Search Results (777)

In the present paper, we use the proximal point method with remotest set control for find an approximate common zero of a finite collection of maximal monotone maps in a real Hilbert space under the presence of computational errors. We prove that the inexact proximal point method generates an approximate solution if these errors are summable. Also, we show that if the computational errors are small enough, then the inexact proximal point method generates approximate solutions Full article

The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a model of RTN obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN. Full article

This work focuses on split feasibility problems in Hilbert spaces. To accelerate the convergent rate of gradient-CQ algorithms, we introduce an inertial term. Additionally, non-monotone stepsizes are employed to adjust the relaxation parameter applied to the original stepsizes, ensuring that these original stepsizes maintain a positive lower bound. Thereby, the efficiency of the algorithms is improved. Moreover, the weak and strong convergence of the proposed algorithms are established through proofs that exhibit a similar symmetry structure and do not require the assumption of Lipschitz continuity for the gradient mappings. Finally, the LASSO problem is presented to illustrate and compare the performance of the algorithms. Full article

Schmidt entropy is used as a common denotation for all Hilbert space entropies that can be defined via the Schmidt decomposition theorem; they include quantum entanglement entropies and classical separability entropies. Exact results about the asymptotic growth in time of such entropies (in the form of Renyi entropies of any order $\ge 1$) are directly derived from the Schmidt decompositions. Such results include a proof that pure point spectra entail boundedness in time of all entropies of order larger than 1; and that slower than exponential transport forbids faster than logarithmic asymptotic growth. Applications to coupled Quantum Kicked Rotors and to Floquet systems are presented. Full article

Aspect-Based Sentiment Analysis (ABSA) has gained significant popularity in recent years, which emphasizes the aspect-level sentiment representation of sentences. Current methods for ABSA often use pre-trained models and graph convolution to represent word dependencies. However, they struggle with long-range dependency issues in lengthy texts, resulting in averaging and loss of contextual semantic information. In this paper, we explore how richer semantic relationships can be encoded more efficiently. Inspired by quantum theory, we construct superposition states from text sequences and utilize them with quantum measurements to explicitly capture complex semantic relationships within word sequences. Specifically, we propose an attention-based semantic dependency fusion method for ABSA, which employs a quantum embedding module to create a superposition state of real-valued word sequence features in a complex-valued Hilbert space. This approach yields a word sequence density matrix representation that enhances the handling of long-range dependencies. Furthermore, we introduce a quantum cross-attention mechanism to integrate sequence features with dependency relationships between specific word pairs, aiming to capture the associations between particular aspects and comments more comprehensively. Our experiments on the SemEval-2014 and Twitter datasets demonstrate the effectiveness of the quantum-inspired attention-based semantic dependency fusion model for the ABSA task. Full article

Studies of subradiance in a chain N two-level atoms in the single excitation regime focused mainly on the complex spectrum of the effective Hamiltonian, identifying subradiant eigenvalues. This can be achieved by finding the eigenvalues N of the Hamiltonian or by evaluating the expectation value of the Hamiltonian on a generalized Dicke state, depending on a continuous variable k. This has the advantage that the sum above N can be calculated exactly, such that N becomes a simple parameter of the system and no longer the size of the Hilbert space. However, the question remains how subradiance emerges from atoms initially excited or driven by a laser. Here we study the dynamics of the system, solving the coupled-dipole equations for N atoms and evaluating the probability to be in a generalized Dicke state at a given time. Once the subradiant regions have been identified, it is simple to see if subradiance is being generated. We discuss different initial excitation conditions that lead to subradiance and the case of atoms excited by switching on and off a weak laser. This may be relevant for future experiments aimed at detecting subradiance in ordered systems. Full article

This paper provides a new proof of the operator norm identity $\parallel Q\parallel = \parallel I-Q\parallel$, where Q is a bounded idempotent operator on a complex Hilbert space, and I is the identity operator. We also derive explicit lower and upper bounds for the distance from an arbitrary idempotent operator to the set of orthogonal projections. Our approach simplifies existing proofs. Full article

This paper presents a mathematical formalism for generative artificial intelligence (GAI). Our starting point is an observation that a “histories” approach to physical systems agrees with the compositional nature of deep neural networks. Mathematically, we define a GAI system as a family of sequential joint probabilities associated with input texts and temporal sequences of tokens (as physical event histories). From a physical perspective on modern chips, we then construct physical models realizing GAI systems as open quantum systems. Finally, as an illustration, we construct physical models realizing large language models based on a transformer architecture as open quantum systems in the Fock space over the Hilbert space of tokens. Our physical models underlie the transformer architecture for large language models. Full article

We analyse the results of three information retrieval tests on conceptual combinations that we have recently performed using corpora of Italian language. Each test has the form of a ‘Bell-type test’ and was aimed at identifying ‘quantum entanglement’ in the combination, or composition, of two concepts. In the first two tests, we studied the Italian translation of the combination The Animal Acts, while in the third test, we studied the Italian translation of the combination The Animal eats the Food. We found a significant violation of Bell’s inequalities in all tests. Empirical patterns confirm the results obtained with corpora of English language, which indicates the existence of deep structures in concept formation that are language independent. The systematic violation of Bell’s inequalities suggests the presence of entanglement, and indeed, we elaborate here a ‘quantum model in Hilbert space’ for the collected data. This investigation supports our theoretical hypothesis about entanglement as a phenomenon of ‘contextual updating’, independent of the nature, micro-physical or conceptual-linguistic, of the entities involved. Finally, these findings allow us to further clarify the mutual relationships between entanglement, Cirel’son’s bound, and no-signalling in Bell-type situations. Full article

This paper extends the previous work by the author on m-null pairs of operators in Hilbert space. If an elementary operator L has elementary symbols A and B that are p-null and q-null, respectively, then L is $(p+q-1)$-null. Here, we prove the converse under strictness conditions, modulo some nonzero multiplicative constant—if L is strictly $(p+q-1)$-null, then a scalar $\lambda \ne 0$ exists such that $\lambda A$ is strictly p-null and ${\lambda }^{-1}B$ is strictly q-null. Our constructive argument relies essentially on algebraic and combinatorial methods. Thus, the result obtained by Gu on m-isometries is recovered without resorting to spectral analysis. For several operator classes that generalize m-isometries and are subsumed by m-null operators, the result is new. Full article

Air quality modeling has become a strategic area within data science, particularly in urban contexts where pollution exhibits high variability and nonlinear dynamics. This study provides a mathematical and computational comparison between two predictive paradigms: the classical Long Short-Term Memory (LSTM) model, designed for sequential analysis of time series, and the quantum model Quantum Support Vector Machine (QSVM), based on kernel methods applied in Hilbert spaces. Both approaches are applied to real PM2.5 concentration data collected at the Plaza Castilla monitoring station (Madrid) over the period 2017–2024. The LSTM model demonstrates moderate accuracy for smooth seasonal trends but shows limited performance in detecting extreme pollution events. In contrast, the QSVM achieves perfect binary classification through a quantum kernel based on angle encoding, with significantly lower training time and computational cost. Beyond the empirical results, this work highlights the growing potential of Quantum Artificial Intelligence as a hybrid paradigm capable of extending the boundaries of classical models in complex environmental prediction tasks. The implications point toward a promising transition to quantum-enhanced predictive systems aimed at advancing urban sustainability. Full article

The inverse mixed variational inequality problem comes from classical variational inequality, and it has many applications. In this paper, we propose new algorithms to study the inverse mixed variational inequality problems in Hilbert spaces, and these algorithms are based on the generalized projection operator. Next, we establish convergence theorems under inverse strong monotonicity conditions. In addition, we also provide inertial-type algorithms for the inverse mixed variational inequality problems with conditions that differ from the above convergence theorems. Full article

Variational inequality problems (VIPs) provide a versatile framework for modeling a wide range of real-world applications, including those in economics, engineering, transportation, and image processing. In this paper, we propose a novel iterative algorithm for solving VIPs in real Hilbert spaces. The method integrates a double-inertial mechanism with the two-subgradient extragradient scheme, leading to improved convergence speed and computational efficiency. A distinguishing feature of the algorithm is its self-adaptive step size strategy, which generates a non-monotonic sequence of step sizes without requiring prior knowledge of the Lipschitz constant. Under the assumption of monotonicity for the underlying operator, we establish strong convergence results. Numerical experiments under various initial conditions demonstrate the method’s effectiveness and robustness, confirming its practical advantages and its natural extension of existing techniques for solving VIPs. Full article

Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects. Full article

This paper addresses the critical challenge of point cloud down-sampling for digital twin creation, where reducing data volume while preserving geometric fidelity remains an ongoing research problem. We propose a novel down-sampling approach that combines Low-Discrepancy Sequences (LDS) with Hilbert curve ordering to create a method that preserves both global distribution characteristics and local geometric features. Unlike traditional methods that impose uniform density or rely on computationally intensive feature detection, our LDS-Hilbert approach leverages the complementary mathematical properties of Low-Discrepancy Sequences and space-filling curves to achieve balanced sampling that respects the original density distribution while ensuring comprehensive coverage. Through four comprehensive experiments covering parametric surface fitting, mesh reconstruction from basic closed geometries, complex CAD models, and real-world laser scans, we demonstrate that LDS-Hilbert consistently outperforms established methods, including Simple Random Sampling (SRS), Farthest Point Sampling (FPS), and Voxel Grid Filtering (Voxel). Results show parameter recovery improvements often exceeding 50% for parametric models compared to the FPS and Voxel methods, nearly 50% better shape preservation as measured by the Point-to-Mesh Distance (than FPS) and up to 160% as measured by the Viewpoint Feature Histogram Distance (than SRS) on complex real-world scans. The method achieves these improvements without requiring feature-specific calculations, extensive pre-processing, or task-specific training data, making it a practical advance for enhancing digital twin fidelity across diverse application domains. Full article