Novel subgradient extragradient methods for equilibrium problems in Hilbert spaces

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Light carries both spin (polarization) and orbital angular momentum. Combining these degrees of freedom produces hybrid spin-orbit states that live in a high-dimensional Hilbert space, offering greater information capacity and robustness for optical communication, quantum technologies, and metrology. However, generating arbitrary states in these spaces and characterizing them efficiently has remained difficult. Here we show a compact metasurface that generates arbitrary spin-orbit states in a four-dimensional Hilbert space, visualized on a Poincaré hypersphere, with straightforward scalability to higher dimensions. Using a tetratomic unit cell, the single-layer device precisely controls complex amplitude, phase, and polarization. We further introduce an efficient interferometric scheme that reconstructs the full density matrix of any N-dimensional spin-orbit state using only three interferograms. This approach uncovers an intrinsic spin-orbit parity order that governs the symmetry of projected intensity patterns, independent of the weighting of the eigenmodes, and enables controlled mode transformations through higher-order geometric phases. These advances establish a versatile platform for high-dimensional photonic technologies.

Contextuality, incompatibility, and intra-system entanglement of mental markers: From cognition and decision making to medicine.

Open quantum system dynamics traditionally described by a quantum master equation can be effectively reformulated in the Liouville space of vectorised density matrices. Despite an extensive use of the Liouville space framework to describe quantum states evolution, its application to quantum metrology is not well explored. Several papers have studied a Liouville-space generalisation of the quantum Fisher information, called the dissipative quantum Fisher information (DQFI). We perform a comparative study between the quantum Fisher information (QFI) in Hilbert space and DQFI, deriving an explicit relation between the two quantities for qubit, qudit, and harmonic oscillator systems. The derived relations are nontrivial, and depend on the state's purity. We show that for a single qubit, the QFI can be efficiently recovered from the DQFI via a simple mapping, while in higher dimensions the expression becomes convoluted and does not convert to a compact form. We find examples where the DQFI is neither an upper nor lower bound for the Hilbert space QFI, and suggest that the informational content of the DQFI itself cannot be straightforwardly interpreted. Our results clarify the limitations of the Liouville space based approach and structural differences with the QFI in the context of quantum parameter estimation problem.

We present a deterministic optimization framework for neural network quantum states (NQS) designed to bypass the sampling variance and slow mixing issues inherent in stochastic optimization. By projecting a neural backflow ansatz onto dynamically evolving configuration subspaces, our method provides a systematic route for optimizing the selected variational component of the wave function and estimating residual correlation through a posthoc second-order perturbative correction. The implementation utilizes a hybrid CPU-GPU architecture that shows empirical sublinear wall-time scaling with respect to the subspace size over the tested range, enabling the calculation of strongly correlated systems, such as the chromium dimer, within Hilbert spaces of 1023 configurations. Benchmarks on molecular bond dissociations demonstrate that this deterministic approach yields stable convergence and accuracies comparable to selected reference methods in the tested systems.

Orbital angular momentum (OAM) of light has acquired the interest of scientists due to its novel properties. The high-dimensional Hilbert space and entanglement that OAM offers make it a valuable DOF for many quantum optical protocols. However, conventional measurement of OAM state, which exploits coincidence count-based quantum state tomography (QST), suffers from low-brightness and subsequent need for data accumulation times. Such problems are a serious obstacle, especially for high-dimensional OAM qudits. In this work, we suggest stimulated emission tomography (SET) as a solution for bright and efficient measurement of OAM entangled photons. We show that SET can successfully extract information of SPDC photons by measuring the spiral bandwidth and reconstructing the density matrix of SPDC photons. The fidelity and linear entropy of the reconstructed density matrix are [Formula: see text] and [Formula: see text], respectively, while the time required for each projective measurement is 1s. Our results show the potential of SET as an efficient alternative to conventional QST.

As a type of non-Abelian anyons, Yang-Lee anyons have topological quantum computing potential comparable to that of Fibonacci anyons, and their non-unitary braiding property is conducive to the construction of non-unitary quantum gates. Any braiding process of Yang-Lee anyons can be realized by combining two basic braiding matrices, which are generated by the R-matrix of the Yang-Baxter equation. However, one of the basic braiding matrices of the braiding process of Yang-Lee anyons is difficult to be simulated directly because of its non-unitary property. To address this challenge, we propose, using the linear combination of unitaries (LCU) framework, the first probabilistic simulation scheme for the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding. We design two feasible quantum circuits in a qubit-qutrit hybrid system and in a pure-qubit system. Both schemes achieve simulation through controlled operations and ancillary qubit measurements, and the success probability is jointly determined by the input state, the dimensions of the total Hilbert space and the normalization factor. This work presents an approach to simulating the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding, which can serve as a building block for non-unitary quantum information processing.

In this paper, we would establish the existence and stability of periodic solutions to the Benjamin–Bona–Mahony–Burgers (BBM-Burgers) equation in H 0 1 ( [ 0 , 1 ] ) , whose medium interior is applied with a time-periodic force f ( x , t ) with period θ. High regularity analysis has been conducted in Hilbert spaces 1 $ ]]> H ℓ , ℓ > 1 . We also consider periodic solution to the same initial boundary value problem (IBVP) scenario of a pseudo-parabolic-regularized equation as an extension of the BBM-Burgers in H ℓ , ℓ = { 1 , 2 } .

The purpose of this paper is to design a novel iterative algorithm to solve a generalized split feasibility and fixed point problem with multiple output sets (GSFFPPM) in the framework of Hilbert spaces. The proposed algorithm combines inertial extrapolation and the S-iterative methodology to accelerate convergence, Tikhonov regularization to ensure stability, and viscosity approximation to guarantee strong convergence to a solution of the GSFFPPM. Due to the generality of our model, we demonstrate the applicability of our iterative method to important classes of problems, including split variational inclusions and split equilibrium problems. To illustrate the practical relevance and computational efficiency of the proposed method, we present numerical experiments on real-world models such as Nash–Cournot semi-oligopolistic market equilibria and signal recovery tasks. These numerical experiments demonstrate the robustness and effectiveness of the proposed method.

We present a streamlined implementation of a computational framework for constructing and analyzing reduced dynamical maps for complex system-bath models at finite temperature. The methodology is based on three established ingredients of quantum dynamics: the Choi-Jamiołkowski isomorphism for the representation of quantum channels, thermofield (TFD) purification of thermal environments, and tensor-train (TT) propagation of the resulting enlarged pure state. The reduced map is obtained from a single unitary propagation in a thermofield-doubled Hilbert space and represented in matrix form through the Choi-Jamiołkowski isomorphism. The TFD evolution is implemented in the TT representation, enabling efficient propagation of high-dimensional purified thermal states. We illustrate the methodology for exciton transfer in the Fenna-Matthews-Olson complex with site-dependent structured spectral densities represented by discretized bosonic environments. The resulting maps are used to analyze decoherence, relaxation, and finite-memory effects, and to assess the crossover to an effectively time-local description. The proposed approach provides a route to compute reduced propagators and to post-process them into memory kernels, transfer tensors, and effective kinetic rate descriptions for complex molecular systems.

Quantum kernel methods map clinical features into exponentially large Hilbert spaces where overlapping biological markers can become more separable than in fixed-dimensional classical feature spaces, but existing work evaluates single quantum feature maps, ignores barren-plateau failure modes, and relies on single train-test splits vulnerable to data leakage. Classical diagnostics for Parkinson's disease, breast cancer, and diabetes remain limited by the Specificity-Recall trade-off that fixed-dimensional kernels impose on overlapping biomarker distributions. We propose an adaptive hybrid quantum framework routing clinical data through three distinct quantum feature maps, namely Angle, Amplitude, and ZZ-entanglement, computing fidelity-based Gram matrices for Quantum SVM and Quantum KNN classifiers. A Logistic Regression meta-learner, trained on strictly out-of-fold predictions from nested cross-validation (5-fold inner, 10-fold outer), learns which quantum kernel generalizes on each dataset and suppresses those that do not. Evaluated on Parkinson's (195 patients), Breast Cancer (569), and Diabetes (768) with 1,000-iteration bootstrapping, the ensemble raised Parkinson's Specificity from 0.585 to 0.813 ([Formula: see text]) while maintaining Recall above 0.95, matched classical RBF-SVM on Breast Cancer (all [Formula: see text]), and improved Diabetes Recall from 0.553 to 0.621 ([Formula: see text]). A standalone Variational Quantum Classifier failed on all datasets (ROC-AUC 0.51 to 0.57), confirming barren plateau limitations. Explainability via SHAP, LIME, and Permutation Importance revealed dataset-dependent kernel trust: the meta-learner suppressed QKNN Amplitude on Diabetes (coefficient [Formula: see text]) while amplifying it on Parkinson's (1.761). PCA-based feature backtracking recovered established biomarkers including Insulin and Glucose for Diabetes, vocal perturbation measures for Parkinson's, and nucleus geometry for Breast Cancer. Noise simulations confirmed graceful degradation under NISQ conditions. The framework performs data-driven kernel selection, removing the need to pre-specify an encoding strategy.

By applying the reproducing kernel approach of Jury [Jury MT. Reproducing kernels, de-Branges-Rovnyak spaces, and norm of weighted composition operators. Proc Am Math Soc. 2007;135:3669–3675], we give norm estimates for a class of composition operators on Hilbert spaces of holomorphic functions on the unit ball or polydisk. The results roughly assert that if a certain multiplier norm of a symbol φ is bounded by 1 or a kernel related to φ is positive semi-definite, then the squared norm of the composition operator C φ is bounded by ( 1 + | φ ( 0 ) | ) / ( 1 − | φ ( 0 ) | ) . This generalizes the classical result on the Hardy and weighted Bergman spaces of the disk and Jury's result on Drury-Arveson space, Hardy and weighted Bergman spaces of the unit ball to a class of reproducing kernel spaces of holomorphic functions including Dirichlet space of the unit ball and the Hardy space of the polydisk.

This paper presents an alternated inertial subgradient extragradient projection method for solving variational inequalities in real Hilbert spaces. The proposed algorithm employing an adaptive halfspace correction parameter at each iteration, has the advantage of improving stability and enhancing convergence. Under suitable conditions, weak convergence and strong convergence are established. Numerical experiments validate the effectiveness of the proposed method compared against existing related algorithms.

The main aim of this research is to study the sufficient conditions for the existence of an integral-form mild solution and approximate controllability results for a new class of the nonlinear Ψ-Caputo fractional Sobolev-type delayed stochastic multivalued system with extended impulsive dynamics in a separable Hilbert space. The Ψ-Caputo fractional derivative offers a versatile and unifying framework that extends the classical fractional differential operators through an appropriate choice of the kernel function Ψ. This adaptability enables more accurate modelling of memory and hereditary characteristics inherent in complex dynamical systems. Firstly, the proposed control system is transferred into an equivalent fixed point problem using the Ψ-Riemann-Liouville fractional integral operator. Then, the Karlin fixed point theorem is implemented to establish the existence of mild solution. Furthermore, the approximate controllability result of the proposed control system is investigated under the consideration that the corresponding linear system is approximate controllable. The main results are derived using fractional calculus, theory of multivalued map, the concepts of stochastic analysis, and fixed point approach. At the end of the paper, a concrete example is provided to illustrate the theoretical findings.

Quark-like modes of fractional orbital angular momentum.

We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the equality of two nonatomic (Borel) probability measures on a locally compact uncountable Polish space is equivalent to testing for the singularity between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an information-theoretic perspective than nonparametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This "separation of measure phenomenon" appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods.

Kernelized linear principal component discriminant analysis.

The extended core inverse was presented for square complex matrices as an extension of the core inverse based on the sum and difference of known generalized inverses. Unlike of existing generalizations of the core inverse, the extended core inverse is an inner inverse of the matrix which need not necessarily be the null matrix of a nilpotent matrix. The aim of this paper is to consider a generalization of the system for defining the extended core inverse based on a weight and to introduce a weighted extended core inverse for operators between two Hilbert spaces. Thus, we consider a new type of weighted generalized inverses. Properties, characterizations and expressions for the weighted extended core inverse are established. The dual version of the weighted extended core inverse is studied too and it presents a new extension of the Moore-Penrose inverse. We apply the weighted extended core inverse and its dual to solve certain systems of linear equations and minimization problems.

Entanglement generation is a cornerstone of quantum information science, yet its speed in Hermitian systems is fundamentally constrained by the coupling strength, a restriction known as the quantum speed limit. Here we demonstrate that this bound can be beaten by exploiting the unique topology of non-Hermitian systems near exceptional points (EPs). Using a pair of trapped ions, we engineer a parity-time symmetric Hamiltonian where the coalescence of eigenstates near the EP distorts the Hilbert space geometry, providing a shortcut for quantum state evolution. We observe that, as the system approaches the EP, the time required to generate a maximally entangled state is markedly reduced with respect to the limits imposed by the equivalent Hermitian interaction. We further uncover a fundamental physical trade-off whereby the acceleration of entanglement is intrinsically coupled to a reduction in the success probability, revealing the information cost of non-Hermitian speedup. Our results suggest that tailored dissipation, rather than being a source of decoherence, can serve as a powerful resource for accelerating quantum dynamics, offering a new paradigm for designing high-speed quantum gates and sensors in hardware-constrained platforms.