Hilbert Space Research Articles

Hyers–Ulam Stability of the Coefficient Multipliers on Analytic Hilbert Spaces

Hyers–Ulam Stability of the Coefficient Multipliers on Analytic Hilbert Spaces

Glassy Word Problems: Ultraslow Relaxation, Hilbert Space Jamming, and Computational Complexity

We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector q exhibit almost no relaxation until times O(exp(1/q)), at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions. Published by the American Physical Society 2024

Floquet engineering of Hilbert space fragmentation in Stark lattices

Floquet engineering of Hilbert space fragmentation in Stark lattices

Convergence of two-step inertial Tseng’s extragradient methods for quasimonotone variational inequality problems

This paper introduces two-step inertial Tseng’s extragradient methods with self-adaptive step sizes for solving quasimonotone variational inequalities in real Hilbert spaces. Under suitable conditions on the iterative parameters, weak convergence of the sequence generated by the proposed algorithms is obtained. Numerical results demonstrate the efficiency of the methods compared to others available ones in literature.

A unifying quantum speed limit for time-independent Hamiltonian evolution

Quantum speed limit (QSL) is the study of fundamental limits on the evolution time of quantum systems. For instance, under the action of a time-independent Hamiltonian, the evolution time between an initial and a final quantum state obeys various mutually complementary lower bounds. They include the Mandelstam–Tamm (MT) bound, the Margolus–Levitin (ML) bound, the Luo–Zhang bound, the Lee–Chau (LC) bound together with the dual ML bound introduced by Ness and coworkers. Here we show that the MT bound can be obtained by optimizing the LC bound over a certain parameter. More importantly, we report a QSL that includes all the above bounds as special cases before optimizing over the physically meaningless reference energy level of a quantum system. This unifying bound depends on a certain parameter p. For any fixed p, we find all pairs of time-independent Hamiltonian and initial pure quantum state that saturate this unifying bound. More importantly, these pairs allow us to compute this bound accurately and efficiently using an oracle that returns certain pth moments related to the absolute value of energy of the quantum state. Moreover, this oracle can be simulated by a computationally efficient and accurate algorithm for finite-dimensional quantum systems as well as for certain infinite-dimensional quantum states with bounded and continuous energy spectra. This makes our computational method feasible in a lot of practical situations. We further compare the performance of this bound for the case of a fixed p as well as the case of optimizing over p with existing QSLs. We find that if the dimension of the underlying Hilbert space is ≲2000 , our unifying bound optimized over p can be computed accurately in a few minutes using Mathematica code with just-in-time compilation in a typical desktop. Besides, this optimized unifying QSL is at least as good as all the existing ones combined and can occasionally be a few percent to a few times better.

The Kalton-Peck space as a spreading model

The so-called Kalton-Peck space $Z_2$ is a twisted Hilbert space induced, using complex interpolation, by $c_0$ or $\ell _p$ for any $1\leq p\neq 2<\infty $. Kalton and Peck developed a scheme of results for $Z_2$ showing that it is a very rigid space. For example, every normalized basic sequence in $Z_2$ contains a subsequence which is equivalent to either the Hilbert copy $\ell _2$ or the Orlicz space $\ell _M$. Recently, new examples of twisted Hilbert spaces, which are induced by asymptotic $\ell _p$-spaces, have appeared on the stage. Thus, our aim is to extend the Kalton-Peck theory of $Z_2$ to twisted Hilbert spaces $Z(X)$ induced by asymptotic $c_0$ or $\ell _p$-spaces $X$ for $1\leq p<\infty $. One of the novelties is to use spreading models to gain information on the isomorphic structure of the subspaces of a twisted Hilbert space. As a sample of our results, the only spreading models of $Z(X)$ are $\ell _2$ and $\ell _M$, whenever $X$ is as above and $p\neq 2$.

A class of closed manifolds in nonseparable Hilbert spaces

A class of closed manifolds in nonseparable Hilbert spaces

Unitary Anchored Planar Algebras

In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{C}^*$$\\end{document}-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces.

The Separability Problem in Molecular Quantum Systems: Information-Theoretic Framework for Atoms in Molecules.

Even though molecules are fundamentally quantum entities, the concept of a molecule retains certain classical attributes concerning its constituents. This includes the empirical separability of a molecule into its three-dimensional, rigid structure in Euclidean space, a framework often obtained through experimental methods like X-Ray crystallography. In this work, we delve into the mathematical implications of partitioning a molecule into its constituent parts using the widely recognized Atoms-In-Molecules (AIM) schemes, aiming to establish their validity within the framework of Information Theory concepts. We have uncovered information-theoretical justifications for employing some of the most prevalent AIM schemes in the field of Chemistry, including Hirshfeld (stockholder partitioning), Bader's (topological dissection), and the quantum approach (Hilbert's space definition). In the first approach we have applied the generalized principle of minimum relative entropy derived from the Sharma-Mittal two-parameter functional, avoiding the need for an arbitrary selection of reference promolecular atoms. Within the ambit of topological-information partitioning, we have demonstrated that the Fisher information of Bader's atoms conform to a comprehensive theory based on the Principle of Extreme Physical Information avoiding the need of employing the Schwinger's principle, which has been proven to be problematic. For the quantum approach we have presented information-theoretic justifications for conducting Löwdin symmetric transformations on the density matrix to form atomic Hilbert spaces generating orthonormal atomic orbitals with maximum occupancy for a given wavefunction.

Automatic State Interaction with Large Localized Active Spaces for Multimetallic Systems.

The localized active space self-consistent field method factorizes a complete active space wave function into an antisymmetrized product of localized active space wave function fragments. Correlation between fragments is then reintroduced through localized active space state interaction (LASSI), in which the Hamiltonian is diagonalized in a model space of LAS states. However, the optimal procedure for defining the LAS fragments and LASSI model space is unknown. We here present an automated framework to explore systematically convergent sets of model spaces, which we call LASSI[r, q]. This method requires the user to select only r, the number of electron hops from one fragment to another, and q, the number of fragment basis functions per Hilbert space, which converges to CASCI in the limit of r, q → ∞. Numerical tests of this method on the trimetal oxo-centered complexes [Fe(III)Al(III)Fe(II)(μ3-O)(HCOO)6] and [Fe(III)2Fe(II)(μ3-O)(HCOO)6] show efficient convergence to the CASCI limit with 4-10 orders of magnitude fewer states than CASCI.

Solving Bilevel quasimonotone variational inequality problem in Hilbert spaces

In this paper, we propose and study a Bilevel quasimonotone Variational Inequality Problem (BVIP) in the frame work of Hilbert space. We introduce a new modified inertial iterative technique with self-adaptive step size for approximating a solution of the BVIP. In addition, we established a strong convergence result of the proposed iterative technique with an adaptive step-size conditions without prior knowledge of Lipschitz’s constant of the cost operators as well as the strongly monotonicity coefficient under some standard mild assumptions. Finally, we provide some numerical experiments to demonstrate the efficiency of our proposed methods in comparison with some recently announced results in the literature.

ℓ1-bounded sets

A subset M of a separable Hilbert space H is ℓ1-bounded if there exists a Riesz basis F={en}n∈N for H such that supx∈M⁡∑n∈N|〈x,en〉|<∞. A similar definition for ℓ1-frame-bounded sets is made by replacing Riesz bases with frames. This paper derives properties of ℓ1-bounded sets, operations on the collection of ℓ1-bounded sets, and the relation between ℓ1-boundedness and ℓ1-frame-boundedness. Some open problems are stated, several of which have intriguing implications.

Complexity growth and the Krylov-Wigner function

For any state in a D-dimensional Hilbert space with a choice of basis, one can define a discrete version of the Wigner function — a quasi-probability distribution which represents the state on a discrete phase space. The Wigner function can, in general, take on negative values, and the amount of negativity in the Wigner function has an operational meaning as a resource for quantum computation. In this note, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. We introduce the Krylov-Wigner function, i.e., the Wigner function defined with respect to the Krylov basis (with appropriate phases), and show that this choice of basis minimizes the early time growth of Wigner negativity in the large D limit. We take this as evidence that the Krylov basis (with appropriate phases) is ideally suited for a dual, semi-classical description of chaotic quantum dynamics at large D. We also numerically study the time evolution of the Krylov-Wigner function and its negativity in random matrix theory for an initial pure state. We observe that the negativity broadly shows three phases: it rises gradually for a time of OD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ O\\left(\\sqrt{D}\\right) $$\\end{document}, then hits a sharp ramp and finally saturates close to its upper bound of D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{D} $$\\end{document}.

Master constraint approach to quantum-reduced loop gravity

We introduce a master constraint operator on the kinematical Hilbert space of loop quantum gravity representing a set of gauge conditions which classically fix the densitized triad to be diagonal. We argue that the master constraint approach provides a natural and systematic way of carrying out the quantum gauge-fixing procedure which underlies the model known as quantum-reduced loop gravity. The Hilbert space of quantum-reduced loop gravity is obtained as a particular space of solutions of the gauge-fixing master constraint operator. We give a concise summary of the fundamental structure of the quantum-reduced framework, and consider several possible extensions thereof, for which the master constraint formulation provides a convenient starting point. In particular, we propose a generalization of the standard Hilbert space of quantum-reduced loop gravity, which may be relevant in the application of the quantum-reduced model to physical situations in which the Ashtekar connection is not diagonal.

Constructive approach and randomization of a two-parameter chaos system for securing data

Secure communication techniques are important due to the increase in the number of technology users across the world. Likewise, a more random encryption algorithm suitable to secure data from unauthorised users is highly expected. This paper proposes a two-parameter nonlinear chaos map that is sensitive to the trio seed (s0, \alpha, \lambda) and has better information encryption. We introduce the parameter \alpha to linearise the conventional chaos system, which in turn brings a delay in the cryptosystems. The delay is a phenomenon that changes the chaotic features of a system. A small delay in the system leads to more aperiodicity and the unpredictability of the chaotic attractions. We normalise the new chaos map and use the Lipschitz and pseudo-contractive operators to obtain its irregularity region in Hilbert spaces. We also analyse the chaos map in terms of trajectory, Lyapunov exponent, complexity, and information entropy. Results obtained show that the new chaos map has a wide chaotic range and better statistical properties. It also maintains low complexity due to its linearity and produces more key spaces than most existing chaotic maps.

Numerical Ranges of Antilinear Operators

We study numerical ranges of antilinear operators on both Hilbert and Banach spaces. We prove that the numerical range of an antilinear operator on at least two-dimensional space is always a disc, and so a convex set. This improves existing results. We also study other properties known for numerical ranges of linear operators and discuss similarities and differences in the antilinear setting.

Kernel Angle Dependence Measures in Metric Spaces

Measuring and testing dependence between data in separable metric spaces is of great importance in modern statistics. Most existing work relied on the distance between random variables, which inevitably required the moment conditions to guarantee the distance is well-defined. Based on the geometry element “angle”, we develop a novel class of nonlinear dependence measures for data in metric space that can avoid such conditions. Specifically, by making use of the reproducing kernel Hilbert space equipped with Gaussian measure, we introduce kernel angle covariances that can be applied to various types of data, including low dimensional vector, high dimensional vectors, non-Euclidean data like symmetric positive definite matrices, and compositional data. We estimate kernel angle covariances based on U-statistic and establish the corresponding independence tests via gamma approximation. Our kernel angle independence tests, imposing no-moment conditions on kernels, are robust with heavy-tailed random variables. We conduct comprehensive simulation studies and apply our proposed methods to a facial recognition task. Our kernel angle covariances-based tests show remarkable performances in dealing with image data. All the codes and proofs are included in the Supplementary Materials.

Partial Smoothing of the Stochastic Wave Equation and Regularization by Noise Phenomena

AbstractWe establish partial smoothing properties of the transition semigroup $$(P_t)$$ ( P t ) associated to the linear stochastic wave equation driven by a cylindrical Wiener noise on a separable Hilbert space. These new results allow the study of related vector-valued infinite-dimensional PDEs in spaces of functions which are Hölder continuous along special directions. As an application we prove strong uniqueness for semilinear stochastic wave equations involving nonlinearities of Hölder type. We stress that we are able to prove well-posedness although the Markov semigroup $$(P_t)$$ ( P t ) is not strong Feller.

Spectral structure of Moran Sierpinski-type measure on R2

Let Mn=diag[3pn,3qn] with pn,qn⩾1 for all n⩾1 and let D={(0,0)t,(1,0)t,(0,1)t} . One can generate a Borel probability measure μ{Mn},D=δM1−1D∗δ(M2M1)−1D∗δ(M3M2M1)−1D∗⋯. Such measure μ{Mn},D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L2(μ{Mn},D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L2(μ{Mn},D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L2(μ{Mn},D) or not.

Optimal control problem for higher-order non-instantaneous impulsive fractional system with non-local conditions

The purpose of this article is to study the optimal control problem of higher-order non-instantaneous impulsive fractional system with non-local conditions by virtue of cosine operator theory in the Hilbert space. Initially, the sufficient conditions for the existence of a mild solution of the proposed system are verified by applying the fixed point theorem in infinite dimensional space. Further, we formulate and prove the existence results for optimal control of the proposed system with the corresponding cost functional by using Balder's theorem. Finally, an example is provided to illustrate the main results.