Two-dimensional Subspace Research Articles

Optimal violation of a Bell inequality in two-dimensional state spaces for an asymmetric orbital angular momentum system

The symmetric orbital angular momentum (OAM) entanglement in the subspace of { ± ℓ} has been extensively exploited in quantum information science. Here we investigate instead the Bell inequalities for the two-dimensional subspace spanned by asymmetric OAM modes of ℓ1 and ℓ2, usually corresponding to the quantum property of nonmaximal entanglement. We demonstrate, both theoretically and experimentally, the optimal violation of a suitable Clauser-Horne-Shimony-Holt Bell inequality for the asymmetric OAM modes, thus manifesting the entanglement for asymmetric OAM states space.

Two-dimensional and absolutely entanglement-breaking subspaces

Entanglement-breaking (EB) subspaces determine the additivity of entanglement of formation (EOF), which is a long-standing issue in quantum information. We explicitly construct the two-dimensional EB subspaces of any bipartite system, when system dimensions are equal, and we apply the subspaces to construct EB spaces of arbitrary dimensions. We also present partial construction when system dimensions are different. Then, we present the notion and properties of EB subspaces for some systems, and in particular the absolute EB subspaces. We construct some examples of absolute EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states.

Adaptive two-dimensional subspace identification for monitoring batch processes with limited batch data

Data-driven based batch process monitoring is of critical importance in ensuring stable operating processes and consistent product quality. For long-duration batch processes, it is unrealistic to involve expensive data to train a statistical model for monitoring. To model the inherently batch-wise and variable-wise dynamics, nonlinearity, and time-varying characteristics, this paper proposes a local learning-based two-dimensional subspace identification (LL-2D-SID) scheme based on the similarity between the ongoing batch and the previous batches. The similarity is estimated by the extended extrapolative time-warping. Unlike the conventional statistical models using rich batch data, LL-2D-SID through online optimizing mechanism using limited batch data still has good prediction performance. The application of the sintering process in the polytetrafluoroethylene production has demonstrated that the LL-2D-SID based process monitoring scheme can not only accurately track temperature changes but also timely give fault alarms with a lower error alarm rate than the other SID-based process monitoring schemes.

Geometry of entanglement and separability in Hilbert subspaces of dimension up to three

We present a complete classification of the geometry of the mutually complementary sets of entangled and separable states in three-dimensional Hilbert subspaces of bipartite and multipartite quantum systems. Our analysis begins by finding the geometric structure of the pure product states in a given three-dimensional Hilbert subspace, which determines all the possible separable and entangled mixed states over the same subspace. In bipartite systems, we characterise the 14 possible qualitatively different geometric shapes for the set of separable states in any three-dimensional Hilbert subspace (5 classes which also appear in two-dimensional subspaces and were found and analysed by Boyer et al. (Phys Rev A 95:032308, 2017. https://doi.org/10.1103/PhysRevA.95.032308), and 9 novel classes which appear only in three-dimensional subspaces), describe their geometries, and provide figures illustrating them. We also generalise these results to characterise the sets of fully separable states (and hence the complementary sets of somewhat entangled states) in three-dimensional subspaces of multipartite systems. Our results show which geometrical forms quantum entanglement can and cannot take in low-dimensional subspaces.

Transport of energetic particles in turbulent space plasmas: pitch-angle scattering, telegraph, and diffusion equations

Introduction: In this article, we revisit the pitch-angle scattering equation describing the propagation of energetic particles through magnetized plasma. In this case, solar energetic particles and cosmic rays interact with magnetohydrodynamic turbulence and experience stochastic changes in the pitch-angle. Since this happens over an extended period of time, a pitch-angle isotropization process occurs, leading to parallel spatial diffusion. This process is described well by the pitch-angle scattering equation. However, the latter equation is difficult to solve analytically even when considering special cases for the scattering coefficient.Methods: In the past, a so-called subspace approximation was proposed, which has important applications in the theory of perpendicular diffusion. Alternatively, an approach based on the telegraph equation (also known as telegrapher’s equation) has been developed. We show that two-dimensional subspace approximation and the description based on the telegraph equation are equivalent. However, it is also shown that the obtained distribution functions contain artifacts and inaccuracies that cannot be found in the numerical solution to the problem. Therefore, an N-dimensional subspace approximation is proposed corresponding to a semi-analytical/semi-numerical approach. This is a useful alternative compared to standard numerical solvers.Results and Discussion: Depending on the application, the N-dimensional subspace approximation can be orders of magnitude faster. Furthermore, the method can easily be modified so that it can be used for any pitch-angle scattering equation.

A clustering-based partially stratified sampling for high-dimensional structural reliability assessment

Assessing structural reliability problem with high-dimensional random inputs is still challenging due to the “curse of dimensionality”. In this paper, this challenge is addressed by extending the Generalized Distribution Reconstruction Method via Characteristic Function Inversion (GDRM-CFI). Specifically, a clustering-based partially stratified sampling method is proposed for selecting high-dimensional points to numerically evaluate the characteristic function (CF) curve of complex high-dimensional problems. An improved number-theoretical method (i-NTM) is used to establish a uniform, efficient point set, ensuring determinism and reducing variability. Subsequently, a partial stratification approach partitions the high-dimensional space into orthogonal two-dimensional subspaces. The fundamental point set is projected into each subspace, and the k-means clustering algorithm identifies centroids within each, acting as representative points. The complete set of representative points from all subspaces formulates the high-dimensional point set. Numerical examples are investigated, which demonstrate the proposed method is effective for high-dimensional structural reliability assessment.

Planes in Schatten-3

We prove that any two-dimensional real subspace of Schatten-3 can be linearly isometrically embedded into L3. This resolves the case p=3 of Hanner's inequality for Schatten classes, conjectured by Ball, Carlen and Lieb. We conjecture that similar isometric embedding is possible for any p≥1.

Designing pareto-optimal selection systems for multiple minority subgroups and multiple criteria.

Currently used Pareto-optimal (PO) approaches for balancing diversity and validity goals in selection can deal only with one minority group and one criterion. These are key limitations because the workplace and society at large are getting increasingly diverse and because selection system designers often have interest in multiple criteria. Therefore, the article extends existing methods for designing PO selection systems to situations involving multiple criteria and multiple minority groups (i.e., multiobjective PO selection systems). We first present a hybrid multiobjective PO approach for computing selection systems that are PO with respect to (a) a set of quality objectives (i.e., criteria) and (b) a set of diversity objectives where each diversity objective relates to a different minority group. Next, we propose three two-dimensional subspace procedures that aid selection designers in choosing between the PO systems in case of a high number of quality and diversity objectives. We illustrate our novel multiobjective PO approaches via several example applications, thereby demonstrating that they are the first to reveal the complete gamut of eligible PO selection designs and to faithfully capture the Pareto trade-off front in case of more than two objectives. In addition, a small-scale cross-validation study confirms that the resulting PO selection designs retain an advantage over alternative designs when applied in new validation samples. Finally, the article provides a link to an executable code to perform the new multiobjective PO approaches. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Exceptional points and quantum dynamics in a non-Hermitian two-qubit system

We study the exceptional-point (EP) structure and the associated quantum dynamics in a system consisting of a non-Hermitian qubit and a Hermitian qubit. We find that the system possesses two sets of EPs, which divide the system-parameter space into PT -symmetry unbroken, partially broken and fully broken regimes, each with distinct quantum-dynamics characteristics. Particularly, in the partially broken regime, while the PT -symmetry is generally broken in the whole four-dimensional Hilbert space, it is preserved in a two-dimensional subspace such that the quantum dynamics in the subspace are similar to those in the PT -symmetry unbroken regime. In addition, we reveal that the competition between the inter-qubit coupling and the intra-qubit driving gives rise to a complex pattern in the EP variation with system parameters.

Notes on a surface defect in the O(N) model

We study a surface defect in the free and critical O(N) vector models, defined by adding a quadratic perturbation localized on a two-dimensional subspace of the d-dimensional CFT. We compute the beta function for the corresponding defect renormalization group (RG) flow, and provide evidence that at long distances the system flows to a nontrivial defect conformal field theory (DCFT). We use epsilon and large N expansions to compute several physical quantities in the DCFT, finding agreement across different expansion methods. We also compute the defect free energy, and check consistency with the so-called b-theorem for RG flows on surface defects.

Robust moving target detection for FDA-MIMO radar with steering vector mismatches

This paper studies a robust moving target detection problem for frequency diverse array multiple-input multiple-output (FDA-MIMO) radar with a target embedded in Gaussian clutter with an unknown covariance matrix. We focus on the case of signal mismatch where the errors exist in the transmit-receive vector and Doppler steering vector. To solve the above detection problem, we adopt the subspace approach, where the true and nominal transmit-receive and Doppler steering vectors are assumed to lie in four-dimensional and two-dimensional subspaces, respectively, for the special framework for FDA-MIMO radar. At the detector design stage, we propose three robust adaptive detectors based on the one-step generalized likelihood ratio test (GLRT), two-step GLRT, and Wald test, respectively, with the training data free. The proposed detectors own the constant false alarm rate (CFAR) property against the clutter covariance matrix. The numerical examples validate the effectiveness of the proposed detectors.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Two-dimensional playability maps for single-reed woodwind instruments

One method to analyze musical instrument playability is by visualizing a two-dimensional subspace of the musician's control parameters. This hasbeen widely used in the form of Schelleng diagrams for bowed string instruments. There, it is possible to identify regions within the bow force—bowing position subspace where Helmholtz motion is achieved. Such diagrams may be populated either on the basis of experimental measurements, or via physical modeling. In fact, physical modeling is particularly suited to this task, since playing parameters can be directly controlled. It has been recently suggested (Woodhouse, in Proc. SMAC 2023) that similar diagrams may be used for analyzing wind instrument playability. This study aims at exploring this direction using a physical model which has been previously validated against experimental measurements. Initially, an informed decision is made on which parameters should be chosen for plotting an equivalent playability diagram. Subsequently, various diagram versions are generated, focusing on different aspects of the generated tones. As already pointed out by Woodhouse, similarities appear between the diagrams for bowed-string and woodwind instruments. It remains to examine how valuable information regarding woodwind playability may be extracted from such studies.

A Bi-Directional Two-Dimensional Deep Subspace Learning Network with Sparse Representation for Object Recognition

A principal component analysis network (PCANet), as one of the representative deep subspace learning networks, utilizes principal component analysis (PCA) to learn filters that represent the dominant structural features of objects. However, the filters used in PCANet are linear combinations of all the original variables and contain complex and redundant principal components, which hinders the interpretability of the results. To address this problem, we introduce sparse constraints into a subspace learning network and propose three sparse bi-directional two-dimensional PCANet algorithms, including sparse row 2D2PCANet (SR2D2PCANet), sparse column 2D2PCANet (SC2D2PCANet), and sparse row–column 2D2PCANet (SRC2D2PCANet). These algorithms perform sparse operations on the projection matrices in the row, column, and row–column direction, respectively. Sparsity is achieved by utilizing the elastic net to shrink the loads of the non-primary elements in the principal components to zero and to reduce the redundancy in the projection matrices, thus improving the learning efficiency of the networks. Finally, a variety of experimental results on ORL, COIL-100, NEC, and AR datasets demonstrate that the proposed algorithms learn filters with more discriminative information and outperform other subspace learning networks and traditional deep learning networks in terms of classification and run-time performance, especially for less sample learning.

Investigations of complex systems’ dynamics, based on reduced amount of information: introduction to the method

Controlling dynamics of complex systems is one of the most important issues in science and engineering. Thus, there is continuous need to study and develop numerical algorithms of control methods. In this paper, we would like to present our introductory study of a new simple method of investigations of such systems based on vector field properties and reduced amount of applied information. Firstly, we present the basis of our approach for extraction of nonlinear indicators of two-dimensional systems. We show that basing on simplified analyses and exploiting half of commonly applied information, we can precisely estimate widely applied indicators. We prove that our method is simpler, more efficient and more accurate than commonly applied algorithms. After the introductory analysis, we extend our studies and apply the presented method in investigations of complex systems, based on the analysis that we discussed in the first part of the article and carried out in two-dimensional subspaces. We present simplicity and effectiveness of our approach and demonstrate how it simplifies investigations of complex dynamical phenomena. We verify our method studying the example of synchronization and chimera phenomena in the chosen set of coupled oscillators.

A quantum logic gate for free electrons

The topological charge m of vortex electrons spans an infinite-dimensional Hilbert space. Selecting a two-dimensional subspace spanned by m=±1, a beam electron in a transmission electron microscope (TEM) can be considered as a quantum bit (qubit) freely propagating in the column. A combination of electron optical quadrupole lenses can serve as a universal device to manipulate such qubits at the experimenter's discretion. We set up a TEM probe forming lens system as a quantum gate and demonstrate its action numerically and experimentally. High-end TEMs with aberration correctors are a promising platform for such experiments, opening the way to study quantum logic gates in the electron microscope.

Spinning (A)dS black holes with slow-rotation approximation in dynamical Chern-Simons modified gravity

One of the most crucial areas of gravity research, after the direct observation of gravitational waves, is the possible modification of General Relativity at ultraviolet and infrared scales. In particular, the possibility of parity violation should be considered in strong field regime. The Chern-Simons gravity takes into account parity violation in strong gravity regime. For all conformally flat spacetimes and spacetimes with a maximally symmetric two-dimensional subspace, Chern-Simons gravity is identical to General Relativity. Specifically, the Anti-de Sitter (A)dS-Kerr/Kerr black hole is not a solution for Chern-Simons gravity. The slow-rotating BH and the quadratic order in spin solutions are some of the known solutions to quadratic order in spin and they are rotating solutions in the frame of dynamical Chern-Simons gravity. In the present study, for the (A)dS slow-rotating situation (correct to the first order in spin), we derive the linear perturbation equations controlling the metric and the dynamical Chern-Simons field equation corrected to the linear order in spin and to the second order in the Chern-Simons coupling parameter. We show that the black hole of the (A)dS-Kerr solution is stronger (i.e. more compact and energetic) than the Kerr black hole solution and the reason for this feature comes form contributions at Planck scales. Moreover, we calculate the thermodynamical quantities related to this black hole. Finally, we calculate the geodesic equation and derive the effective potential of the black hole.

Phase matching in quantum search algorithm

We systematically generalize the Grover algorithm in a density matrix formalism by exploiting the underlying two-dimensional subspace of the problem. Using this, we derive analytic expressions for the success probability after arbitrary iterations of the generalized Grover operator with two generic phase angles . We show for the phase matching condition with three iterations, success probability can be achieved only with knowledge about the lower bound , where λ is the ratio of marked to total number of states in the database. This result will improve the quantum search algorithm when applied to databases with unknown number of marked states in the specified regime of λ, at the cost of decreased efficiency in the smaller λ region.

Comparison of the efficiency of zero and first order minimization methods in neural networks

To minimize the objective function in neural networks, first-order methods are usually used, which involve the repeated calculation of the gradient. The number of variables in modern neural networks can be many thousands and even millions. Numerous experiments show that the analytical calculation time of an N variable function’s gradient is approximately N/5 times longer than the calculation time of the function itself. The article considers the possibility of using zero-order methods to minimize the function. In particular, a new zero-order method for function minimization, descent over two-dimensional spaces, is proposed. The convergence rates of three different methods are compared: standard gradient descent with automatic step selection, coordinate descent with step selection for each coordinate, and descent over two-dimensional subspaces. It has been shown that the efficiency of properly organized zero-order methods in the considered problems of training neural networks is not lower than the gradient ones.

Mathematical Modeling of Four-dimensional Genetic Regulatory Networks Using a Logistic Function

Mathematical modeling is a universal tool for the study of complex systems. In this paper formulas for characteristic numbers of critical points for the systems of order four (4D) are considered. We show how an unstable focus-focus can appear in a four-dimensional system. Projections of 4D trajectories on two-dimensional and threedimensional subspaces are shown. In the considered four-dimensional system the logistic function is used. The research aims to investigate the four-dimensional system, find critical points of the system, calculate the characteristic numbers, and calculate Lyapunov exponents.