Geometric Entropy Research Articles

Geometric quantum states beyond the AdS/CFT correspondence

We characterize the quantum states dual to entanglement wedges in arbitrary spacetimes, in settings where the matter entropy can be neglected compared to the geometric entropy. In AdS/CFT, such states obey special entropy inequalities known as the holographic entropy cone. In particular, the mutual information of CFT subregions is monogamous (MMI). We extend this result to arbitrary spacetimes, using a recent proposal for the generalized entanglement wedge e(a) of a gravitating region a. Given independent input regions a, b, and c, we prove MMI: Area[e(a)]+Area[e(b)]+Area[e(c)]−Area[e(ab)]−Area[e(bc)]−Area[e(ca)]+Area[e(abc)]≤0. We expect that the full holographic entropy cone can be extended to arbitrary spacetimes using similar methods. Published by the American Physical Society 2024

Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter

Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter

Seismic noise attenuation method based on low‐rank adaptive symplectic geometry decomposition

AbstractThe basic assumption of low‐rank methods is that noise‐free seismic data can be represented as a low‐rank matrix. Effective noise reduction can be achieved through the low‐rank approximation of Hankel matrices composed of the data. However, selecting the appropriate rank parameter and avoiding expensive singular value decomposition are two challenges that have limited the practical application of this method. In this paper, we first propose symplectic geometric decomposition that avoids singular value decomposition. The symplectic similarity transformation preserves the essence of the original time sequence as well as the signal's basic characteristics and maintains the approximation of the Hankel matrix. To select an appropriate rank, we construct the symplectic geometric entropy according to the distribution of eigenvalues and search for high‐contributing eigenvalues to determine the needed rank parameter. Therefore, we provide an adaptive approach to selecting the rank parameter by the symplectic geometric entropy method. The synthetic examples and field data results show that our method significantly improves the computational efficiency while adaptively retaining more effective signals in complex structures. Therefore, this method has practical application value.

Geometric entropy of plant leaves: A measure of morphological complexity.

Shape is an objective characteristic of an object. A boundary separates a physical object from its surroundings. It defines the shape and regulates energy flux into and from an object. Visual perception of a definite shape (geometry) of physical objects is an abstraction. While the perceived geometry at an object's sharp interface (macro) creates a Euclidian illusion of actual shape, the notion of diffuse interfaces (micro) allows an understanding of the realistic form of objects. Here, we formulate a dimensionless geometric entropy of plant leaves (SL) by a 2-D description of a phase-field function. We applied this method to 112 tropical plant leaf images. SL was estimated from the leaf perimeter (P) and leaf area (A). It correlates positively with a fractal dimensional measure of leaf complexity, viz., segmental fractal complexity. Leaves with a higher P: A ratio have higher SL and possess complex morphology. The univariate cluster analysis of SL reveals the taxonomic relationship among the leaf shapes at the genus level. An increase in SL of plant leaves could be an evolutionary strategy. The results of morphological complexity presented in this paper will trigger discussion on the causal links between leaf adaptive stability/efficiency and complexity. We present SL as a derived plant trait to describe plant leaf complexity and adaptive stability. Integrating SL into other leaf physiological measures will help to understand the dynamics of energy flow between plants and their environment.

The action of geometric entropy in topologically massive gravity

Due to the presence of a gravitational anomaly in topologically massive gravity (TMG), the geometric entropy is no longer simply the Hubeny-Rangamani-Takayanagi (HRT) area; instead, it is given by the HRT area plus an anomalous contribution. We study the action of this geometric entropy on the covariant phase space of classical solutions for TMG with matter fields whose action is algebraic in the metric. The result agrees precisely with the action of HRT area operators in Einstein-Hilbert gravity given in [8], i.e., it is a boundary-condition-preserving kink transformation. Furthermore, we show our result to be consistent with direct computations of semiclassical commutators of geometric entropies in pure TMG spacetimes asymptotic to planar AdS, as computed in [21].

Entropy measures of the metal–organic network via topological descriptors

Abstract A family of chemical compounds known as metal–organic networks (MONs) is composed mainly of clusters of metal ions with organic ligands. It can increase volatility or make substances soluble in organic solvents. By using these salient features, organic compounds generate applications in material sciences for sol–gel processing. A graph’s entropy is utilized as a complexity indicator and is interpreted as the structural information content of the graph. Investigating the entropies of relationship systems is a common occurrence in discrete mathematics, computer science, information theory, statistics, chemistry, and biology. In this article, we investigated the degree-based entropies: geometric arithmetic entropy, atom bond connectivity entropy, general Randic′ entropy, and general sum connectivity entropy for MONs. Furthermore, we created tables for all expressions by using 1–10 values for the s s parameter of these entropies.

Boltzmann-Shannon interaction entropy: A normalized measure for continuous variables with an application as a subsample quality metric.

The recent introduction of geometric partition entropy offered an alternative to differential Shannon entropy for the quantification of uncertainty as estimated from a sample drawn from a one-dimensional bounded continuous probability distribution. In addition to being a fresh perspective for the basis of continuous information theory, this new approach provided several improvements over traditional entropy estimators including its effectiveness on sparse samples and a proper incorporation of the impact from extreme outliers. However, a complimentary relationship exists between the new geometric approach and the basic form of its frequency-based predecessor that is leveraged here to define an entropy measure with no bias toward the sample size. This stable normalized measure is named the Boltzmann-Shannon interaction entropy (BSIE)) as it is defined in terms of a standard divergence between the measure-based and frequency-based distributions that can be associated with the two historical figures. This parameter-free measure can be accurately estimated in a computationally efficient manner, and we illustrate its utility as a quality metric for subsampling in the context of nonlinear polynomial regression.

Markov chain Monte Carlo methods for graph refinement in spinfoam cosmology

We study the behavior of the Lorentzian Engle-Pereira-Rovelli-Livine spinfoam amplitude with homogeneous boundary data, under a graph refinement going from five to twenty boundary tetrahedra. This can be interpreted as a wave function of the Universe, for which we compute boundary geometrical operators, correlation functions, and entanglement entropy. The numerical calculation is made possible by adapting the Metropolis-Hastings algorithm, along with recently developed computational methods appropriate for the deep quantum regime. We confirm that the transition amplitudes are stable against such refinement. We find that the average boundary geometry does not change, but the new degrees of freedom correct the quantum fluctuations of the boundary and the correlations between spatial patches. The expectation values are compatible with their geometrical interpretation and the correlations between neighboring patches decay when computed across different spinfoam vertices.

Spatial entanglement in two-dimensional QCD: Renyi and Ryu-Takayanagi entropies

We derive a general formula for the replica partition function in the vacuum state, for a large class of interacting theories with fermions, with or without gauge fields, using the equal-time formulation on the light front. The result is used to analyze the spatial entanglement of interacting Dirac fermions in two-dimensional QCD. A particular attention is paid to the issues of infrared cut-off dependence and gauge invariance. The Renyi entropy for a single interval, is given by the rainbow dressed quark propagator to order ${\cal O}(N_c)$. The contributions to order ${\cal O}(1)$, are shown to follow from the off-diagonal and off mass-shell mesonic T-matrix, with no contribution to the central charge. The construction is then extended to mesonic states on the light front, and shown to probe the moments of the partonic PDFs for large light-front separations. In the vacuum and for small and large intervals, the spatial entanglement entropy following from the Renyi entropy, is shown to be in agreement with the Ryu-Takayanagi geometrical entropy, using a soft-wall AdS$_3$ model of two-dimensional QCD.

Optimal Demodulation Band Extraction Method for Bearing Faults Diagnosis Based on Weighted Geometric Cyclic Relative Entropy

Optimal demodulation band extraction is a significant step in rolling bearing fault analysis. However, existing methods, primarily based on global indexes and neglecting negative local outliers, cannot identify compound faults in intense noise environments. To address this problem, a novel demodulation band extraction method based on weighted geometric cyclic relative entropy (WGCRE) is proposed. WGCRE is defined on the cyclic sub-bands model of the logarithmic envelope spectrum (LES) to fully consider the bearing characteristic frequency of pseudo-cyclostationarity. In detail, local and global thresholds are separately set by the white noise parameter and harmonic-to-noise ratio to exclude the exogenous noise outliers. On this basis, the WGCRE is defined as a geometrically weighted index of several different fault types to avoid harmonic interference and improve the identification of composite faults. WGCRE–gram, similar to fast kurtogram (FK), is then constructed by replacing kurtosis with WGCRE to extract the optimal demodulation band. Compared with FK and another LES-based method, logarithmic-cycligram, the proposed method is more robust for accurately identifying single and compound faults under external noise. The effectiveness of this method is verified through simulations and actual tests. Simulation experiments of different kinds and intensities of exogenous noise interference preliminarily determine the superior robustness of WGCRE in the face of solid noise. The inner ring, outer ring, and composite fault experiments further confirmed the robust adaptability of WGCRE in the face of complex working conditions.

Geometric Partition Entropy: Coarse-Graining a Continuous State Space

Entropy is re-examined as a quantification of ignorance in the predictability of a one dimensional continuous phenomenon. Although traditional estimators for entropy have been widely utilized in this context, we show that both the thermodynamic and Shannon’s theory of entropy are fundamentally discrete, and that the limiting process used to define differential entropy suffers from similar problems to those encountered in thermodynamics. In contrast, we consider a sampled data set to be observations of microstates (unmeasurable in thermodynamics and nonexistent in Shannon’s discrete theory), meaning, in this context, it is the macrostates of the underlying phenomenon that are unknown. To obtain a particular coarse-grained model we define macrostates using quantiles of the sample and define an ignorance density distribution based on the distances between quantiles. The geometric partition entropy is then just the Shannon entropy of this finite distribution. Our measure is more consistent and informative than histogram-binning, especially when applied to complex distributions and those with extreme outliers or under limited sampling. Its computational efficiency and avoidance of negative values can also make it preferable to geometric estimators such as k-nearest neighbors. We suggest applications that are unique to this estimator and illustrate its general utility through an application to time series in the approximation of an ergodic symbolic dynamics from limited observations.

Expansive Lie Group Actions

In this work we introduce a concept of expansiveness for actions of connected Lie groups. We study some of its properties and investigate some implications of expansiveness. We study the centralizer of expansive actions and introduce CW-expansiveness for pseudo-group actions. As an application, we prove positiveness of geometric entropy for expansive foliations and expansive group actions.

Möbius mirrors

An accelerating boundary (mirror) acts as a horizon and black hole analog, radiating energy with some particle spectrum. We demonstrate that a Möbius transformation on the null coordinate advanced time mirror trajectory uniquely keeps invariant not only the energy flux but the particle spectrum. We clarify how the geometric entanglement entropy is also invariant. The transform allows generation of families of dynamically distinct trajectories, including -symmetric ones, mapping from the eternally thermal mirror to the de Sitter horizon, and different boundary motions corresponding to Kerr or Schwarzschild black holes.

Geometric and topological entropies of sphere packing.

We present a statistical mechanical description of randomly packed spherical particles, where the average coordination number is treated as a macroscopic thermodynamic variable. The overall packing entropy is shown to have two contributions: geometric, reflecting statistical weights of individual configurations, and topological, which corresponds to the number of topologically distinct states. Both of them are computed in the thermodynamic limit for isostatic and weakly underconstrained packings in 2D and 3D. The theory generalizes concepts of granular and glassy configurational entropies for the case of nonjammed systems. It is directly applicable to sticky colloids and predicts an asymptotic phase behavior of sticky spheres in the limit of strong binding.

On Curvilinear Regression Analysis via Newly Proposed Entropies for Some Benzene Models

To avoid exorbitant and extensive laboratory experiments, QSPR analysis, based on topological descriptors, is a very constructive statistical approach for analyzing the numerous physical and chemical properties of compounds. Therefore, we presented some new entropy measures which are based on the sum of the neighborhood degree of the vertices. Firstly, we made the partition of the edges of benzene derivatives which are based on the degree sum of neighboring vertices and then computed the neighborhood version of entropies. Secondly, we made use of the software SPSS for developing a correlation between newly introduced entropies and the physicochemical properties of benzene derivatives. Our obtained results demonstrated that the critical temperature , critical pressure , and critical volume can be predicted through fifth geometric arithmetic entropy, second SK entropy, and fifth ND entropy, respectively. Other remaining physical characteristics include Gibb’s energy (qℰ), log(P), molar refractivity (ℳℛ), and Henry’s law (ℋℒ) that can be predicted by using sixth ND entropy.

Quantum and Classical Ergotropy from Relative Entropies.

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.

Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy

The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity.

Entropic uncertainty principle, partition function and holographic principle derived from Liouville’s Theorem

An entropic version of Liouville’s Theorem is defined in terms of the conjugate variables (“hyperbolic position” and “entropic momentum”) of an entropic Hamiltonian. It is used to derive the Holographic Principle as applied to holomorphic structures that represent maximum entropy configurations. The Bekenstein-Hawking expression for black hole entropy is a consequence. Based on the entropic commutator derived from Liouville’s Theorem and the same entropic conjugate variables, an entropic Uncertainty Principle (in units of Boltzmann’s constant) isomorphic to the kinematic Uncertainty Principle (in units of Planck’s constant) is also derived. These formal developments underpin the previous treatment of Quantitative Geometrical Thermodynamics (QGT) which has established (entirely on geometric entropy grounds) the stability of the double-helix, the double logarithmic spiral, and the sphere. Since in the QGT formalism the Boltzmann and Planck constants are quanta of quantities orthogonal to each other in Minkowski spacetime, a solution of the Schrödinger Equation is demonstrated isomorphic to a probability term of an entropic Partition Function, where both are defined by path integrals obeying the stationary principle: this isomorphism represents an important symmetry of the formalism. The geometry of a holomorphic structure must also exhibit at least C2 symmetry.

Research on the Fault Feature Extraction of Rolling Bearings Based on SGMD-CS and the AdaBoost Framework

Symplectic geometric mode decomposition (SGMD) is a newly proposed signal processing method. Because of its superiority, it has gained more and more attention in the field of fault diagnosis. However, the similar component reorganization problem involved in this method has not been clearly stated. Aiming at this problem, this paper proposes the SGMD-CS method based on the SGMD method and the cosine similarity (CS) and has been compared and verified on the simulation signal and the actual rolling bearing signal. In addition, in order to realize the intelligent diagnosis of the wind turbine bearing fault, the symplectic geometric entropy (SymEn) is extracted as the fault feature and input it into the AdaBoost classification model. In summary, this paper proposes a new wind turbine fault feature extraction method based on the SGMD-CS and AdaBoost framework, and the validity of the method is verified by the rolling bearing vibration data of the Electrical Engineering Laboratory of Case Western Reserve University.

Area law of connected correlation function in higher dimensional conformal field theory

We present a new area law which is associated with the correlator of OPE blocks in higher dimensional conformal field theories (CFTs). The area law shows similar behaviour as black hole entropy or geometric entanglement entropy. It includes a leading term which is proportional to the area of the entanglement surface, and a logarithmic subleading term with degree q. We extract the UV cutoff independent coefficients and discuss various properties of the coefficients.