Hilbert Space Research Articles

Asymptotics of the sticky particles evolution

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.

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

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.

Black hole firewalls and quantum mechanics

Abstract Firewalls in black holes are easiest to understand by imposing time reversal
invariance, together with a unitary evolution law. The best approach seems to
be to split up the time span of a black hole into short periods, during which no
firewalls can be detected by any observer. Then, gluing together subsequent
time periods, firewalls seem to appear, but they can always be transformed
away. At all times we need a Hilbert space of a finite dimension, as long as
particles far separated from the black hole are ignored. Our conclusion con tradicts other findings, as these assume that information that entered into a
black hole, cannot re-emerge. But re-emergence of that information is exactly
what our version of firewalls is supposed to ensure. Indeed, the firewall trans formation removes the entanglement between very early and very late in- and out-particles, in a far-from-trivial way.

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.

Towards minimal self-testing of qubit states and measurements in prepare-and-measure scenarios

Abstract Self-testing is a promising approach to certifying quantum states or measurements. Originally, it relied solely on the output statistics of the measurements involved in a device-independent setup. Extra physical assumptions about the system make the setup semi-device-independent. In the latter setup, we consider a prepare-and-measure scenario in which the dimension of the mediating particle is assumed to be two. In a setup involving four (three) preparations and three (two) projective measurements in addition to the target, we exemplify how to self-test any four- (three-) outcome extremal positive operator-valued measure using a linear witness. One of our constructions also achieves self-testing of any number of states with the help of as many projective measurements as the dimensionality of the space spanned by the corresponding Bloch vectors. These constructions are conjectured to be minimal in terms of the number of preparations and measurements required. In addition, we implement one of our constructions in the prepare-and-measure setup on IBM and IonQ quantum processors and certify the existence of a complex qubit Hilbert space based on the data obtained from these experiments.

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.

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.

Algorithms for split equality variational inequality and fixed point problems

ABSTRACT This study presents algorithms for addressing split equality variational inequality and fixed point problems in real Hilbert spaces that are inertial-like subgradient extragradient and inertial-like Tseng extragradient, respectively. We prove that the resulting sequences of the proposed algorithms converge strongly to solutions of the problem provided that the underlying mappings are quasi-monotone, uniformly continuous and quasi-nonexpansive mappings under some mild conditions. Furthermore, numerical experiments are shown to demonstrate the effectiveness of our techniques.

Frames and Dual Frames for Krein Spaces

In last decades, the operator theory of Krein spaces, Krein space approaches, and various generalizations of frames have interested many mathematicians due to their potential applications in mathematics and engineering. This paper addresses the frame theory for Krein spaces. We present some properties of J-orthonormal bases, Parseval frames and frames for Krein spaces, and a parametric expression of all duals of an arbitrarily given frame in Krein spaces. This study shows that the frame theory for Krein spaces is not a direct generalization of the frame theory for Hilbert spaces.

Fuzzy Soft Paranormal Operator in Fuzzy Soft Hilbert Space

This paper defines the fuzzy soft paranormal operator and discusses several fundamental fuzzy soft paranormal operator properties in fuzzy soft hilbert space. Some concepts relevant to the fuzzy soft paranormal operator have been defined in fuzzy soft Hilbert space.

Valuation of a financial claim contingent on the outcome of a quantum measurement

Abstract We consider a rational agent who at time 0 enters into a financial contract for which the payout is determined by a quantum measurement at some time T > 0. The state of the quantum system is given in the Heisenberg representation by a known density matrix p. How much will the agent be willing to pay at time 0 to enter into such a contract? In the case of a finite dimensional Hilbert space H, each such claim is represented by an observable X where the eigenvalues of X determine the amount paid if the corresponding outcome is obtained in the measurement. We prove, under reasonable axioms, that there exists a pricing state q which is equivalent to the physical state p such that the pricing function Π takes the linear form Π (X) = P_0T tr(q X) for any claim X, where P_0T is the one-period discount factor. By “equivalent” we mean that p and q share the same null space: thus, for any |ξ〉∈ H one has p|ξ〉 = 0 if and only if q|ξ〉 = 0. We introduce a class of optimization problems and solve for the optimal contract payout structure for a claim based on a given measurement. Then we consider the implications of the Kochen-Specker theorem in this setting and we look at the problem of forming portfolios of such contracts. Finally, we consider multi-period contracts.