Hilbert Space Research Articles

Some Results of n-EP Operators on Hilbert Spaces

Some Results of n-EP Operators on Hilbert Spaces

Quantum duality in electromagnetism and the fine structure constant

We describe the interplay between electric-magnetic duality and higher symmetry in Maxwell theory. When the fine structure constant is rational, the theory admits noninvertible symmetries which can be realized as composites of electric-magnetic duality and gauging a discrete subgroup of the one-form global symmetry. These noninvertible symmetries are approximate quantum invariances of the natural world which emerge in the infrared below the mass scale of charged particles. We construct these symmetries explicitly as topological defects and illustrate their action on local and extended operators. We also describe their action on boundary conditions and illustrate some consequences of the symmetry for Hilbert spaces of the theory defined in finite volume. Published by the American Physical Society 2024

Stochastic Partial Differential Equations and Invariant Manifolds in Embedded Hilbert Spaces

AbstractWe provide necessary and sufficient conditions for stochastic invariance of finite dimensional submanifolds for solutions of stochastic partial differential equations (SPDEs) in continuously embedded Hilbert spaces with non-smooth coefficients. Furthermore, we establish a link between invariance of submanifolds for such SPDEs in Hermite Sobolev spaces and invariance of submanifolds for finite dimensional SDEs. This provides a new method for analyzing stochastic invariance of submanifolds for finite dimensional Itô diffusions, which we will use in order to derive new invariance results for finite dimensional SDEs.

On A-normaloid d-tuples of operators and related questions

Let ß A (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T 1 ,…, T d ) ∈ ß A (ℍ) d is called jointly A-normaloid if rA (T) = ∥T∥ A , where rA (T) and ∥T∥ A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple . Here . Finally, several related questions are explored.

Simulating Open Quantum Systems Using Hamiltonian Simulations

We present a novel method to simulate the Lindblad equation, drawing on the relationship between Lindblad dynamics, stochastic differential equations, and Hamiltonian simulations. We derive a sequence of unitary dynamics in an enlarged Hilbert space that can approximate the Lindblad dynamics up to an arbitrarily high order. This unitary representation can then be simulated using a quantum circuit that involves only Hamiltonian simulation and tracing out the ancilla qubits. There is no need for additional postselection in measurement outcomes, ensuring a success probability of one at each stage. Our method can be directly generalized to the time-dependent setting. We provide numerical examples that simulate both time-independent and time-dependent Lindbladian dynamics with accuracy up to the third order. Published by the American Physical Society 2024

Convergence Results for History-Dependent Variational Inequalities

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations.

Does Quantum Mechanics Require “Conspiracy”?

Quantum states containing records of incompatible outcomes of quantum measurements are valid states in the tensor-product Hilbert space. Since they contain false records, they conflict with the Born rule and with our observations. I show that excluding them requires a fine-tuning to an extremely restricted subspace of the Hilbert space that seems “conspiratorial”, in the sense that (1) it seems to depend on future events that involve records (including measurement settings) and on the dynamical law (normally thought to be independent of the initial conditions), and (2) it violates Statistical Independence, even when it is valid in the context of Bell’s theorem. To solve the puzzle, I build a model in which, by changing the dynamical law, the same initial conditions can lead to different histories in which the validity of records is relative to the new dynamical law. This relative validity of the records may restore causality, but the initial conditions still must depend, at least partially, on the dynamical law. While violations of Statistical Independence are often seen as non-scientific, they turn out to be needed to ensure the validity of records and our own memories and, by this, of science itself. A Past Hypothesis is needed to ensure the existence of records and turns out to require violations of Statistical Independence. It is not excluded that its explanation, still unknown, ensures such violations in the way needed by local interpretations of quantum mechanics. I suggest that an as-yet unknown law or superselection rule may restrict the full tensor-product Hilbert space to the very special subspace required by the validity of records and the Past Hypothesis.

Temporal approximation of stochastic evolution equations with irregular nonlinearities

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong errorEk∞:=(Esupj∈{0,…,Nk}‖U(tj)-Uj‖Xp)1/p→0(k→0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_{k}^{\\infty } {:}{=}\\Big (\\mathbb {E}\\sup _{j\\in \\{0, \\ldots , N_k\\}} \\Vert U(t_j) - U^j\\Vert _X^p\\Big )^{1/p} \\rightarrow 0\\quad (k \\rightarrow 0), \\end{aligned}$$\\end{document}where p∈[2,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [2,\\infty )$$\\end{document}, U is the mild solution, Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^j$$\\end{document} is obtained from a time discretisation scheme, k is the step size, and Nk=T/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k = T/k$$\\end{document} for final time T>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T>0$$\\end{document}. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong errorEk:=(supj∈{0,…,Nk}E‖U(tj)-Uj‖Xp)1/p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_k {:}{=}\\bigg (\\sup _{j\\in \\{0,\\ldots ,N_k\\}}\\mathbb {E}\\Vert U(t_j) - U^{j}\\Vert _X^p\\bigg )^{1/p}, \\end{aligned}$$\\end{document}which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

Fast Fourier transforms and fast Wigner and Weyl functions in large quantum systems

Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space D=dn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d^n$$\\end{document} with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space D=d0...dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=d_0...d_{n-1}$$\\end{document} with d0,...,dn-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0,...,d_{n-1}$$\\end{document} odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from O(D2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D^2)$$\\end{document} to O(DlogD)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(D\\log D)$$\\end{document}. The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Systematic Study of Different Types of Interactions in α-, β- and γ-Cyclodextrin: Quantum Chemical Investigation

In this work, comprehensive ab initio quantum chemical calculations using the DFT level of theory were performed to characterize the stabilization interactions (H-bonding and hyperconjugation effects) of two stable symmetrical conformations of α-, β-, and γ-cyclodextrins (CDs). For this purpose, we analyzed the electron density using “Atom in molecules” (AIM), “Natural Bond Orbital” (NBO), and energy decomposition method (CECA) in 3D and in Hilbert space. We also calculated the H-bond lengths and OH vibrational frequencies. In every investigated CD, the quantum chemical descriptors characterizing the strength of the interactions between the H-bonds of the primary OH (or hydroxymethyl) and secondary OH groups are examined by comparing the same quantity calculated for ethylene glycol, α-d-glucose (α-d-Glcp) and a water cluster as reference systems. By using these external standards, we can characterize more quantitatively the properties of these bonds (e.g., strength). We have demonstrated that bond critical points (BCP) of intra-unit H-bonds are absent in cyclodextrins, similar to α-d-Glcp and ethylene glycol. In contrast, the CECA analysis showed the existence of an exchange (bond-like) interaction between the interacting O…H atoms. Consequently, the exchange interaction refers to a chemical bond, namely the H-bond between two atoms, unlike BCP, which is not suitable for its detection.

Join multiple Riemannian manifold representation and multi‐kernel non‐redundancy for image clustering

AbstractImage clustering has received significant attention due to the growing importance of image recognition. Researchers have explored Riemannian manifold clustering, which is capable of capturing the non‐linear shapes found in real‐world datasets. However, the complexity of image data poses substantial challenges for modelling and feature extraction. Traditional methods such as covariance matrices and linear subspace have shown promise in image modelling, and they are still in their early stages and suffer from certain limitations. However, these include the uncertainty of representing data using only one Riemannian manifold, limited feature extraction capacity of single kernel functions, and resulting incomplete data representation and redundancy. To overcome these limitations, the authors propose a novel approach called join multiple Riemannian manifold representation and multi‐kernel non‐redundancy for image clustering (MRMNR‐MKC). It combines covariance matrices with linear subspace to represent data and applies multiple kernel functions to map the non‐linear structural data into a reproducing kernel Hilbert space, enabling linear model analysis for image clustering. Additionally, the authors use matrix‐induced regularisation to improve the clustering kernel selection process by reducing redundancy and assigning lower weights to identical kernels. Finally, the authors also conducted numerous experiments to evaluate the performance of our approach, confirming its superiority to state‐of‐the‐art methods on three benchmark datasets.

On Berezin norm and Berezin number inequalities for sum of operators

Abstract The aim of this study is to obtain several inequalities involving the Berezin number and the Berezin norm for various combinations of operators acting on a reproducing kernel Hilbert space. First, we present some bounds regarding the Berezin number associated with W * Q + W * Q ′ {W}^{* }Q+{W}^{* }Q^{\prime} , where W W , Q Q , and Q ′ Q^{\prime} are three bounded linear operators. Next, several Berezin norm and Berezin number inequalities for the sum of n n operators are established.

The expanding universe of the geometric mean

AbstractIn this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital $$C^*$$ C ∗ -algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the $$C^*$$ C ∗ -algebra of operators on a Hilbert space, even to general unital $$C^*$$ C ∗ -algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

This study focuses on optimal feedback control for stochastic Caputo fractional evolution systems in separable Hilbert spaces. Initially, we deal with the existence of a mild solution, establishing it through the application of the generalised Clarke's subdifferential and a fixed point theorem for multivalued maps. Then, we utilise the Cesari property and the Filippov theorem to demonstrate the existence of feasible pairs. Additionally, we provide pairs of suitable feedback controls under specific conditions. Finally, we present a simple illustration to support our theoretical findings.

Optimal control results for second‐order semilinear integro‐differential systems via resolvent operators

AbstractIn the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings.

New Construction of Mutually Unbiased Bases for Odd-dimensional State Space

Abstract We study the construction of mutually unbiased bases in Hilbert space for composite dimensions d which are not prime powers. We explore the results for composite dimensions which are true for prime power dimensions. And then provide a method for selecting mutually unbiased vectors from the eigenvectors of generalized Pauli matrices to construct mutually unbiased bases. In particular, we present four mutually unbiased bases in C15.

Stability of convergence rates: kernel interpolation on non-Lipschitz domains

Abstract Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\varOmega \subset{\mathbb{R}} ^{d}$, thus allowing for non-Lipschitz domains including e.g., cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tilde{\varOmega } \subset \varOmega \subset{\mathbb{R}} ^{d}$, the convergence rate does not deteriorate—i.e., the convergence rates are stable with respect to going to a subset. We obtain this by leveraging an analysis of greedy kernel algorithms. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $\varOmega $ has an impact on the approximation properties. Numerical experiments illustrate and confirm the analysis.

Tseng-type splitting projection algorithms for equilibrium problems in Hilbert spaces

In this work, based on the subgradient-type projection method considered by Santos and Scheimberg (Optimization, 2017), we propose some Tseng-type splitting algorithms for solving equilibrium problems whose bifunction is decomposed into the sum of two bifunctions in Hilbert spaces. We establish the weak and strong convergence of the proposed algorithm under standard assumptions. Numerical results in Nash–Cournot oligopolistic electricity market equilibrium model and comparisons with other related methods demonstrate the effectiveness of the proposed algorithm.

Regular variation in Hilbert spaces and principal component analysis for functional extremes

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm ‖X‖ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal ‘directions’ of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert–Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

The rates of non-adiabatic processes in large molecular systems: Toward an effective full-dimensional quantum mechanical approach.

Two computational approaches for computing the rates of internal conversions in molecular systems where a large set of nuclear degrees of freedom plays a role are discussed and compared. One approach is based on the numerical solution of the time-dependent Schrödinger equation and allows us to include almost the whole set of vibrational coordinates, thanks to the employment of effective procedures for selecting those elements of the Hilbert space which play a significant role in dynamics. The other approach, based on the time-dependent perturbation theory and limited to the use of the harmonic approximation, allows us to include the whole Hilbert space spanned by the vibrational states of the system. The two approaches are applied to the photophysics of azulene, whose anti-Kasha behavior caused by anomalous internal conversion rates is well assessed. The calculated rates for the decays of the first two excited singlet states are in very good agreement with experimental data, indicating the reliability of both methodologies.