Relative Entropy Research Articles

A family of subfactors arising from a pair of complex Hadamard matrices

In this paper, we define a new equivalence relation ‘[Formula: see text]’ on the set of all Hadamard inequivalent complex Hadamard matrices of order [Formula: see text] and show that pairs [Formula: see text] of equivalent matrices [Formula: see text] produce an infinite family of potentially new subfactors of the hyperfinite type [Formula: see text] factor [Formula: see text]. All these subfactors are irreducible with the Jones index [Formula: see text], including all possibilities. We also show that this family contains infinitely many infinite-depth subfactors. As an application, we compute the Connes–Størmer relative entropy and the angle between the pair [Formula: see text] of spin model subfactors arising from the pair [Formula: see text] of equivalent matrices. On the other hand, pairs [Formula: see text] of inequivalent matrices [Formula: see text] lead to subalgebras of [Formula: see text] with infinite Pimsner–Popa index.

From interacting agents to Boltzmann-Gibbs distribution of money

Abstract We investigate the unbiased model for money exchanges: agents give at random time a dollar to one another (if they have one). Surprisingly, this dynamics eventually leads to a geometric distribution of wealth (shown empirically by Dragulescu and Yakovenko, and rigorously by several follow-up papers). We prove a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. This deterministic description is then analyzed by taking advantage of several entropy–entropy dissipation inequalities and we provide a quantitative almost-exponential rate of convergence toward the equilibrium (geometric distribution) in relative entropy.

Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

AbstractWe prove a quantitative convergence result of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow. The proof uses gradient flow calibrations and the relative entropy method, which has been used in the recent literature to prove weak–strong uniqueness results for mean curvature flow and convergence of the Allen–Cahn equation. A crucial difference in this work is a new notion of gradient flow calibrations. We add a tangential component to the velocity field in order to prove the Gronwall estimate for the relative energy. This allows us to derive the optimal convergence rate without having to show the closeness of the Lagrange-multipliers.

Entropy-based geometric measures of correlations for bell diagonal states via measurement

Abstract The classification and quantification of quantum correlations are the important issues in quantum information theory. Quantum relative entropy and Rényi relative entropy are both non-negative, and can therefore be regarded as generalized distances between quantum states. We first introduced the method proposed by Brodutch and Modi for constructing measurements of classical and quantum correlations. Subsequently, we use this method to construct measurement-induced geometric classical and quantum correlations based on quantum relative entropy and Rényi− 1 2 relative entropy. For two-qubit Bell diagonal states, the analytic expressions for these geometric classical correlations and quantum correlations are obtained.

On the nonequilibrium dynamics of gravitational algebras

Abstract We explore nonequilibrium features of certain operator algebras which appear in quantum gravity. The algebra of observables in a black hole background is a Type II ∞ von Neumann algebra. We discuss how this algebra can be coupled to the algebra of observable of an infinite reservoir within the canonical ensemble, aiming to induce nonequilibrium dynamics. The resulting dynamics can lead the system towards a nonequilibrium steady state which can be characterized through modular theory. Within this framework we address the definition of entropy production and its relationship to relative entropy, alongside exploring other applications.

Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy

Quantum relative entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior-point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy. We design a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package Domain-Driven Solver (DDS). In addition to handling QRE constraints, DDS accepts any combination of several other conic and nonconic convex constraints. Our comprehensive numerical experiments encompass several parts, including (1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, (2) using DDS 2.2 for combining QRE constraints with various other constraint types, and (3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the National Science Foundation [Grant CMMI-2347120] and Discovery Grants from the Natural Sciences and Engineering Research Council of Canada. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0570 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0570 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Evaluation of CMIP5 and CMIP6 Models Based on Weather Types Applied to the South Atlantic Ocean

ABSTRACTChanges in climate in the South Atlantic region and adjacent regions have been described in numerous works using projections from global climate models from CMIP5 and CMIP6. This paper presents an evaluation of the ability of these models to reproduce the atmospheric circulation patterns (weather types) and their seasonal and inter‐annual variability. The analyses are performed based on the probability of occurrence of weather types in the historical period and in future projections. The scatter index and the relative entropy are the statistical parameters used to evaluate the models' performance in the historical period. Future projections consist of RCP2.6, 4.5 and 8.5 scenarios for the CMIP5 models and the SSP126, 245, 370 and 585 scenarios for the CMIP6 and are assessed at different time intervals: short term (2015–2039), mid‐term (2040–2069) and long term (2070–2100). The performance of projections is measured by analysing their consistency, that is, based on the similarity between projections of the same scenario in different models. The results show that the reproduction of the probability of occurrence of historical weather types and their seasonal and interannual variability was better performed by ACCESS1‐0, HadGEM2‐ES, HadGEM2‐CC, CMCC‐CM and MPI‐ESM‐P when assessing the models from CMIP5, and by HadGEM3‐GC31‐MM, ACCESS‐ESM1‐5, ACCESS‐ CM2 and MRI‐ESM‐P when assessing the models from CMIP6. As for future projections, only the BESM‐AO2‐5, GFDL‐ESM4 and HadGEM3‐GC31‐MM models showed inconsistency in one or more scenarios.

The Representative Points of Generalized Alpha Skew-t Distribution and Applications

Assuming the underlying statistical distribution of data is critical in information theory, as it impacts the accuracy and efficiency of communication and the definition of entropy. The real-world data are widely assumed to follow the normal distribution. To better comprehend the skewness of the data, many models more flexible than the normal distribution have been proposed, such as the generalized alpha skew-t (GAST) distribution. This paper studies some properties of the GAST distribution, including the calculation of the moments, and the relationship between the number of peaks and the GAST parameters with some proofs. For complex probability distributions, representative points (RPs) are useful due to the convenience of manipulation, computation and analysis. The relative entropy of two probability distributions could have been a good criterion for the purpose of generating RPs of a specific distribution but is not popularly used due to computational complexity. Hence, this paper only provides three ways to obtain RPs of the GAST distribution, Monte Carlo (MC), quasi-Monte Carlo (QMC), and mean square error (MSE). The three types of RPs are utilized in estimating moments and densities of the GAST distribution with known and unknown parameters. The MSE representative points perform the best among all case studies. For unknown parameter cases, a revised maximum likelihood estimation (MLE) method of parameter estimation is compared with the plain MLE method. It indicates that the revised MLE method is suitable for the GAST distribution having a unimodal or unobvious bimodal pattern. This paper includes two real-data applications in which the GAST model appears adaptable to various types of data.

Stability of the composite wave of two planar viscous shock waves for the compressible Navier–Stokes system

We are concerned with the nonlinear stability of the composite wave consisting of two planar viscous shock waves to the three-dimensional compressible Navier–Stokes system. It is shown that if the shock strengths are suitably small but not necessarily of the same order of magnitude, and the initial perturbations are suitably small without the zero-mass conditions, then there exists a unique, globally strong solution in time to the compressible Navier–Stokes system, which asymptotically approaches the corresponding composite wave up to time-dependent shifts in L^{\infty} norm. The proof employs the weighted relative entropy method, an L^{2} -contraction technique with time-dependent shifts to the shocks developed by Kang and Vasseur [J. Eur. Math. Soc. (JEMS) 23 (2021), 585–638; Invent. Math. 224 (2021), 55–146]. We perform the stability analysis within the original H^{2} -perturbation framework instead of using the anti-derivative technique. Compared with the previous work of the author and Wang [J. Eur. Math. Soc. (JEMS) (2023), DOI 10.4171/JEMS/1486] for the single shock case, a major difficulty is the construction of shifts to ensure that the two shock waves are well separated.

Near-infrared light stimulation regulates neural oscillation and memory behavior of mice with Alzheimer's disease.

Photobiomodulation (PBM) is a non-invasive neuromodulation technique for the brain. Low-intensity near-infrared light (1-500 mw) has demonstrated the ability to improve memory in Alzheimer's disease (AD) model mice, suggesting its potential for AD treatment. However, the impact of PBM on neural oscillations in the hippocampal region affected by AD remains unknown. In this study, AD model mice were subjected to PBM for 60 days and then tested using novel object recognition behavior (NOR) experiments. During behavioral experiments, local field potential signals (LFP) of the mice was recorded using a single electrode in the CA1 region to analyze memory ability and neural oscillation characteristics. The results revealed that mice stimulated with PBM exhibited significantly higher new object differentiation indices compared to the Sham group (p < 0.01). Furthermore, PBM stimulation led to a significant increase in relative power and sample entropy of theta and gamma bands (p < 0.01). The coupling intensities of θ-low-γ and θ-high-γ were also significantly higher in the PBM group compared to the Sham group (p < 0.01). In conclusion, these findings suggest that PBM may improve memory ability in AD mice through regulation of neural oscillation characteristics, providing a theoretical basis for utilizing PBM as a treatment modality for Alzheimer's disease.

Coupling by change of measure for conditional McKean–Vlasov SDEs and applications

The couplings by change of measure are applied to establish log-Harnack inequality(equivalently the entropy-cost estimate) for conditional McKean–Vlasov SDEs and derive the quantitative conditional propagation of chaos in relative entropy for mean field interacting particle system with common noise. For the log-Harnack inequality, two different types of couplings will be constructed for non-degenerate conditional McKean–Vlasov SDEs with multiplicative noise. As to the quantitative conditional propagation of chaos in relative entropy, the initial distribution of interacting particle system is allowed to be singular with that of limit equation. The above results are also extended to conditional distribution dependent stochastic Hamiltonian system.

Should the multi-layer transportation network structure be reduced?

ABSTRACT The increasing diversity of transportation modes and the rapid expansion of transportation networks present significant challenges for modeling multi-layer comprehensive transportation networks. It is crucial to determine whether aggregating certain layers is a viable option for balancing complexity reduction and information preservation. This decision defines the layered structures and informs subsequent analyses of these networks. Two-dimensional factors, namely topological structures and transportation attributes, are considered to enhance understanding of the similarities among network layers. The relative entropy and the Gini index are employed as metrics to assess information gain or loss resulting from layer aggregation or segregation, guiding decisions on network reduction. Furthermore, an integrated similarity measure, based on the quantum Jensen-Shannon divergence and the Gower distance, is utilized to identify the optimal aggregation sequences. Two real-world transportation networks serve as case studies. Results demonstrate that these transportation networks are more effectively maintained with layer-separated structures, preserving maximum information.

Uncertainty relations for quantum coherence with respect to mutually unbiased equiangular tight frames

The role of quantum coherence as an information resource has attracted increasing attention in recent years. Many protocols of quantum information science deal with specially selected states. Complete sets of mutually unbiased bases and symmetric informationally complete measurements are widely used in this regard. Other constructions were found to be useful, including projective designs and equiangular tight frames. As a rule, there are additional restrictions on the probabilities generated by quantum measurements assigned to such sets. Inequalities in terms of coherence quantifiers allow one to examine complementarity with respect to different sets. This study aims to examine uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased equiangular tight frames. To quantify the amount of coherence, quantum coherence quantifiers of the Tsallis type and the geometric coherence are used. The first case is induced by the Tsallis relative entropies. The derived inequalities are exemplified with equiangular tight frames of a ququart.

MethylSeqLogo: DNA methylation smart sequence logos

BackgroundSome transcription factors, MYC for example, bind sites of potentially methylated DNA. This may increase binding specificity as such sites are (1) highly under-represented in the genome, and (2) offer additional, tissue specific information in the form of hypo- or hyper-methylation. Fortunately, bisulfite sequencing data can be used to investigate this phenomenon.MethodWe developed MethylSeqLogo, an extension of sequence logos which includes new elements to indicate DNA methylation and under-represented dimers in each position of a set binding sites. Our method displays information from both DNA strands, and takes into account the sequence context (CpG or other) and genome region (promoter versus whole genome) appropriate to properly assess the expected background dimer frequency and level of methylation. MethylSeqLogo preserves sequence logo semantics—the relative height of nucleotides within a column represents their proportion in the binding sites, while the absolute height of each column represents information (relative entropy) and the height of all columns added together represents total informationResultsWe present figures illustrating the utility of using MethylSeqLogo to summarize data from several CpG binding transcription factors. The logos show that unmethylated CpG binding sites are a feature of transcription factors such as MYC and ZBTB33, while some other CpG binding transcription factors, such as CEBPB, appear methylation neutral.ConclusionsOur software enables users to explore bisulfite and ChIP sequencing data sets—and in the process obtain publication quality figures.

On application of the relative entropy concept in reliability assessment of some engineering cable structures

The main research problem studied in this work is an uncertain response and reliability assessment of the spatial cable structures due to the environmental stochasticity as well as material and geometrical imperfections. Some popular cable structures are analyzed for this purpose using the Stochastic Finite Element Method (SFEM) implemented with the use of three different techniques, namely the iterative generalized perturbation method, semi-analytical approach as well as the Monte-Carlo simulation. Uncertainty quantification delivered in this study is based on the series of FEM analyses of both static and dynamic structural problems. They enable the Least Squares Method determination of the structural polynomial responses linking extreme stresses and deformations with several uncorrelated uncertainty sources. Reliability assessment, fundamental in durability and Structural Health Monitoring, is completed using a comparison of the First Order Reliability Method (FORM) with probabilistic distance formulated by Bhattacharyya. Input uncertainties are assumed to be Gaussian according to the Maximum Entropy Principle. They have specific expected values following engineering design demands or the provisions of designing codes, whereas their standard deviations do not exceed the 10% level. The methods presented and the results obtained in this study may serve for further reliability analyses of large-scale civil engineering structures completed with both steel cables and also reinforced concrete plates like suspended bridges, for instance.

Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond

Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond

Martingale Pricing and Single Index Models: Unified Approach with Esscher and Minimal Relative Entropy Measures

In this paper, we explore the connection between a single index model under the real-world probability measure and martingale pricing via minimal relative entropy or Esscher transform, within the context of a one-period market model, possibly incomplete, with multiple risky assets and a single risk-free asset. The minimal relative entropy martingale measure and the Esscher martingale measure coincide in such a market, provided they both exist. From their Radon–Nikodym derivative, we derive a portfolio of risky assets in a natural way, termed portfolio G. Our analysis shows that pricing using the Esscher or minimal relative entropy martingale measure is equivalent to a single index model (SIM) incorporating portfolio G. In the special case of elliptical returns, portfolio G coincides with the classical tangency portfolio. Furthermore, in the case of jointly normal returns, Esscher or minimal relative entropy martingale measure pricing is equivalent to CAPM pricing.

Investigations on collision entropy and its variant for machine condition monitoring

Abstract The efficacy of sparsity measures and complexity measures in machine condition monitoring is well acknowledged. However, a comprehensive exploration of their theoretical underpinnings and interrelations remains incomplete. This study endeavors to address this gap by employing collision entropy as an illustrative case and establishing its relationship with sparsity measures. Through mathematical derivations, it is demonstrated that collision entropy exhibits a phenomenon termed as “bilateral reduction,” a characteristic indicative of complexity measures. Motivated by foundational concepts such as Shannon entropy and relative entropy, this paper introduces the concept of variation in collision entropy, which serves as a sparsity measure. Notably, both Shannon entropy and collision entropy, as specific cases of Rényi entropy, manifest the “bilateral reduction” effect and can be transformed into sparsity measures through certain methodologies. These insights contribute significantly to the exploration of the theoretical properties of Rényi entropy and its conversion into a cluster of sparse measures. This research augments the understanding of sparsity measures and complexity measures, offering theoretical insights crucial for their practical deployment in real-world applications.

Two novel pure-state coherence measures in quantifying coherence

Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with \(l_{1}\)-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.

Distribution reconstruction and reliability assessment of complex LSFs via an adaptive Non-parametric Density Estimation Method

Complex limit state functions (LSFs), often characterized by strong nonlinearity, non-smoothness, or discontinuity, pose challenges for structural reliability analysis in engineering practices. Conventional methods for uncertainty propagation and reliability assessment may struggle to handle these issues effectively. This paper introduces a novel approach to adaptively reconstruct the unknown distributions of complex LSFs. The Non-parametric Density Estimation Method based on Harmonic Transform (NDEM-HT) is employed as the tool for this purpose. An adaptive strategy is then proposed to determine the number of harmonic moments required in NDEM-HT for achieving high accuracy. Specifically, the Adaptive Kernel Density Estimation (AKDE) method is also adopted to provide an initial estimation of the rough distribution. Subsequently, the optimal number of harmonic moments is determined by minimizing the relative entropy between the distributions obtained by AKDE and NDEM-HT. The efficacy of the proposed method is demonstrated through five numerical examples, considering various types of complex LSFs. Comparative results are also provided employing MCS along with both conventional and state-of-the-art methods.