Information dynamics: temporal behavior of uncertainty measures

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tools are the information theory methods and inequalities satisfied by Fisher and Shannon information measures. We focus on the compatibility of these inequalities with the prescribed (deterministic, random or quantum) temporal behavior of pertinent probability densities.      

Entropy,Fisher Information and Variance with Frost-Musulin Potenial

This study presents the Shannon and Renyi information entropy for both position and momentum space and the Fisher information for the position-dependent mass Schrödinger equation with the Frost-Musulin potential. The analysis of the quantum mechanical probability has been obtained via the Fisher information. The variance information of this potential is equally computed. This controls both the chemical properties and physical properties of some of the molecular systems. We have observed the behaviour of the Shannon entropy. Renyi entropy, Fisher information and variance with the quantum number n respectively.     

Dynamics and Entropy Analysis of a Frictionally Loaded Pendulum

We use friction to simultaneously damp and excite a pendulum system. A Froude pendulum attached to a suspension shaft is subjected to a frictional load. We investigate two types of response of the system: regular and chaotic responses, depending on the excitation frequency. A transient chaotic solution was also obtained. We identify the motions using phase portraits, Poincaré maps, and Fourier spectra. Finally, the composite multiscaled entropy was estimated for the specified cases to confirm the preliminary classification.     

Quantum Fisher Information and Uncertainty Relations

It is well known that the Cramér–Rao inequality places a lower bound for quantum Fisher information in terms of the variance of any quantum measurement. We establish an upper bound for quantum Fisher information of a parameterized family of density operators in terms of the variance of the generator. These two bounds together yield a generalization of the Heisenberg uncertainty relations from statistical estimation perspective.     

Maximum Shannon Entropy, Minimum Fisher Information, and an Elementary Game

We formulate an elementary statistical game which captures the essence of some fundamental quantum experiments such as photon polarization and spin measurement. We explore and compare the significance of the principle of maximum Shannon entropy and the principle of minimum Fisher information in solving such a game. The solution based on the principle of minimum Fisher information coincides with the solution based on an invariance principle, and provides an informational explanation of Malus' law for photon polarization. There is no solution based on the principle of maximum Shannon entropy. The result demonstrates the merits of Fisher information, and the demerits of Shannon entropy, in treating some fundamental quantum problems. It also provides a quantitative example in support of a general philosophy: Nature intends to hide Fisher information, while obeying some simple rules.     

Entropy and Random Vectors

Barron⁽¹⁾ produced a proof of the Central Limit Theorem for real-valued IID random variables, in the sense of convergence in relative entropy. Here, we establish a similar result for independent real-valued random vectors, not necessarily identically distributed. The main developments required are a generalisation of De Bruijn's identity, and various inequalities proposed in ref. 2.     

Entropy Method for Decision-Making: Uncertainty Cycles in Tourism Demand

A new methodology is presented for measuring, classifying and predicting the cycles of uncertainty that occur in temporary decision-making in the tourist accommodation market (apartments and hotels). Special attention is paid to the role of entropy and cycles in the process under the Adaptive Markets Hypothesis. The work scheme analyses random cycles from time to time, and in the frequency domain, the linear and nonlinear causality relationships between variables are studied. The period analysed is from January 2005 to December 2018; the following empirical results stand out: (1) On longer scales, the periodicity of the uncertainty of decision-making is between 6 and 12 months, respectively, for all the nationalities described. (2) The elasticity of demand for tourist apartments is approximately 1% due to changes in demand for tourist hotels. (3) The elasticity of the uncertainty factor is highly correlated with the country of origin of tourists visiting Spain. For example, it has been empirically shown that increases of 1% in uncertainty cause increases in the demand for apartments of 2.12% (worldwide), 3.05% (UK), 1.91% (Germany), 1.78% (France), 7.21% (Ireland), 3.61% (The Netherlands) respectively. This modelling has an explanatory capacity of 99% in all the models analysed.     

Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.     

Entropy and information characteristics of qubit states

Shannon entropy, Rényi entropy, and Tsallis entropy are discussed for the tomographic probability distributions of qubit states. Relative entropy and its properties are considered for the tomographic probability distribution describing the states of multi-spin systems. New inequalities for Hermite polynomials are obtained.     

Analysis of dynamics in magnetotelluric data by using the Fisher-Shannon method

The Fisher-Shannon information (FS) plane, defined by the Fisher information measure and the Shannon entropy power, is used to investigate the complex dynamics of magnetotelluric data of three stations in Taiwan. In the FS plane the electric and magnetic components are significantly separated, characterized by different degrees of order. Further investigation shows that signals measured in areas with very high level of seismic activity are well discriminated.     

The Fisher information measure and Shannon entropy for particulate matter measurements

We investigated the dynamics of particulate matter data, recorded in Tito, a small industrial area of southern Italy. The analysis of these signals was performed using the Fisher information measure (FIM), which is a powerful tool for investigating complex and nonstationary signals, and the Shannon entropy, which is a well-known tool for investigating the degree of disorder in dynamical systems. Our results point to an increase of disorder and complexity from fine to coarse particulates.     

Quantum Information and Entropy

Thermodynamic entropy is not an entirely satisfactory measure of information of a quantum state. This entropy for an unknown pure state is zero, although repeated measurements on copies of such a pure state do communicate information. In view of this, we propose a new measure for the informational entropy of a quantum state that includes information in the pure states and the thermodynamic entropy. The origin of information is explained in terms of an interplay between unitary and non-unitary evolution. Such complementarity is also at the basis of the so-called interaction-free measurement.     

Spherical-Symmetry and Spin Effects on the Uncertainty Measures of Multidimensional Quantum Systems with Central Potentials

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state’s angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies–Thakkar, Lieb–Thirring, Redheffer–Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies.     

Informational analysis of seismic sequences by applying the Fisher Information Measure and the Shannon entropy: An application to the 2004–2010 seismicity of Aswan area (Egypt)

The interevent-time (IET) and interevent-distance (IED) series of seismic events occurred at Aswan area (Egypt) from 2004 to 2010 were investigated by means of the Fisher Information Measure and the Shannon entropy. The analysis was performed varying the depth and the magnitude thresholds. The results point out to an increase of level of organization and order with the decrease of magnitude threshold and the increase of depth threshold for the IET series, while the IED series are characterized by a level of uncertainty approximately constant with the threshold magnitude. The complexity measure, calculated as the product of the Fisher Information Measure and the Shannon entropy power, presents very similar pattern for both the types of seismic series, indicating an increasing complexity with the decrease of the threshold magnitude and the increase of the threshold depth.     

Principle of Maximum Entropy and Reduced Dynamics

A new method to obtain a series of reduced dynamics at various stages of coarse-graining is proposed. This ranges from the most coarse-grained one which agrees with the deterministic time evolution equation for averages of the relevant variables to the least coarse-grained one which is the generalized Fokker-Planck equation for the probability distribution function of the relevant variables. The method is based on the extention of the Kawasaki-Gunton operator with the help of the principle of maximum entropy.     

Uncertainty Relations in Terms of Fisher Information

By virtue of the well-known concept of Fisher information in the theory of statistical inference, we obtain an inequality chain which generalizes and refines the conventional Heisenberg uncertainty relations.     

Power spectrum and Fisher–Shannon information plane analysis of tidal records

The time dynamics of tidal fluctuations measured in three sites in Bahia Blanca Estuary (central Argentina), Ingeniero White, Puerto Belgrano and Torre Mareografica, are analyzed. To investigate the time series of the tidal waves we used two different approaches: the power spectral density (PSD) and the Fisher–Shannon (FS) information plane. The PSD permitted: (i) the identification of diurnal, semi-diurnal and higher frequency cycles in all the three tidal signals, and (ii) the detection of two different dynamical regimes (scaling and white-noise), involving respectively timescales lower and higher than about 2–2.5 days. The FS method, which allows to gain insight into the complex structure of a time series, quantifying its degree of organization and order, was applied to the residual tidal series (after removing the main cycles) and permitted to identify a period of low organization in the tidal signal measured at Puerto Belgrano.     

Calculating Entropy of Plane Symmetry Black Hole via Generalized Uncertainty Relation

The generalized uncertainty relation is introduced to calculate entropy of the black hole. By using quantum statistical method, we directly obtain the partition function of Bose and Fermi field on the background of the plane symmetry black hole. Then we calculate the entropy of Bose and Fermi field on the background of black hole near the horizon of the black hole. In our calculation, we need not introduce cutoff. There are not the left out term and the divergent logarithmic term in the original brick-wall method. And it is obtained that the entropy of the black hole is proportional to the area of the horizon. The inherent contact between the entropy of black hole and the area of horizon is opened out. Further it is shown the entropy of black hole is entropy of quantum state on the surface of horizon. The black hole’s entropy is the intrinsic property of the black hole. The entropy is a quantum effect.