set of an [Formula: see text]-dimensional long memory stationary process having separable dependence structure. As [Formula: see text] and [Formula: see text], we establish the asymptotics and the joint CLT for [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th largest eigenvalue of [Formula: see text] and [Formula: see text] is a fixed integer. For the CLT, we first study the case where the entries of [Formula: see text] are Gaussian, and then we generalize the result to some more generic cases. This result substantially extends our previous result in Merlevède et al. 2019, where we studied [Formula: see text] in the case where [Formula: see text] and [Formula: see text] with [Formula: see text] having Gaussian entries. In order to establish this CLT, we are led to study the first order asymptotics of the largest eigenvalues and the associated eigenvectors of some deterministic Toeplitz matrices related to long memory stationary processes. We prove multiple spectral gap properties for the largest eigenvalues and a delocalization property for their associated eigenvectors.

A simple counterexample for the permanent-on-top conjecture

The permanent-on-top conjecture states that the largest eigenvalue of the Schur power matrix of a positive semi-definite Hermitian matrix H is per(H). A counterexample has been found with the help of computers, but here, I present another counterexample that can be checked by hand. My method is to use linear representations of groups to connect the spectrum of the Schur power matrix with the spectra of the entrywise(Hadamard) product matrices of permanental compound matrices. By that, we are able to study the properties of the spectrum of the Schur power matrix through the entrywise(Hadamard) product matrices of permanental compound matrices. The counterexample we find is in fact also a counterexample to a weaker conjecture related to permanental compound matrices. This conjecture was also known to be false, but the new counterexample is smaller than the known one.

Statistical inference for Bures–Wasserstein barycenters

In this work we introduce the concept of Bures–Wasserstein barycenter Q∗, that is essentially a Fréchet mean of some distribution P supported on a subspace of positive semi-definite d-dimensional Hermitian operators H+(d). We allow a barycenter to be constrained to some affine subspace of H+(d), and we provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of Q∗ in both Frobenius norm and Bures–Wasserstein distance, and explain, how the obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.

Nonnegativity for hafnians of certain matrices

We show that a complex symmetric matrix of the form , where B is Hermitian positive semidefinite, has a nonnegative hafnian. Some positive scalar multiples of matrices are encodable in a Gaussian boson sampler. Further, the hafnian of this matrix is non-decreasing in B in the sense that if .

Maximizing products of linear forms, and the permanent of positive semidefinite matrices

We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden’s conjecture for HPSD matrices, where the polynomial optimization problem is interpreted as a relaxation of the permanent.

Group Invariant Dictionary Learning

The dictionary learning problem concerns the task of representing data as sparse linear sums drawn from a smaller collection of basic building blocks. In application domains where such techniques are deployed, we frequently encounter datasets where some form of symmetry or invariance is present. Motivated by this observation, we develop a framework for learning dictionaries for data under the constraint that the collection of basic building blocks remains invariant under such symmetries. Our procedure for learning such dictionaries relies on representing the symmetry as the action of a matrix group acting on the data, and subsequently introducing a convex penalty function so as to induce sparsity with respect to the collection of matrix group elements. Our framework specializes to the convolutional dictionary learning problem when we consider integer shifts. Using properties of positive semidefinite Hermitian Toeplitz matrices, we develop an extension that learns dictionaries that are invariant under continuous shifts. Our numerical experiments on synthetic data and ECG data show that the incorporation of such symmetries as priors are most valuable when the dataset has few data-points, or when the full range of symmetries is inadequately expressed in the dataset.

Space‐Projected Conductivity and Spectral Properties of the Conduction Matrix

Herein, the Kubo–Greenwood formula is utilized to project the electronic conductivity into real space, and a Hermitian positive semidefinite matrix Γ is discussed, which is called the conduction matrix, that reduces the computation of spatial conduction activity to a diagonalization. It is shown that for low‐density amorphous carbon, connected sp2 rings and sp chains are conduction‐active sites in the network. In amorphous silicon, transport involves hopping through tail states mediated by the defects near the Fermi level. It is found that for liquid silicon, thermal fluctuations induce spatial and temporal conductivity fluctuations in the material. The frequency‐dependent absorption of light as a function of wavelength in an amorphous silicon suboxide (a‐SiO1.3) is also studied. It is shown that the absorption is strongly frequency dependent and selects out different oxygen vacancy subnetworks depending on the frequency. Γ is diagonalized to obtain conduction eigenvalues and eigenvectors, and it is shown that the density of states of the eigenvalues for FCC aluminum has an extended spectral tail that distinguishes metals from insulators and semiconductors. The method is easy to implement with any electronic structure code, providing suitable estimates for single‐particle electronic states and energies.

Operators on positive semidefinite inner product spaces

Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form 〈⋅,⋅〉. If a linear operator A:U→U is bounded (i.e., ‖Au‖⩽c‖u‖ for some c∈R and all u∈U), then the subspace U0:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A0:U0→U0 and A1:U/U0→U/U0.Let A be an indecomposable bounded operator on U such that 0≠U0≠U. Let λ be an eigenvalue of A0. We prove that the algebraic multiplicity of λ in A1 is not less than the geometric multiplicity of λ in A0, and the geometric multiplicity of λ in A1 is not less than the number of Jordan blocks Jt(λ) of each fixed size t×t in the Jordan canonical form of A0.We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U.For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.

The Choquet integral as an approximation to density matrices with incomplete information

A total set of n states |i〉 and the corresponding projectors Π(i)=|i〉〈i| are considered, in a quantum system with d-dimensional Hilbert space H(d). A partially known density matrix ρ with given p(i)=Tr[ρΠ(i)] (where i=1,…,n and d≤n≤d2−1) is considered, and its ranking permutation is defined. It is used to calculate the Choquet integral C(ρ) which is a positive semi-definite Hermitian matrix. Comonotonicity is an important concept in the formalism, which is used to formalize the vague concept of physically similar density matrices. It is shown that C(ρ)∕Tr[C(ρ)] is a density matrix which is a good approximation to the partially known density matrix ρ.

Non-Abelian Berry phase for open quantum systems: Topological protection versus geometric dephasing

We provide a detailed theoretical analysis of the adiabatic evolution of degenerate open quantum systems, where the dynamics is induced by time-dependent fluctuating loop paths in control parameter space. For weak system-bath coupling, the fluctuations around a deterministic base path obey Gaussian statistics, where we assume that the quantum adiabatic theorem is also satisfied for fluctuating paths. We show that universal non-Abelian geometric dephasing (NAGD) contributions are contained in the fluctuation-averaged evolution operator. This operator plays a key role in all experimental protocols proposed in this work for the detection of NAGD. In particular, we formulate interference measurements providing full access to NAGD. Apart from a factor due to the dynamic phase and its fluctuations, the averaged evolution operator contains the averaged Berry matrix. A polar decomposition of this non-unitary matrix into the product of a unitary, $V$, and a positive semi-definite Hermitian matrix, $R$, reveals the physics of the NAGD contributions. Unlike the conventional dynamic dephasing rate, the NAGD eigenrates encoded by $R$ depend on the loop orientation sense and, in particular, change sign under a reversal of the direction. A negative rate then implies amplification of coherences as compared to the ones when only dynamic dephasing is present. The non-Abelian character of geometric dephasing is rooted in the non-commutativity of $V$ and $R$ and causes smoking-gun signatures in interference experiments. We also clarify why systems subject to topological protection do not show geometric dephasing. Without full protection, however, geometric dephasing can arise and has to be taken into account. As concrete application, we propose spin-echo NAGD detection protocols for modified Majorana braiding setups.

Experimental linear optical computing of the matrix permanent

Linear optical computing (LOC) with thermal light has recently gained attention because the problem is connected to the permanent of a Hermitian positive semidefinite matrix (HPSM), which is of importance in the computational complexity theory. Despite the several theoretical analyses on the computational structure of an HPSM in connection to LOC, the experimental demonstration and the computational complexity analysis via the linear optical system have not been performed yet. We present, herein, experimental LOC for estimating the permanent of an HPSM. From the linear optical experiments and theoretical analysis, we find that the LOC efficiency for a multiplicative error is dependent on the value of the permanent and that the lower bound of the computation time scales exponentially. Furthermore, our results are generalized and applied to LOC of permanents of unitary matrices, which can be implemented with a multi-port quantum interferometer involving single-photons at the input ports. We find that LOC with single-photons, for the permanent estimation, is on average less efficient than the most efficient classical algorithm known to date, even in ideal conditions.

Synthesis of passive fractional‐order LCn‐port with three element orders

Fractional-order circuits find a widespread use in different engineering applications. The problem of realising fractional-order circuits has been discussed by several authors, however, it is far from being solved. Realising fractional-order resistorless passive network with three element orders is been studied. At first, this study extends the two-variable reactance matrix synthesis method to three-variable case, and then proposes a synthesis method of fractional-order reactance matrix with three element orders by variable substitution. The process in above methods mainly involves variable substitution, decomposition of three-variable reactance matrix, extraction of unit inductors, Laurent series expansion, spectral factorisation of two-variable positive semidefinite Hermitian matrix and synthesis of univariable reactance matrix. Then the above-mentioned synthesis process is illustrated by two examples.

An Upper Bound for the Largest Eigenvalue of a Positive Semidefinite Block Banded Matrix

The new upper bound $$ {\uplambda}_{\mathrm{max}}(A)\le \sum \limits_{k=1}^{p+1}i\equiv {k}_{\left(\operatorname{mod}p+1\right)}^{\mathrm{max}}{\uplambda}_{\mathrm{max}}\left({A}_{ii}\right) $$ for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (Aij ) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the bound $$ {\uplambda}_{\mathrm{max}}(A)\le p+1, $$ depending on p only, which improves the bounds established for such matrices earlier and extends the bound $$ {\uplambda}_{\mathrm{max}}(A)\le 2, $$ old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1.

An Efficient and Fast Quantum State Estimator With Sparse Disturbance.

A pure or nearly pure quantum state can be described as a low-rank density matrix, which is a positive semidefinite and unit-trace Hermitian. We consider the problem of recovering such a low-rank density matrix contaminated by sparse components, from a small set of linear measurements. This quantum state estimation task can be formulated as a robust principal component analysis (RPCA) problem subject to positive semidefinite and unit-trace Hermitian constraints. We propose an efficient and fast inexact alternating direction method of multipliers (I-ADMM), in which the subproblems are solved inexactly and hence have closed-form solutions. We prove global convergence of the proposed I-ADMM, and the theoretical result provides a guideline for parameter setting. Numerical experiments show that the proposed I-ADMM can recover state density matrices of 5 qubits on a laptop in 0.69 s, with 6 ×10-4 accuracy (99.38% fidelity) using 30% compressive sensing measurements, which outperforms existing algorithms.