Hilbert Space Research Articles

A study on the hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.

Two-step inertial bregman projection iterative algorithm for solving the split feasibility problem

In this paper, we propose a new two-step inertial Bregman projection iterative algorithm by Halphern iterative method to solve the split feasibility problem in real Hilbert spaces. We give two selection strategies of step-sizes and prove that the proposed iterative sequence converges strongly to solution of the split feasibility problem. Our results extend and improve the corresponding results.

Scaled forward-backward algorithm and the modified superiorized version for solving the split monotone variational inclusion problem

In this paper, we propose an inexact scaled forward-backward algorithm to solve the split monotone variational inclusion problem in a real Hilbert space and prove the strong convergence of it under appropriate conditions. Based on this, we discuss the bounded perturbation resilience of the exact algorithm for introducing the corresponding superiorized version and the superiorization algorithm with restarted perturbations. The numerical experiments illustrate that the proposed algorithms perform well and the superiorization version with restarted perturbations has advantage in decreasing the number of the iterations.

Integrated preparation and manipulation of high-dimensional flying structured photons

The hope for a futuristic global quantum internet that provides robust and high-capacity quantum information transfer lies largely on qudits, the fundamental quantum information carriers prepared in high-dimensional superposition states. However, preparing and manipulating N-dimensional flying qudits as well as subsequently establishing their entanglement are still challenging tasks, which require precise and simultaneous maneuver of 2 (N-1) parameters across multiple degrees of freedom. Here, using an integrated approach, we explore the synergy from two degrees of freedom of light, spatial mode and polarization, to generate, encode, and manipulate flying structured photons and their formed qudits in a four-dimensional Hilbert space with high quantum fidelity, intrinsically enabling enhanced noise resilience and higher quantum data rates. The four eigen spin–orbit modes of our qudits possess identical spatial–temporal characteristics in terms of intensity distribution and group velocity, thereby preserving long-haul coherence within the entirety of the quantum data transmission link. Judiciously leveraging the bi-photon entanglement, which is well preserved in the integrated manipulation process, we present versatile spin–orbit cluster states in an extensive dimensional Hilbert space. Such cluster states hold the promise for quantum error correction which can further bolster the channel robustness in long-range quantum communication.

RELAXED DOUBLE INERTIA AND VISCOSITY ALGORITHMS FOR THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM WITH MULTIPLE OUTPUT SETS

In this paper, we investigate a multiple-sets split feasibility problem with multiple output sets in infinite-dimensional Hilbert spaces. To address this problem, we propose relaxed inertial self-adaptive algorithms that do not use the least squares method and prove strong convergence results for the sequences generated by these algorithms. Finally, we validate the performances of the proposed algorithms by using numerical examples.

Information encoding and encryption in acoustic analogues of qubits

Cryptography is crucial in protecting sensitive information and ensuring secure transactions in a time when data security and privacy are major concerns. Traditional cryptography techniques, which depend on mathematical algorithms and secret keys, have historically protected against data breaches and illegal access. With the advent of quantum computers, traditional cryptography techniques are at risk. In this work, we present a cryptography idea using logical phi-bits, which are classical analogues of quantum bits (qubits) and are supported by driven acoustic metamaterials. The state of phi-bits displays superpositions similar to quantum bits, with complex amplitudes and phases. We present a representation of the state vector of single and multi-phi-bit systems. The state vector of multiple phi-bits system lies in a complex exponentially scaling Hilbert space and is used to encode information or messages. By changing the driving conditions of the metamaterial, the information can be encrypted with exceptional security and efficiency. We illustrate experimentally the practicality and effectiveness of encoding and encryption of a message using a 5 phi-bits system and emphasize the scalability of this approach to an N phi-bits system with the same processing time.

Computational decomposition and composition technique for approximate solution of nonstationary problems

Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit–implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition–composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).

Factoring non-negative operator valued trigonometric polynomials in two variables

AbstractIt is shown using Schur complement techniques that on finite dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two variables with degree $$(d_1,d_2)$$ ( d 1 , d 2 ) can be written as a sum of hermitian squares of at most $$2d_2$$ 2 d 2 analytic polynomials.

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams

AbstractWe prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for $$1\le p\le 2$$ 1 ≤ p ≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on $$\mathbb {R}^2$$ R 2 . When $$p > 1$$ p > 1 , Wasserstein space on $$\mathbb {R}^2$$ R 2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

Theoretical and numerical study of case reporting rate with application to epidemiology

A clear understanding of actual infection rate is imperative for control and prevention of diseases. In particular, it helps in formulating effective vaccination strategies and in assessing the level of herd immunity required to contain the virus. In this paper, we conduct theoretical and numerical study of a novel optimization procedure aimed at stable estimation of incidence reporting rate and time-dependent effective reproduction number from real data on new incidence cases, daily new deaths, and vaccination percentages. The iteratively regularized optimization algorithm can be applied to a broad class of data fitting problems constrained by various biological models, where one has to account for under-reporting of cases. To that end, general nonlinear observation operators in real Hilbert spaces are considered in the proposed convergence analysis. To illustrate theoretical findings, numerical simulations with SVIsIvRD compartmental model and real data for Delta variant of COVID-19 pandemic in different states of the US are conducted.

A numerical computational technique for solving controllability of impulsive Hilfer fractional integro-differential equation with order [formula omitted]

The primary focus of this paper is the concept of controllability criteria for the impulsive Hilfer fractional integro-differential equation (IHFrIDE) of order 0≤ϖ≤1 and 1<ϑ<2 in Hilbert space. We use the iterative process and the strongly cosine family to obtain a given system outcome. We use the degree method to demonstrate the existence of a solution for a given dynamical system. We obtained uniqueness results from Gronwall’s inequality and also addressed the controllability criteria for our given problem. The outputs of numerical computations demonstrate the efficiency of our present method.

Symmetric operator means

AbstractThe purpose of this paper is to extend symmetric means of multi-variable positive matrices to those of multi-variable positive operators on an infinite dimensional Hilbert space. We also consider the situations where symmetric means of operators A, B, C become linear combinations of them.

Eliminating Qubit-Type Cross-Talk in the omg Protocol.

The omg protocol is a promising paradigm that uses multiple, application-specific, qubit subspaces within the Hilbert space of each single atom during quantum information processing. A key assumption for omg operation is that a subspace can be accessed independently without deleterious effects on information stored in other subspaces. We find that intensity noise during laser-based quantum gates in one subspace can cause decoherence in other subspaces, potentially complicating omg operation. We show, however, that a magnetic-field-induced vector light shift can be used to eliminate this source of decoherence. As this technique simply requires choosing a specific, magnetic field-dependent polarization for the gate lasers, itis straightforward to implement and potentially helpful for omg-based quantum technology.

Managing temperature in open quantum systems strongly coupled with structured environments.

In non-perturbative non-Markovian open quantum systems, reaching either low temperatures with the hierarchical equations of motion (HEOM) or high temperatures with the Thermalized Time Evolving Density Operator with Orthogonal Polynomials Algorithm (T-TEDOPA) formalism in Hilbert space remains challenging. We compare different ways of modeling the environment. Sampling the Fourier transform of the bath correlation function, also called temperature dependent spectral density, proves to be very effective. T-TEDOPA [Tamascelli et al., Phys. Rev. Lett. 123, 090402 (2019)] uses a linear chain of oscillators with positive and negative frequencies, while HEOM is based on the complex poles of an optimized rational decomposition of the temperature dependent spectral density [Xu et al., Phys. Rev. Lett. 129, 230601 (2022)]. Resorting to the poles of the temperature independent spectral density and of the Bose function separately is an alternative when the problem due to the huge number of Bose poles at low temperatures is circumvented. Two examples illustrate the effectiveness of the HEOM and T-TEDOPA approaches: a benchmark pure dephasing case and a two-bath model simulating the dynamics of excited electronic states coupled through a conical intersection. We show the efficiency of T-TEDOPA to simulate dynamics at a finite temperature by using either continuous spectral densities or only all the intramolecular oscillators of a linear vibronic model calibrated from abinitio data of a phenylene ethynylene dimer.

Calibration-independent bound on the unitarity of a quantum channel with application to a frequency converter

We report on a method to certify a unitary operation with the help of source and measurement apparatuses whose calibration throughout the certification process needs not be trusted. As in the device-independent paradigm our certification method relies on a Bell test and requires no assumption on the underlying Hilbert space dimension, but it removes the need for high detection efficiencies by including the single additional assumption that non-detected events are independent of the measurement settings. The relevance of the proposed method is demonstrated experimentally by bounding the unitarity of a quantum frequency converter. The experiment starts with the heralded creation of a maximally entangled two-qubit state between a single 40Ca+ ion and a 854 nm photon. Entanglement preserving frequency conversion to the telecom band is then realized with a non-linear waveguide embedded in a Sagnac interferometer. The resulting ion-telecom photon entangled state is assessed by means of a Bell-CHSH test from which the quality of the frequency conversion is quantified. We demonstrate frequency conversion with an average certified fidelity of ≥84% and an efficiency ≥3.1 × 10−6 at a confidence level of 99%. This ensures the suitability of the converter for integration in quantum networks from a trustful characterization procedure.

Exponential stability of non-instantaneous impulsive second-order fractional neutral stochastic differential equations with state-dependent delay

The aim of this manuscript mainly concerned with a new class of exponential stability for non-instantaneous impulsive second-order fractional neutral stochastic differential equations (NIIFNSDEs) with state-dependent delay driven by Poisson jump in a separable Hilbert spaces. Firstly, a more appropriate concept of mild solution is introduced. Secondly, sufficient conditions are derived for the existence of mild solution by means stochastic analysis, fractional calculus approach and Mönch fixed point theorem with appropriate hypotheses on non-linear continuous functions combined with solution operator. Finally, stability results are derived based on the pth moment exponential stable with the help of new impulsive integral inequality techniques. In addition, an example is provided to validate the theoretical results. This manuscript is the unique combination of the new theoretical investigation which is simulated numerically. Our work extends the works of Arthi et al. (2014), Das et al. (2016), Huang et al. (2018), Pandey et al. (2014), Wang et al. (2018), Yan and Jia (2016).

A Novel Construction of the g-Riesz Decomposition in Hilbert Spaces

The concept of the g-frame, a generalized frame in Hilbert spaces, has garnered attention in recent research. While numerous properties of g-frames have been explored, certain aspects remain unexamined, including a novel construction approach for the g-Riesz decomposition in Hilbert spaces. While prior works such as [12] presented the equivalence conditions for g-Riesz decomposition and Khosravi [10] proposed a new construction method for g-frames, neither addressed a new construction method for g-Riesz decomposition. This paper aims to fill this gap by investigating a novel construction method for the specialized g-frame-g-Riesz decomposition. Leveraging operator theory from generalized functional analysis and function space techniques in complex Hilbert spaces, we establish necessary and sufficient conditions for constructing g-Riesz decompositions, an area insufficiently explored by Khosravi and [12]. Furthermore, we introduce two annotations and provide proofs demonstrating that g-Riesz bases are equivalent to Riesz bases, aligning with the findings of W.C. Sun in [6]. This underscores the significance of our research. The newly proposed g-Riesz decomposition not only contributes to mathematical inquiry but also holds promise for various applications, particularly in signal and image processing.

An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization

It is known that many problems in science and technology are reduced to the calculation of singular or regular integrals. Basically, these integrals are calculated approximately using quadrature and cubature formulas. In the present paper we develop an algorithm for construction of optimal quadrature formulas in some Hilbert spaces based on discrete analogues of the linear differential operators. Then we apply the algorithm to construct optimal quadrature formulas which are exact for hyperbolic functions and polynomials. We get explicit expressions for the coefficients of the optimal quadrature formulas. The obtained optimal quadrature formulas have m-th order of convergence.

Orthonormal rational functions on a semi-infinite interval

In this paper we propose a novel family of weighted orthonormal rational functions on a semi-infinite interval. We write a sequence of integer-coefficient polynomials in several forms and derive their corresponding differential equations. These equations do not form Sturm-Liouville problems. We overcome this disadvantage by multiplying some factors, resulting in a sequence of irrational functions. We deduce various generating functions of this sequence and find the associated Sturm-Liouville problems, which bring orthogonality. Then we establish a Hilbert space of functions defined on a semi-infinite interval with an inner product induced by a weight function determined by the Sturm-Liouville problems mentioned above. The even subsequence of the irrational function sequence above forms an orthonormal basis for this space. The Fourier expansions of some power functions are deduced. The sequence of non-positive integer power functions forms a non-orthogonal basis for this Hilbert space. We give two examples, one example of Fourier expansion and one example of interpolation as applications.