Hilbert Space Research Articles

Some properties of (p, q)-continuous fusion frames in Hilbert spaces

Some properties of (p, q)-continuous fusion frames in Hilbert spaces

Symmetry-Protected Topological Phases of Mixed States in the Doubled Space

The interplay of symmetry and topology in quantum many-body mixed states has recently garnered significant interest. In a phenomenon not seen in pure states, mixed states can exhibit average symmetries—symmetries that act on component states while leaving the ensemble invariant. In this work, we systematically characterize symmetry-protected topological (SPT) phases of short-range entangled (SRE) mixed states of spin systems—protected by both average and exact symmetries—by studying their pure Choi states in a doubled Hilbert space, where the familiar notions and tools for SRE and SPT pure states apply. This advantage of the doubled space comes with a price: extra symmetries as well as subtleties around how hermiticity and positivity of the original density matrix constrain the possible SPT invariants. Nevertheless, the doubled-space perspective allows us to obtain a systematic classification of mixed-state SPT (MSPT) phases. We also investigate the robustness of MSPT invariants under symmetric finite-depth quantum channels, the bulk-boundary correspondence for MSPT phases, and the consequences of the MSPT invariants for the separability of mixed states and the symmetry-protected sign problem. In addition to MSPT phases, we study the patterns of spontaneous symmetry breaking (SSB) of mixed states, including the phenomenon of exact-to-average SSB, and the order parameters that detect them. Mixed-state SSB is related to an ingappability constraint on symmetric Lindbladian dynamics. Published by the American Physical Society 2025

Quantum combinatorial optimization beyond the variational paradigm: Simple schedules for hard problems

Advances in quantum algorithms suggest a tentative scaling advantage on certain combinatorial optimization problems. Recent work, however, has also reinforced the idea that barren plateaus render variational algorithms ineffective on large Hilbert spaces. Hence, finding annealing protocols by variation ultimately appears to be difficult. Similarly, the adiabatic theorem fails on hard problem instances with first-order quantum phase transitions. Here we show how to use the spin coherent-state path integral to shape the geometry of quantum adiabatic evolution, leading to annealing protocols at polynomial overhead that provide orders-of-magnitude improvements in the probability to measure optimal solutions, relative to linear protocols. These improvements are not obtained on a controllable toy problem but on randomly generated hard instances (Sherrington-Kirkpatrick and maximum 2-satisfiability), making them generic and robust. Our method works for large systems and may thus be used to improve the performance of state-of-the-art quantum devices. Published by the American Physical Society 2025

Tensorial Maclaurin Approximation Bounds and Structural Properties for Mixed-Norm Orlicz–Zygmund Spaces

This article explores two distinct function spaces: Hilbert spaces and mixed-Orlicz–Zygmund spaces with variable exponents. We first examine the relational properties of Hilbert spaces in a tensorial framework, utilizing self-adjoint operators to derive key results. Additionally, we extend a Maclaurin-type inequality to the tensorial setting using generalized convex mappings and establish various upper bounds. A non-trivial example involving exponential functions is also presented. Next, we introduce a new function space, the mixed-Orlicz–Zygmund space ℓq(·)logβLp(·), which unifies Orlicz–Zygmund spaces of integrability and sequence spaces. We investigate its fundamental properties including separability, compactness, and completeness, demonstrating its significance. This space generalizes the existing structures, reducing to mixed-norm Lebesgue spaces when β=0 and to classical Lebesgue spaces when q=∞,β=0. Given the limited research on such spaces, our findings contribute valuable insights to the functional analysis.

RELATIVE POSITION BETWEEN A PAIR OF SPIN MODEL SUBFACTORS

Abstract Jones [‘Two subfactors and the algebraic decomposition of bimodules over $II_1$ factors’, Acta Math. Vietnam33(3) (2008), 209–218] proposed the study of ‘two subfactors’ of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. Motivated by this, we initiate a systematic study of a special class of two subfactors, namely a pair of spin model subfactors. We characterize which pairs of distinct complex Hadamard matrices in $M_n(\mathbb {C})$ give rise to distinct spin model subfactors. Then, a detailed investigation is carried out for $n=2$ , where the spin model subfactors correspond to $\mathbb {Z}_2$ -actions on the hyperfinite type $II_1$ factor R. We observe that the intersection of the pair of spin model subfactors in this case is a nonirreducible vertex model subfactor and we characterize it as a diagonal subfactor. A few key invariants for the pair of spin model subfactors are computed to understand their relative positions.

Stochastic bundles, new classes of Gaussian processes and white noise-space analysis indexed by measures

Starting from a fixed measure space (X,F,μ), with μ a positive sigma-finite measure defined on the sigma-algebra F, we continue here our study of a generalization W(μ) of Brownian motion, and introduce a corresponding white-noise process. In detail, the generalized Brownian motion is a centered Gaussian process W(μ), indexed by the elements A in F of finite μ measure, and with covariance function μ(A∩B). The purpose of our present paper is to make precise and study the corresponding white-noise process, i.e., a point-wise process which is indexed by X, and which arises as a generalized μ derivative of W(μ). A key tool in our definition and analysis of this pair is a construction of three operators between the underlying Hilbert spaces. One of these operators is a stochastic integral, the second is a gradient associated with the measure μ, and the third is a mathematical expectation in the underlying probability space. We show that, with the setting of families of processes indexed by sets of measures μ, our results lead to new stochastic bundles. They serve in turn to extend the tool set for stochastic calculus.

Homogeneous Functions on Hilbert Spaces and Quasiconformal Transformations of the Sphere

Homogeneous Functions on Hilbert Spaces and Quasiconformal Transformations of the Sphere

Existence and convergence of approximate solutions for nonlinear measure driven differential systems

We investigate the existence and convergence of approximate solutions for measure driven nonlinear differential systems defined on a separable Hilbert space. The system is restricted to finite-dimensional subspaces by using projection operators. We utilize analytic semigroup theory to construct the space in which existence and convergence are studied. In particular, the approximate solutions in finite dimensions are considered, and their convergence is verified. We apply our major findings to a partial differential equation.

δ-Rescaled pure super greedy algorithm with respect to Riesz dictionary

We propose the [Formula: see text]-Rescaled Pure Super Greedy Algorithm ([Formula: see text]-RPSGA) for approximation with respect to a dictionary [Formula: see text] in a Hilbert space. We estimate the upper bound of the error for the [Formula: see text]-RPSGA when [Formula: see text] is a Riesz dictionary. Our results show that the convergence rate of the [Formula: see text]-RPSGA on the closure of the convex hull of [Formula: see text] is optimal. Then, we design the [Formula: see text]-Rescaled Pure Super Greedy Learning Algorithm ([Formula: see text]-RPSGLA) for the kernel-based regression in supervised learning. We obtain the convergence rates of the [Formula: see text]-RPSGLA for continuous kernels. When the kernel is infinitely smooth, we derive a convergence rate that can be arbitrarily close to the best rate [Formula: see text] under a mild assumption of the regression function.

Generalization of Titchmarsh's theorem for the linear Canonical Fourier–Bessel transform

Let an f function belongs to the Hilbert space L 2 ( R ) , and let f ^ be the Fourier transform of f. The classical theorem of E. C. Titchmarsh states that a function f belongs to the Lipschitz class Lip ( α , 2 ) , ( 0 < α < 1 ) , if and only if ∫ | λ | ≥ s | f ^ ( λ ) | 2 d λ = O ( s − 2 α ) as s → + ∞ . In this paper, we generalize this result for the case when the function f belongs to the canonical Bessel–Lipschitz class Li p M ( α , 2 ) .

Vertex operators for the kinematic algebra of Yang-Mills theory

The kinematic algebra of Yang-Mills theory can be understood in the framework of homotopy algebras: the L∞ algebra of Yang-Mills theory is the tensor product of the color Lie algebra and a kinematic space that carries a C∞ algebra. There are also hidden structures that generalize Batalin-Vilkovisky algebras, which explain color-kinematics duality and the double copy but are only partially understood. We show that there is a representation of the C∞ algebra, in terms of vertex operators, on the Hilbert space of a first-quantized worldline theory. To this end we introduce A∞ morphisms, which define the vertex operators and which inject the C∞ algebra into the strictly associative algebra of operators on the Hilbert space. We also take the first steps to represent the hidden structures on the same space. Published by the American Physical Society 2025

Simulating Electronic Structure on Bosonic Quantum Computers.

Quantum harmonic oscillators, or qumodes, provide a promising and versatile framework for quantum computing. Unlike qubits, which are limited to two discrete levels, qumodes have an infinite-dimensional Hilbert space, making them well-suited for a wide range of quantum simulations. In this work, we focus on the molecular electronic structure problem. We propose an approach to map the electronic Hamiltonian into a qumode bosonic problem that can be solved on bosonic quantum devices using the variational quantum eigensolver (VQE). Our approach is demonstrated through the computation of ground potential energy surfaces for benchmark model systems, including H2 and the linear H4 molecule. The preparation of trial qumode states and the computation of expectation values leverage universal ansatzes based on the echoed conditional displacement (ECD), or the selective number-dependent arbitrary phase (SNAP) operations. These techniques are compatible with circuit quantum electrodynamics (cQED) platforms, where microwave resonators coupled to superconducting transmon qubits can offer an efficient hardware realization. This work establishes a new pathway for simulating many-fermion systems, highlighting the potential of hybrid qubit-qumode quantum devices in advancing quantum computational chemistry.

Interconnected systems for the Lyapunov’s linear operator equations in the Hilbert space

Interconnected systems for the Lyapunov’s linear operator equations in the Hilbert space

An extragradient-type algorithm for solving a nonmonotone equilibrium problem over the fixed point set in a Hilbert space

An extragradient-type algorithm for solving a nonmonotone equilibrium problem over the fixed point set in a Hilbert space

Characterizations of the Crandall–Pazy Class of C0-semigroups on Hilbert Spaces and Their Application to Decay Estimates

Characterizations of the Crandall–Pazy Class of C0-semigroups on Hilbert Spaces and Their Application to Decay Estimates

On spaces of functions associated to the canonical commutation relation

Topological generalized eigenvectors have a pivotal role in the rigged Hilbert space formalism of quantum mechanics. In this paper, we take a step back and deal with algebraic generalized eigenvectors of the position and momentum operators on the simplest algebraic irreducible representation of the Heisenberg algebra. Then we investigate the spaces of functions that can be associated to P and Q for diagonalisation. In particular, we shall see that a Krein space can be naturally associated to the momentum operator.

Reproducing kernel Hilbert space method for high-order linear Fredholm integro-differential equations with variable coefficients

Reproducing kernel Hilbert space method for high-order linear Fredholm integro-differential equations with variable coefficients

A unified criterion for semiglobal and global finite/fixed-time stability of stochastic systems

In this paper, semiglobal and global finite/fixed-time stability of a class of semilinear stochastic systems in Hilbert spaces are studied. By employing the stopping time technique and applying the infinite-dimensional Itô formula, a unified Lyapunov criterion is established under which global finite-time, global fixed-time, semiglobal finite-time and semiglobal fixed-time stability can all be obtained. This fills the research gaps in the finite/fixed-time stability of stochastic distributed parameter systems and the semiglobal finite/fixed-time stability of stochastic systems. Moreover, compared to existing results on finite/fixed-time stability for stochastic systems, our Lyapunov criterion can still work well under more stringent conditions. This is done by adding a term which can reflect the stabilising effect of the diffusion term and replacing original fractional power functions with a class of piecewise continuous fractional power functions that are no longer required to be increasing. Finally, illustrative examples are given and the obtained theoretical results are verified by numerical simulations.

Quantum-Inspired Latent Variable Modeling in Multivariate Analysis

Latent variables play a crucial role in psychometric research, yet traditional models often struggle to address context-dependent effects, ambivalent states, and non-commutative measurement processes. This study proposes a quantum-inspired framework for latent variable modeling that employs Hilbert space representations, allowing questionnaire items to be treated as pure or mixed quantum states. By integrating concepts such as superposition, interference, and non-commutative probabilities, the framework captures cognitive and behavioral phenomena that extend beyond the capabilities of classical methods. To illustrate its potential, we introduce quantum-specific metrics—fidelity, overlap, and von Neumann entropy—as complements to correlation-based measures. We also outline a machine-learning pipeline using complex and real-valued neural networks to handle amplitude and phase information. Results highlight the capacity of quantum-inspired models to reveal order effects, ambivalent responses, and multimodal distributions that remain elusive in standard psychometric approaches. This framework broadens the multivariate analysis theoretical and methodological toolkit, offering a dynamic and context-sensitive perspective on latent constructs while inviting further empirical validation in diverse research settings.

Calculating the spectral response strength of non-Hermitian systems with an exceptional point directly from wave simulations

Exceptional points are spectral degeneracies in open systems, which have attracted considerable attention in recent years. One reason for the attention is that physical systems with an exceptional point exhibit a strong spectral response to perturbations. This response can be quantified by a single quantity known as the spectral response strength. Up to now, the theory for the spectral response strength is restricted to effective non-Hermitian Hamiltonians acting on a finite-dimensional Hilbert space, which often appear naturally within coupled-mode theory. We introduce here a scheme for the computation of the spectral response strength directly from numerical results of wave simulations, which correspond to an infinite Hilbert-space dimension. Our approach is based on the relation of the spectral response strength to the Petermann factors of eigenstates in a system near the exceptional point. To illustrate our theory, we consider three different photonic systems: a microring dimer, waveguide-coupled microrings, and a weakly deformed microdisk. A comparison to results provided by effective Hamiltonians based on coupled-mode theory and perturbation theory demonstrates very good agreement. Published by the American Physical Society 2025