Correlation Tensor Research Articles

Generative Multi-Label Correlation Learning

In real-world applications, a single instance could have more than one label. To solve this task, multi-label learning methods emerged in recent years. It is a more challenging problem for many reasons, such as complex label correlation, long-tail label distribution, and data shortage. In general, overcoming these challenges and bettering learning performance could be achieved by utilizing more training samples and including label correlations. However, these solutions are expensive and inflexible. Large-scale, well-labeled datasets are difficult to obtain, and building label correlation maps requires task-specific semantic information as prior knowledge. To address these limitations, we propose a general and compact Multi-Label Correlation Learning (MUCO) framework. MUCO explicitly and effectively learns the latent label correlations by updating a label correlation tensor, which provides highly accurate and interpretable prediction results. In addition, a multi-label generative strategy is deployed to handle the long-tail label distribution challenge. It borrows the visual clues from limited samples and synthesizes more diverse samples. All networks in our model are optimized simultaneously. Extensive experiments illustrate the effectiveness and efficiency of MUCO. Ablation studies further prove the effectiveness of all the modules.

Analytical method of multiqubit entanglement robustness in correlated quantum channels

In practice, the interaction between quantum system and environment will lead to the decoherence of entanglement. By quantifying the entanglement through the correlation tensor, we systematically investigate the entanglement robustness of two-qubit states under the four types of noisy channels and corresponding correlated channels. We use an analytical method to find all of the most robust and fragile states under the noisy channels. The puzzling phenomenon that the most robust and fragile states have the same entanglement under noisy channels has been investigated. Moreover, we compare the entanglement robustness of a three-qubit Greenberger-Horne-Zeilinger state with a $W$ state. Our method shows the behavior of entanglement under the noisy channels and characterizes the entanglement robustness of the multiqubit system analytically.

Two Types of Trilocality of Probability and Correlation Tensors

In this work, we discuss two types of trilocality of probability tensors (PTs) P=〚P(a1a2a3)〛 over an outcome set Ω3 and correlation tensors (CTs) P=〚P(a1a2a3|x1x2x3)〛 over an outcome-input set Δ3 based on a triangle network and described by continuous (integral) and discrete (sum) trilocal hidden variable models (C-triLHVMs and D-triLHVMs). We say that a PT (or CT) P is C-trilocal (resp. D-trilocal) if it can be described by a C-triLHVM (resp. D-triLHVM). It is proved that a PT (resp. CT) is D-trilocal if and only if it can be realized in a triangle network by three shared separable states and a local POVM (resp. a set of local POVMs) performed at each node; a CT is C-trilocal (resp. D-trilocal) if and only if it can be written as a convex combination of the product deterministic CTs with a C-trilocal (resp. D-trilocal) PT as a coefficient tensor. Some properties of the sets consisting of C-trilocal and D-trilocal PTs (resp. C-trilocal and D-trilocal CTs) are proved, including their path-connectedness and partial star-convexity.

Divergence-free turbulent inflow data from realistic covariance tensor

Scale-resolving computational fluid dynamics (CFD) methods require carefully constructed boundary conditions to produce accurate results. The inflow data should be unsteady and the successive realizations must follow specific statistics while ideally having a particular correlation in both space and time. A method for generating synthetic correlated stochastic data from uncorrelated sequences is detailed and applied to the problem of inflow turbulence generation for CFD simulations. The technique constructs divergence-free anisotropic random fields with the sensible spectrum and complete complex correlation in space and time. A realistic two-point correlation tensor is inferred from first and second moments and a set of heuristic recommendations based on turbulent flow observations. These statistics are readily available in most practical cases making the technique highly versatile. The approach is computationally efficient with the use of eigendecomposition to reduce the resources required depending on the accuracy needed. Demonstration of the method is provided with the simulation of a turbulent channel flow and a square duct flow, and validation is done against existing numerical data.

Tensor- and spinor-valued random fields with applications to continuum physics and cosmology

In this paper, we review the history, current state-of-art, and physical applications of the spectral theory of two classes of random functions. One class consists of homogeneous and isotropic random fields defined on a Euclidean space and taking values in a real finite-dimensional linear space. In applications to continuum physics, such a field describes the physical properties of a homogeneous and isotropic continuous medium in the situation, when a microstructure is attached to all medium points. The range of the field is the fixed point set of a symmetry class, where two compact Lie groups act by orthogonal representations. The material symmetry group of a homogeneous medium is the same at each point and acts trivially, while the group of physical symmetries may act nontrivially. In an isotropic random medium, the rank 1 (resp., rank 2) correlation tensors of the field transform under the action of the group of physical symmetries according to the above representation (resp., its tensor square), making the field isotropic. Another class consists of isotropic random cross-sections of homogeneous vector bundles over a coset space of a compact Lie group. In applications to cosmology, the coset space models the sky sphere, while the random cross-section models a cosmic background. The Cosmological Principle ensures that the cross-section is isotropic. For the convenience of the reader, a necessary material from multilinear algebra, representation theory, and differential geometry is reviewed in Appendix.

Generalized Resonance Energy Transfer Theory: Applications to Vibrational Energy Flow in Optical Cavities.

A general rate theory for resonance energy transfer (gRET) is formulated to incorporate any degrees of freedom (e.g., rotation, vibration, exciton, and polariton) as well as coherently coupled composite donor or acceptor states. The compact rate expression allows us to establish useful relationships: (i) detailed balance condition when the donor and acceptor are at the same temperature; (ii) proportionality to the product of dipole correlation tensors, which is not necessarily equivalent to spectral overlap; (iii) scaling with the effective coherent size, i.e., the number of coherently coupled molecules or modes; (iv) decomposition of collective rate in homogeneous systems into the monomer and coherence contributions such that the ratio of the two defines the quantum enhancement factor F; (v) spatial and orientational dependences as derived from the interaction potential. For the special case of exciton transfer, the general rate formalism reduces to FRET or its multichromophoric extension. When applied to cavity-assisted vibrational energy transfer between molecules or within a molecule, the general rate expression provides an intuitive explanation of intriguing phenomena such as cooperativity, resonance, and nonlinearity in the collective vibrational strong coupling (VSC) regime, as demonstrated in recent simulations. The relevance of gRET to cavity-catalyzed reactions and intramolecular vibrational redistribution is discussed and will lead to further theoretical developments.

Generation of turbulent inflow data from realistic approximations of the covariance tensor

This study presents a novel synthetic inflow generator capable of producing a random field matching a realistic set of two-point statistics with minimal input. The method is based on two main elements. The first element is a procedure to infer realistic two-point covariance tensors from readily available data (e.g., freestream velocity, boundary layer thickness, and turbulence intensity) by a preliminary Reynolds-averaged Navier–Stokes simulation with an explicit algebraic Reynolds stress model closure. The second element is an efficient eigen-decomposition step of the two-point correlation tensor, which determines a set of modes. The modal decomposition guarantees the spatial correlation in transversal directions, while the temporal correlation/streamwise spatial correlation is obtained by digital filters based on longitudinal and transversal spectra of a realistic shape and Taylor's hypothesis. The instantaneous inlet flow field is obtained by a linear combination of the modes via uncorrelated random weights with unit variance. The modes are generated in a computationally inexpensive pre-processing step. Compared to existing inflow generation methods that try to match given two-point statistics, the proposed method relieves the burden of obtaining data from direct numerical simulation (DNS) or experiments, while the complexity of the eigenvalue problem that needs to be solved is reduced. The proposed method is shown to produce a realistic turbulent channel flow and a realistic turbulent boundary layer by the large-eddy simulation, which contains statistics that are in good agreement with results from DNS. The proposed inflow generator features, cost-effectiveness, robustness, and potential for generalization to complex geometries.

Unsteady responses and correlation characteristics of propeller blades under turbulence excitation

The turbulence ingestion force acting on the propeller is the primary source of noise and vibration for an underwater vehicle. This paper aims to quantitatively investigate the unsteady responses and blade-to-blade correlation characteristics of the rotating blades, which are two critical factors determining the consequent spectra of broadband unsteady forces. The ingested turbulence and unsteady loading of a 10-blade propeller downstream a turbulence-generating grid are numerically predicted based on Large Eddy Simulation. A theoretical approach of the correlation method is adopted to expound better the physical mechanism for the turbulence ingestion force. Through the partition of blade surface meshes in simulation, the extracted unsteady responses of separate blade strips reveal the broadband loadings varying with radii and blade sections. The distributions of flow field and fluctuating pressure demonstrate the cause of this spanwise discrepancy, as well as the relative contributions of different blade strips to the resultant unsteady loading. The correlation analysis on fluctuating velocity by the rotating sampling of simulated incoming turbulence shows the deviation of ingested flow from homogeneous isotropic turbulence. Additionally, the correlation analysis of unsteady forces distinguishes the sources of different broadband loadings. All of this in turn facilitates the modification of lift-curve slopes of blade strips and stretching factors in turbulence correlation tensors, ultimately improving the prediction accuracy of predominant broadband components by the correlation method. This work provides novel ideas for the mechanism study on turbulence ingestion force and beneficial references to the engineering design of noise-reduced propellers.

Classification and measurement of multipartite entanglement by reconstruction of correlation tensors on an NMR quantum processor

Classification and measurement of multipartite entanglement by reconstruction of correlation tensors on an NMR quantum processor

Superstatistical Wind Fields from Pointwise Atmospheric Turbulence Measurements

Accurate models of turbulent wind fields have become increasingly important in the atmospheric sciences, e.g., for the determination of spatiotemporal correlations in wind parks, the estimation of individual loads on turbine rotor and blades, or the modeling of particle-turbulence interaction in atmospheric clouds or pollutant distributions in urban settings. Because of the difficult task of resolving the fields across a broad range of scales, one oftentimes has to invoke stochastic wind field models that fulfill specific, empirically observed, properties. Whereas commonly used Gaussian random field models solely control second-order statistics (i.e., velocity correlation tensors or kinetic energy spectra), we explicitly show that our extended model emulates the effects of higher-order statistics as well. Most importantly, the empirically observed phenomenon of small-scale intermittency, which can be regarded as one of the key features of atmospheric turbulent flows, is reproduced with a very high level of accuracy and at considerably low computational cost. Our method is based on a multipoint statistical description of turbulent velocity fields that consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. We propose a new and efficient sampling algorithm for this Gaussian scale mixture and demonstrate how such “superstatistical” wind fields can be constrained on a certain number of real-world measurement data points from a meteorological mast array.Received 12 April 2022Revised 26 July 2022Accepted 22 August 2022DOI:https://doi.org/10.1103/PRXEnergy.1.023006Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtmospheric scienceBoundary layersShear flowsStochastic processesStructure & turbulence of boundary layersSustainabilityTurbulenceWind energyEnergy Science & TechnologyNonlinear DynamicsFluid DynamicsStatistical Physics

Search for an efficient entanglement witness operator for bound entangled states in bipartite quantum systems

Entanglement detection problem is one of the important problem in quantum information theory. Gurvit (2014) showed that this problem is NP complete and thus this may be the possible reason that only one criterion is not sufficient to detect all entangled states. There are some powerful entanglement detection criterion such as partial transposition criterion, realignment criterion but it may not be possible to implement them successfully in the experiment. This situation can be avoided if the entanglement is detected through the construction of witness operator method. In this work, we take an analytical approach to construct a witness operator. To achieve this task, we first construct a linear map using partial transposition and realignment operation. Then we find some conditions on the parameters of the map for which the map represent a positive map. Further, we have constructed a Choi matrix corresponding to the map and have shown that it is not completely positive. We then construct an entanglement witness operator, which is based on the linear combination of the function of Choi matrix and the identity matrix and it can detect both NPTES and PPTES. Finally, we prove its efficiency by detecting several bipartite bound entangled states which were previously undetected by some well-known separability criteria. We also compared the detection power of our witness operator with three well-known powerful entanglement detection criteria, namely, dV criterion, CCNR criterion and the separability criteria based on correlation tensor (CT) proposed by Sarbicki et. al. and find that our witness operator detects more entangled states than these criterion.

Detection of Multipartite Entanglement Based on Heisenberg-Weyl Representation of Density Matrices

We study entanglement and genuine entanglement of tripartite and four-partite quantum states by using Heisenberg-Weyl (HW) representation of density matrices. Based on the correlation tensors in HW representation, we present criteria to detect entanglement and genuine tripartite and four-partite entanglement. Detailed examples show that our method can detect more entangled states than previous criteria.

Surrogate modelling of wind fields from point-wise atmospheric turbulence measurements

We present an advanced model for the generation of synthetic wind fields that can be understood as an extension of the well-known Mann model. In contrast to such Gaussian random field models which control second-order statistics (i.e., velocity correlation tensors or spectra), we demonstrate that our extended model incorporates the effects of higherorder statistics as well. In particular, the empirically observed phenomenon of small-scale intermittency, a key feature of atmospheric turbulent flows, can be reproduced with high accuracy and at considerably low computational cost. Our method is based on a recently developed multipoint statistical description of a turbulent velocity field [J. Friedrich et al., J. Phys. Complex. 2 045006 (2021)] and consists of a superposition of multivariate Gaussian statistics with fluctuating covariances. Furthermore, we explicitly show how such superstatistical Mann fields can be constraint on a certain number of point-wise measurement data. We give an outlook on the relevance of such surrogate wind fields in the context of fatigue loads on wind turbines.

Theory of length-scale dependent relaxation moduli and stress fluctuations in glass-forming and viscoelastic liquids.

The spatiotemporal correlations of the local stress tensor in supercooled liquids are studied both theoretically and by molecular dynamics simulations of a two-dimensional (2D) polydisperse Lennard-Jones system. Asymptotically exact theoretical equations defining the dynamical structure factor and all components of the stress correlation tensor for low wave-vector q are presented in terms of the generalized (q-dependent) shear and longitudinal relaxation moduli, G(q, t) and K(q, t). We developed a rigorous approach (valid for low q) to calculate K(q, t) in terms of certain bulk correlation functions (for q = 0), the static structure factor S(q), and thermal conductivity κ. The proposed approach takes into account both the thermostatting effect and the effect of polydispersity. The theoretical results for the (q, t)-dependent stress correlation functions are compared with our simulation data, and an excellent agreement is found for qb̄≲0.5 (with b̄ being the mean particle diameter) both above and below the glass transition without any fitting parameters. Our data are consistent with recently predicted (both theoretically and by simulations) long-range correlations of the shear stress quenched in heterogeneous glassy structures.

Crustal Structure across the West Antarctic Rift System from Multicomponent Ambient Noise Surface Wave Tomography

Abstract Approximately 2 yr (2010–2011) of continuous seismic records from a subset of the Antarctic component of the Polar Earth Observing Network (POLENET-ANET) seismic network deployed in West Antarctica are used to compute the nine components of the correlation tensor between each pair of stations in the network. Rayleigh wave velocity information from the vertical and radial components was extracted in the form of group and phase velocity dispersion curves, whereas the transverse component provided complimentary Love wave velocity information. The multicomponent Rayleigh wave measurements (ZZ, RR, ZR, and RZ) were averaged and used to infer the measurement uncertainties. The Rayleigh and Love wave group and phase velocities were then regionalized in space using a 2D deterministic tomography. A transect that spans the West Antarctic rift system was extracted from the tomography at individual periods between 7 and 60 s for the four types of surface wave velocities (i.e., Rayleigh and Love phase and group velocities). A transdimensional Bayesian joint inversion algorithm was used to invert these four datasets for a 1D model of isotropic shear-wave velocity versus depth at each point along the transect. In this way, surface wave dispersion curves from multicomponent noise correlations were used to build a 2D isotropic shear-wave velocity model down to ∼55 km depth. In this model, the top of the large low-velocity zone beneath Marie Byrd Land was imaged (up to a 5% decrease in velocity at ∼50 km depth), which provides further evidence for a mantle hot spot beneath the crust that supports the high topography in this region. We also observed a large velocity contrast in the lower crust beneath Marie Byrd Land at a depth where previous long-period seismicity has been observed. This strong contrast occurs more shallow than in previous crustal models, which compared to our model identify a deeper Moho (∼5–10 km deeper) beneath Marie Byrd Land. This new model has implications for interpreting earthquake locations in this region and perhaps necessitates that we revisit past hypocenter estimation studies using updated velocity models for the region.

Decompositions of n-Partite Nonsignaling Correlation-Type Tensors With Applications

When an n-partite physical system is measured by n observers, the joint probabilities of outcomes conditioned on the observables chosen by the n parties form a nonnegative tensor, called an n-partite correlation tensor (CT). In this paper, we aim to establish some characterizations of nonsignaling and Bell locality of an n-partite CT, respectively. By placing CTs within the linear space of correlation-type tensors (CTTs), we prove that every n-partite nonsignaling CTT can be decomposed as a linear combination of all local deterministic CTs using single-value decomposition of matrices and mathematical induction. As a consequence, we prove that an n-partite CT is nonsignaling (resp. Bell local) if and only if it can be written as a quasi-convex (resp. convex) combination of the outer products of deterministic CTs, implying that an n-partite CT is nonsignaling if and only if it has a local hidden variable model governed by a quasi-probability distribution. As an application of these results, we prove that a CT is nonsignaling if and only if it can be written as a quasi-convex of two Bell local ones, revealing a close relationship between nonsignaling CTs and Bell local ones.

In vivo Correlation Tensor MRI reveals microscopic kurtosis in the human brain on a clinical 3T scanner

Diffusion MRI (dMRI) has become one of the most important imaging modalities for noninvasively probing tissue microstructure. Diffusional Kurtosis MRI (DKI) quantifies the degree of non-Gaussian diffusion, which in turn has been shown to increase sensitivity towards, e.g., disease and orientation mapping in neural tissue. However, the specificity of DKI is limited as different sources can contribute to the total intravoxel diffusional kurtosis, including: variance in diffusion tensor magnitudes (Kiso), variance due to diffusion anisotropy (Kaniso), and microscopic kurtosis (μK) related to restricted diffusion, microstructural disorder, and/or exchange. Interestingly, μK is typically ignored in diffusion MRI signal modelling as it is assumed to be negligible in neural tissues. However, recently, Correlation Tensor MRI (CTI) based on Double-Diffusion-Encoding (DDE) was introduced for kurtosis source separation, revealing non negligible μK in preclinical imaging. Here, we implemented CTI for the first time on a clinical 3T scanner and investigated the sources of total kurtosis in healthy subjects. A robust framework for kurtosis source separation in humans is introduced, followed by estimation of μK (and the other kurtosis sources) in the healthy brain. Using this clinical CTI approach, we find that μK significantly contributes to total diffusional kurtosis both in grey and white matter tissue but, as expected, not in the ventricles. The first μK maps of the human brain are presented, revealing that the spatial distribution of μK provides a unique source of contrast, appearing different from isotropic and anisotropic kurtosis counterparts. Moreover, group average templates of these kurtosis sources have been generated for the first time, which corroborated our findings at the underlying individual-level maps. We further show that the common practice of ignoring μK and assuming the multiple Gaussian component approximation for kurtosis source estimation introduces significant bias in the estimation of other kurtosis sources and, perhaps even worse, compromises their interpretation. Finally, a twofold acceleration of CTI is discussed in the context of potential future clinical applications. We conclude that CTI has much potential for future in vivo microstructural characterizations in healthy and pathological tissue.

Effects of Numerical Forcing on the Two-Time Correlation of Fluid Velocity Differences in Stationary Isotropic Turbulence

In direct numerical simulations (DNS) of homogeneous isotropic turbulence, numerical forcing is needed to achieve statistically stationary velocity fields. The Eulerian two-time correlation tensor of the fluid velocity difference field, Δu(r,t)=u(x+r,t)−u(x,t), characterizes the temporal evolution of turbulent eddies whose sizes scale with separation r=|r|. In this study, we investigate the effects of two spectral forcing schemes on the temporal decay of the Eulerian two-time correlation of fluid velocity differences ⟨Δu(r,t′)Δu(r,t)⟩. Accordingly, DNS of homogeneous isotropic turbulence were performed for two grid sizes, 1283 and 5123, corresponding to the Taylor micro-scale Reynolds numbers Reλ≈80 and 210, respectively. Statistical stationarity was achieved by employing deterministic and stochastic spectral forcing schemes. In the stochastic scheme, one needs to specify the time scale, Tf, of the Uhlenbeck–Ornstein (UO) processes that constitute the forcing. We considered four values of the UO time scale (Tf=TE/4,TE,2TE, and 4TE) for each Reλ, where TE is the large-eddy time scale obtained from the DNS run with deterministic forcing at the same Reλ. It is seen that the correlations ⟨Δu(r,t′)Δu(r,t)⟩ obtained from the deterministic-forcing DNS runs decay more slowly than those from stochastic-forcing DNS runs of all four Tf values. The slower decay of correlations in deterministic DNS runs is more pronounced at larger separations and for higher Reλ.

Detection of genuine tripartite entanglement based on Bloch representation of density matrices

We study the genuine multipartite entanglement in tripartite quantum systems. By using the Schmidt decomposition and local unitary transformation, we convert the general states to simpler forms and consider certain matrices from correlation tensors in the Bloch representation of the simplified density matrices. Using these special matrices, we obtain new criteria for genuine multipartite entanglement. Detail examples show that our criteria are able to detect more tripartite entangled and genuine tripartite entangled states than some existing criteria.

A note on entanglement classification for tripartite mixed states

We study the classification of entanglement in tripartite systems by using Bell-type inequalities and principal basis. By using Bell unctions and the generalized three dimensional Pauli operators, we present a set of Bell inequalities which classifies the entanglement of triqutrit fully separable and bi-separable mixed states. By using the correlation tensors in the principal basis representation of density matrices, we obtain separability criteria for fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example is given to illustrate our criteria in classifying the tripartite entanglement.