Relative Entropy Research Articles

Causal Analysis of Roof Caving on Underground Mine: A New Theory and Optimized DEMATEL Approach

In the context of mines, roof-caving incidents constitute the most common and expensive accidents. To enhance the management and prevention of roof-caving accidents, it is imperative to investigate the factors that contribute to such incidents and comprehend the intricate causal relationships among them. This study aims to classify the causes of these accidents into three categories: basic factors, controllable factors, and sudden factors, based on the mechanism of roof caving. The categorization is primarily determined by two indicators: intervisibility and variability. Furthermore, the study delves into analyzing the mutual influence relationships among these factors and proposes the BCX theory (Basic-Controllable-Sudden causing theory) for roof caving. Subsequently, based on this theory, an index system called BCX is established for roof caving, and the DEMATEL method is employed to analyze the factors within this index system. To attain more accurate results, this study utilizes interval trapezoidal type-2 fuzzy number scale optimization and Tsallis relative entropy to address the limitations of the DEMATEL method. By comparing the outcomes of the traditional and optimal DEMATEL methods, it is observed that the optimal method exhibits superior applicability in the BCX index system of roof caving, with results that align closely with the actual scenario. Therefore, the optimal DEMATEL method’s analysis of centrality, importance, and chain relationships between the factors within the BCX index system will offer valuable guidance for preventing roof-caving accidents in mining operations.

Multiscale groundwater level forecasts with multi-model ensemble approaches: Combining machine learning models using decision theories and bayesian model averaging

Creating precise groundwater level (GWL) prediction models is of crucial significance for the productive use, extended planning, and controlling of limited sub-surface water supplies. In this research, the accuracy of GWL forecasts in Bangladesh was enhanced for three weeks by utilizing ensembles of Machine Learning (ML) models. Six advanced ML-based models were developed and assessed using eight performance indices, and an Overall Ranking (OR) was provided by combining the rankings produced by Grey Relational Analysis (GRA), Variation Coefficient (COV), and Shannon's Entropy (SE). The standalone forecasting models demonstrated excellent performance across the three forecasting horizons, with accuracy values ranging from 0.986 to 0.997 for one-step, 0.971 to 0.999 for two-step, and 0.960 to 0.997 for three-step forecasts at GT3330001. Results also revealed that three ranking techniques (SE, COV, and GRA), as well as their combined ranking (OR), produced different best-performing models at different prediction horizons for different observation wells. Weighted average ensembles of the prediction models were developed by calculating individual model weights using four ensemble modelling techniques: SE, COV, GRA, and Bayesian Model Averaging (BMA). The BMA-based ensemble technique outperformed three benchmark ensemble approaches, achieving R = 0.947, KGE = 0.925, IOA = 0.972, MAE = 0.062 m, and RMSE = 0.123 m for one-step-ahead forecasts at GT3330001. The findings exhibit a consistent trend across other forecasting horizons and observation wells. Finally, the Dempster-Shafer evidence theory was employed to rank the single and composite models. The ranking results demonstrated that the BMA-based ensemble consistently secured the top position (with the weight values of 0.997, 0.991, and 0.987 for one-week, two-weeks, and three-weeks forward forecasts at GT3330001) for all forecasting horizons and observation wells. This study shows that the BMA-based composite model can produce more accurate GWL projections at Bangladesh study location, with potential for application in other regions worldwide.

Steered quantum coherence and entropic uncertainty relation in the cluster Ising model

Abstract In this paper, we investigate the cluster Ising model (CIM) via steered quantum coherence (SQC) and entropic uncertainty relation (EUR). We present the behavior of SQC quantified by the L1 norm, relative entropy and quantum Jensen–Shannon divergence. We also demonstrate the properties of EUR in the CIM. In addition, we provide a comparative analysis of these measures and present detailed numerical results.

Relative entropy based uncertainty principles for graph signals

In physical quantum mechanics, the uncertainty principle in presence of quantum memory [Berta M, Christandl M, Colbeck R,et al., Nature Physics] can reach much lower bound, which has resulted in a huge breakthrough in quantum mechanics. Inspired by this idea, this paper would propose some novel uncertainty relations in terms of relative entropy for signal representation and time-frequency resolution analysis. On one hand, the relative entropy measures the distinguishability between the known (priori) basis and the client basis, which implies that we have partial “memory” of the client basis so that the uncertainty bounds become sharper in some cases. On the other hand, in some cases, if the reference basis along with nearly the same energy distribution could be given, then the uncertainty bound would tend to zero, as shows that there is no uncertainty any longer. These novel uncertainty relationships with sharper bounds would give us the potential advantages over the classical counterpart. In addition, the detailed comparison with classical Shannon entropy based uncertainty principle has been addressed as well via combined uncertainty relations. Finally, the theoretical analysis and numerical experiments on certain application over graph signals have been demonstrated to show the efficiency of these proposed relations.

Non-classical correlations and coherence in a two-dimensional electron gas under the influence of Rashba spin–orbit coupling

This investigation delves into the dynamics of quantum correlations and coherence within a two-dimensional electron gas (2DEG) system, taking into account the influence of Rashba spin–orbit coupling (SOC). We utilize metrics such as logarithmic negativity ([Formula: see text]), local quantum uncertainty (LQU ([Formula: see text])), and relative entropy of coherence ([Formula: see text]) to specifically assess these quantum resources among electron spins within non-interacting electron gases. Our study investigates how the separation distance (R) between electrons and the intensity of Rashba SOC ([Formula: see text]), impact the behavior of quantum properties within the 2DEG system. We find that specific strengths of Rashba SOC can effectively regulate quantum correlations and coherence within this system. Particularly significant is our observation that optimal quantum indicators emerge when electron proximity is minimized, underscoring the substantial influence of electron distance on quantum characteristics. Conversely, as electron separation increases, these quantum metrics diminish. Therefore, by adjusting the Rashba parameter, we can enhance the resilience of these quantum measurements against increasing electron separations, providing valuable insights into the potential for controlling and manipulating quantum behavior in similar 2DEG systems.

Sensitivity Analysis of Excited-State Population in Plasma Based on Relative Entropy.

A highly versatile evaluation method is proposed for transient plasmas based on statistical physics. It would be beneficial in various industrial sectors, including semiconductors and automobiles. Our research focused on low-energy plasmas in laboratory settings, and they were assessed via our proposed method, which incorporates relative entropy and fractional Brownian motion, based on a revised collisional-radiative model. By introducing an indicator to evaluate how far a system is from its steady state, both the trend of entropy and the radiative process' contribution to the lifetime of excited states were considered. The high statistical weight of some excited states may act as a bottleneck in the plasma's energy relaxation throughout the system to a steady state. By deepening our understanding of how energy flows through plasmas, we anticipate potential contributions to resolving global environmental issues and fostering technological innovation in plasma-related industrial fields.

Gradient flows of generalized relative entropy and functional inequalities on graphs

We derive the discrete forms of the porous medium equation and fast diffusion equation with drift terms on a finite graph as gradient flows of the m-relative entropy, with the global existence of solutions. We show the Łojasiewicz inequality to prove the convergence of solutions and some important functional inequalities. More specifically, the Łojasiewicz inequality holds near the stationary solution with exponent 12, which leads to the exactly exponential convergence rate with a finite trajectory length. It can also be applied to show the Talagrand-type inequality and the log-Sobolev-type inequality directly.

Approximation of Classical Two-Phase Flows of Viscous Incompressible Fluids by a Navier–Stokes/Allen–Cahn System

We show convergence of the Navier–Stokes/Allen–Cahn system to a classical sharp interface model for the two-phase flow of two viscous incompressible fluids with same viscosities in a smooth bounded domain in two and three space dimensions as long as a smooth solution of the limit system exists. Moreover, we obtain error estimates with the aid of a relative entropy method. Our results hold provided that the mobility mε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_\\varepsilon >0$$\\end{document} in the Allen–Cahn equation tends to zero in a subcritical way, i.e., mε=m0εβ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_\\varepsilon = m_0 \\varepsilon ^\\beta $$\\end{document} for some β∈(0,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (0,2)$$\\end{document} and m0>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_0>0$$\\end{document}. The proof proceeds by showing via a relative entropy argument that the solution to the Navier–Stokes/Allen–Cahn system remains close to the solution of a perturbed version of the two-phase flow problem, augmented by an extra mean curvature flow term mεHΓt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_\\varepsilon H_{\\Gamma _t}$$\\end{document} in the interface motion. In a second step, it is easy to see that the solution to the perturbed problem is close to the original two-phase flow.

Frozen condition of quantum coherence

Quantum coherence as an important physical resource plays the key role in implementing various quantum tasks, whereas quantum coherence is generally nonincreasing under incoherent operations. In this paper, we analyse under which dynamical conditions the l 1-norm or the relative entropy of coherence can remain unchanged under strictly incoherent operations (freezing coherence). We provide a detailed analysis of their structure together with exact geometric conditions of freezing coherence. It reveals a remarkable feature: any strictly incoherent operation freezing coherence can be decomposed as a convex combination of unitary operations. This partially answers an open question named unitary decomposition of doubly-stochastic quantum operations [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)]. Based on this analysis, we also give a complete classification of coherent states from operational coherence theory. This builds the counterpart of entanglement classification under LOCC.

Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability

Abstract A Cahn–Hilliard–Allen–Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem with respect to the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

Metrological non-classical correlations and quantum coherence in hybrid (1/2,1) system under decoherence channels

Abstract Our research focuses on the interplay between thermal noise and decoherence channels on quantum coherence and nonclassical correlations in a hybrid ( 1 / 2 , 1 ) Heisenberg model. This hybrid system integrates the Dzyaloshinsky–Moriya interaction (DMI) and operates under the influence of an external magnetic field. We use local quantum Fisher information (LQFI) for correlation estimation and relative entropy of coherence for coherence measurement in the considered system. Our investigation encompasses various parameters of the hybrid system, the strength of the DMI and the intensity of the external magnetic field. Our findings underscore that elevated temperature compromises both nonclassical correlations and coherence. On the other hand, the robustness of the DMI mitigates the impact of thermal noise on quantum Fisher information correlations and relative entropy of coherence in the hybrid system. Additionally, we inspect the impact of decohering channels-specifically, dephasing, phase flip, and bit- and trit-flip channels-on thermal coherence and quantum correlations. The introduction of various decoherence processes into the hybrid qubit-qutrit system leads to a competition with thermal fluctuations, thereby giving rise to out-of-equilibrium states. Our results indicate that as the decoherence strength parameter (p) increases, both LQFI and relative entropy of coherence exhibit similar behaviors in the dephasing and phase flip channels. These resources gradually diminish, eventually disappearing entirely at p = 1. In the context of the bit- and trit-flip channels, quantum coherence displays notable distinctions compared to what is observed under dephasing and phase flip channels, revealing that coherence can be preserved if the DMI is strong and the intensity of the external magnetic field is reduced. These findings are important since it is crucial to first understand the decoherence process, arising from the interaction with the environment, and then to find ways to hinder this decoherence in order to avoid the complete loss of the quantum resources necessary to quantum information processing.

Parametric noise reduction and construction of performance degradation indexes for low-voltage switchgear appliances

Aiming at the problem that there is a lot of noise in the feature parameter data of low-voltage switchgear, which makes it difficult to extract useful information about the feature parameters and unable to construct the performance degradation index reasonably. This paper proposes a noise reduction method through Relative Entropy (KL) optimized Variational Mode Decomposition (VMD) combined with Non-Local Mean (NLM) and a method of constructing performance degradation indexes for low-voltage switching appliances through Convolutional Autoencoder (CAE). Firstly, the voltage and current signals are collected based on the full-life test platform of low-voltage switchgear, and the subset of feature parameters is extracted. Then, the feature parameters are decomposed into different components by relative entropy optimization VMD and the noise-dominant and effective components are selected based on the correlation. Secondly, NLM denoising is performed on the noise-dominated signal, and the noise-reduced signals are obtained by signal reconstruction. Finally, the construction of performance degradation indexes for multi-type switchgear is completed based on CAE. The final results show that the two noise reduction evaluation indicators including Noise Rejection Ratio (NRR) and trendiness (T), and the three performance degradation evaluation indexes including Goodness of Fit (R), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE) all show optimal results, indicating that this paper’s method can better complete the construction of noise reduction and performance degradation indicators for the feature parameters of low-voltage switchgear.

Infinite-Volume Gibbs States of the Generalized Mean-Field Orthoplicial Model

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} that are constrained to a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-sphere, equivalently a scaled (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables.

An overview of the development and applications of information entropy

With information entropy gradually taking the lead in modern information theory development, it begins to hold greater influence over multiple research areas as well as technology innovation. This paper aims to clarify peoples confusion with the development of entropy theory and provide a brief overview of the origin of entropy theory, including the original Shannons proposal, variants such as relative entropy and conditional entropy, and entropy concepts proposed by other scientists, such as Rnyi Tsallis entropy. The paper also includes the current application of entropy, studies hotspots, and predicts future entropy development trends. This research paper is able to add more coherence and consistency to information entropys development, helping more people to better understand the concept of entropy and its derivation. At the same time, with hotspots of entropy fields of study, this paper hopes to attract more people to devote themselves to studying entropy-related fields, and boost technological development.

Event-Oriented State Alignment Network for Weakly Supervised Temporal Language Grounding.

Weakly supervised temporal language grounding (TLG) aims to locate events in untrimmed videos based on natural language queries without temporal annotations, necessitating a deep understanding of semantic context across both video and text modalities. Existing methods often focus on simple correlations between query phrases and isolated video segments, neglecting the event-oriented semantic coherence and consistency required for accurate temporal grounding. This can lead to misleading results due to partial frame correlations. To address these limitations, we propose the Event-oriented State Alignment Network (ESAN), which constructs "start-event-end" semantic state sets for both textual and video data. ESAN employs relative entropy for cross-modal alignment through knowledge distillation from pre-trained large models, thereby enhancing semantic coherence within each modality and ensuring consistency across modalities. Our approach leverages vision-language models to extract static frame semantics and large language models to capture dynamic semantic changes, facilitating a more comprehensive understanding of events. Experiments conducted on two benchmark datasets demonstrate that ESAN significantly outperforms existing methods. By reducing false high correlations and improving the overall performance, our method effectively addresses the challenges posed by previous approaches. These advancements highlight the potential of ESAN to improve the precision and reliability of temporal language grounding tasks.

Quantum entropy couples matter with geometry

Abstract We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

Exploring the possibility of a complex-valued non-Gaussianity measure for quantum states of light

We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.

Minimum Mean Squared Error Estimation and Mutual Information Gain

Information theoretic quantities such as entropy, entropy rate, information gain, and relative entropy are often used to understand the performance of intelligent agents in learning applications. Mean squared error has not played a role in these analyses, primarily because it is not felt to be a viable performance indicator in these scenarios. We build on a new quantity, the log ratio of entropy powers, to establish that minimum mean squared error (MMSE) estimation, prediction, and smoothing are directly connected to mutual information gain or loss in an agent learning system modeled by a Markov chain for many probability distributions of interest. Expressions for mutual information gain or loss are developed for MMSE estimation, prediction, and smoothing, and an example for fixed lag smoothing is presented.

Enhancing coherence by local parity-time symmetric operation under the non-Markovian environment

Abstract The dynamic behavior of the relative entropy of coherence for a two-level system is systematically investigated within non-Markovian environment. We derive the exact expressions of relative entropy quantifying coherence for an exactly solving model consisting of single qubit interacting with independent structured reservoir. By employing local non-Hermitian operation, i.e. parity-time ( PT ) symmetric operation, we show that the vanished quantum coherence can be effectively restored under the non-Markovian regimes. Furthermore, the freezing phenomenon of the coherence can be detected by using the optimal PT symmetric operation strength within the non-Markovian environments.

On the sample complexity of quantum Boltzmann machine learning

Quantum Boltzmann machines (QBMs) are machine-learning models for both classical and quantum data. We give an operational definition of QBM learning in terms of the difference in expectation values between the model and target, taking into account the polynomial size of the data set. By using the relative entropy as a loss function, this problem can be solved without encountering barren plateaus. We prove that a solution can be obtained with stochastic gradient descent using at most a polynomial number of Gibbs states. We also prove that pre-training on a subset of the QBM parameters can only lower the sample complexity bounds. In particular, we give pre-training strategies based on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We verify these models and our theoretical findings numerically on a quantum and a classical data set. Our results establish that QBMs are promising machine learning models.