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 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.     

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.     

One-dimensional quantum walks with a position-dependent coin

We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin. The rotation angle, which depends upon the position of a quantum particle, parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classicallike, semi-classical-like, and quantum-like. Further, we study the Shannon entropy associated with position and the coin space of a quantum particle, and compare them with the case of the position-independent coin. Our results show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy by considering a quantum walk with two identical position-dependent entangled coins. We observe that in general, a wave function becomes more localized as compared to the case of the positionindependent coin and hence the corresponding Shannon entropy is lower. Our results show that a position-dependent coin can be used as a controlling tool of quantum walks.     

Real and spurious contributions for the Shannon, Rényi and Tsallis entropies

A two-parameter probability distribution is constructed by dilatation (or contraction) of the escort probability distribution. This transformation involves a physical probability distribution P associated with the system under study and an almost arbitrary reference probability distribution P^′. In contrast to the Shannon and Rényi entropies, the Tsallis entropy does not decompose as the sum of the physical contribution due to P and the reference or spurious part owing to P^′. For solving this problem, a slight modification to the relation between Tsallis and Rényi entropies must be introduced. The procedure in this paper gives rise to a nonconventional one-parameter Shannon entropy and to two-parameter Rényi and Tsallis entropies associated with P. It also contributes to clarify the meaning and role of the escort probabilities set.     

A class of asymmetric pathway distributions and an entropy interpretation

Asymmetric distributions are widely used in probability modeling and statistical analysis. Recently, various asymmetric distributions are being developed by many researchers for modeling various data sets in real life contexts. In the present paper, we introduce a new class of q-type asymmetric distributions which include q-analogues of asymmetric Laplace, exponential power, Weibull etc. and corresponding standard distributions as special cases. Also we show that this pathway model can be obtained by optimizing Mathai’s generalized entropy with more general setup, which is a generalization of various entropy measures due to Shannon and others.     

On entropy and information of quantum states

Shannon entropy and information are applied to study the properties of quantum states of a system in the probability representation of quantum mechanics. Examples of spin states and mixed Gaussian states of the two-mode system are considered. The relationship between the new entropy and the von Neumann entropy is reviewed. Two tomographic maps are considered within the framework of the star-product quantization. The explicit expression of tomographic entropy associated with photon-number tomogram of the two-mode state of photons is obtained in terms of Hermite polynomials of four variables. Based on a contribution to the International Conference “New Trends in Quantum Mechanics. Fundamental Aspects and Applications” (Palermo, Italy, November 2005).     

Tomographic characteristics of spin states

Spin states are studied in the tomographic-probability representation. The standard probability distribution of spin projection onto a direction in space is used instead of the spinor or the density matrix to identify the quantum state. The Shannon entropy and information are associated with the spin tomographic probability. A short review of the probability-theory notions is presented. Analysis of tomographic entropy and tomographic information for the Werner state is considered. The probability representation is used to describe a spin-3/2 particle and two qubits. The connection of tomographic entropy with the von Neumann entropy is discussed.     

Control of Correlation Using Confinement in Case of Quantum System

This article investigates the behavior of a Moshinsky atom in a 1D harmonic trap. Focus is given on the theoretical foundations of confinement and its impact on the correlation between particles in the Moshinsky atom. The investigation begins by illustrating the (de)localization of the probability density function using Shannon entropy. The basics of correlation and interpretation of correlation using tools such as mutual information and statistical correlation coefficients and how these can be quantified are discussed. Then the concept of confinement is explored. The impact of interaction strength and confinement on Shannon entropy, statistical correlation coefficients, and mutual information is investigated. How interaction strength and confinement can be used to induce correlations between previously uncorrelated particles, as well as how they can be used to suppress correlations between previously correlated particles is discussed. Their implications for quantum information processing and quantum simulation are discussed. In conclusion, confinement is a powerful tool for controlling correlations in quantum systems, and its impact on correlation can be understood through theoretical models. The importance of experimental studies in this field, which provide insights into the behavior of quantum systems under confinement and pave the way for future applications in quantum technology is also emphasized.     

Scaling behaviour of Fisher and Shannon entropies for the exponential-cosine screened coulomb potential

The scaling laws are given for the entropies in the information theory, including the Shannon's entropy, its power, the Fisher's information and the Fisher–Shannon product, using the exponential-cosine screened Coulomb potential. The scaling laws are specified, in the r-space, as a function of |μ ? μ_{c, n?}|, where μ is the screening parameter and μ_{c, n?} its critical value for the specific quantum numbers n and ?. Scaling laws for other physical quantities, such as energy eigenvalues, the moments, static polarisability, transition probabilities, etc. are also given. Some of these are reported for the first time. The outcome is compared with the available literatures’ results.     

Probability-Representation Entropy for Spin-State Tomogram

The probability-representation entropy (tomographic entropy) of an arbitrary quantum state is introduced. Using the properties of the spin tomogram as the standard probability-distribution function, the tomographic entropy notion is discussed. The relation of tomographic entropy to Shannon entropy and von Neumann entropy is elucidated.     

符号动力学在心率变异性分析中的参数选择

时间序列的符号动力学信息熵H_k因其计算简单快速,对数据量要求小,而被应用于心率变异性(heart rate variability, HRV)分析,然而符号化的参数选择至今却并未形成统一标准.HRV作为典型的生理信号,存在着极大的个体间差异和非平稳性,要获得稳健的一致性分析,在符号化过程中必须考虑符号化参数α与序列本身均值、标准差的综合影响.文中,首先以仿真噪声序列为对象,考察了3个参数对于H_k的影响及三者相互之间的关联性,研究表明当满足特定关系时,H_k的曲线簇收敛于反映序列动力特性的H_k-up;随后在对15例心跳间隔序列的分析中,验证了H_k-up在消除个体间差异及减弱非平稳干扰影响两方面都优于α取固定值时的研究结果. 关键词： 符号动力学 熵 心率变异性     

Topological permutation entropy

Permutation entropy quantifies the diversity of possible ordering of the successively observed values a random or deterministic system can take, just as Shannon entropy quantifies the diversity of the values themselves. When the observable or state variable has a natural order relation, making permutation entropy possible to compute, then the asymptotic rate of growth in permutation entropy with word length forms an alternative means of describing the intrinsic entropy rate of a source. Herein, extending a previous result on metric entropy rate, we show that the topological permutation entropy rate for expansive maps equals the conventional topological entropy rate familiar from symbolic dynamics. This result is not limited to one-dimensional maps.     

两种群随机动力系统的信息熵和动力学研究

随机种群动力学模型是研究种群间以及种群与不确定性环境间相互作用的动力学行为的数学模型. 本文从概率密度以及信息熵流、熵产生的演化角度探讨了两种群生态系统的Itô (或Statonovich)意义下随机模型的动力学行为.利用Fokker-Planck方程及其边界条件 和信息熵定义导出信息熵流(平均散度)和熵产生的关系式,并通过数值路径积分法捕 捉到熵流的非线性变化趋势以及信息熵的极值点位置与概率密度的快速迁移和分岔的联系. 应用数值路径积分法计算结果表明Itô (或Statonovich)意义下两种随机模型的概率密度 和信息熵的极值点位置不同但演化趋势一致.     

Acoustic classification of Australian anurans based on hybrid spectral-entropy approach

A new hybrid method for automated frog sound identification, using spectral centroid, Shannon entropy and Rényi entropy is proposed. The advantage of using entropy based information theoretic approach for analyzing complexity of bioacoustics signals in animal vocalization is discussed. Sound samples from nine species of Microhylidae frogs are first segmented into syllables. Fourier spectral centroid, Shannon entropy and Rényi entropy of the syllables are then determined. Finally, nonparametric k-th nearest neighbour (k-NN) classifier is used to recognize the frog species based on these three extracted features. Result shows that the k-NN classifier based on these selected features is capable to identify the species of the frogs with an average accuracy of 98%. It is found that the accuracy reduces significantly only when the noise levels higher than −20 dB.     

Conditional entropy in algebraic statistical physics

The notion of conditional entropy as entropy of conditional state on C^*-algebra with respect to its C^*-subalgebra ₁ is introduced. It is proved that for a compatible state σ on (which admits the conditional expectation of Umegaki-Takesaki) the mean conditional entropy in an a priori state σ₁ on ₁ is equal to the difference of the entropy of the state σ on and the entropy of the state σ₁ on ₁. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.     

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.     

Information Between Quantum Systems via POVMs

The concepts of conditional entropy and information between subsystems of a composite quantum system are generalized to include arbitrary indirect measurements (POVMs). Some properties of those quantities differ from those of their classical counterparts; certain equalities and inequalities of classical information theory may be violated. PACS: 03.67.-a.     

等概率符号化样本熵应用于脑电分析

样本熵(或近似熵)以信息增长率刻画时间序列的复杂性,能应用于短时序列,因而在生理信号分析中被广泛采用.然而,一方面由于传统样本熵采用与标准差线性相关的容限,使得熵值易受非平稳突变干扰的影响,另一方面传统样本熵还受序列概率分布的影响,从而导致其并非单纯反映序列的信息增长率.针对上述两个问题,将符号动力学与样本熵结合,提出等概率符号化样本熵方法,并对其物理意义、数学推导及参数选取都做了详细阐述.通过对噪声数据的仿真计算,验证了该方法的正确性及其区分不同强度时间相关的有效性.此方法应用于脑电信号分析的结果表明,在不对信号做人工伪迹去除的前提下,只需要1.25 s的脑电信号即可有效地区分出注意力集中和注意力发散两种状态.这进一步证明了该方法可很好地抵御非平稳突变干扰,能快速获得短时序列的潜在动力学特性,对脑电生物反馈技术具有很大的应用价值.