Fisher information,Rényi entropy power and quantum phase transition in the Dicke model

Fisher information, Rényi entropy power and Fisher–Rényi information product are presented for the Dicke model. There is a quantum phase transition in this quantum optical model. It is pointed out that there is an abrupt change in the Fisher information, Rényi entropy power, the Fisher, Shannon and Rényi lengths at the transition point. It is found that these quantities diverge as the characteristic length: |_λc−λ|^−1/4${|{\lambda }_c-\lambda |}^{-1/4}$ around the critical value of the coupling strength _λc${\lambda }_c$ for any value of the parameter β$\beta$.     

Rényi information of atoms

Position and momentum space Rényi information of order α has been studied within a Hartree-Fock framework for 103 neutral atoms, 54 singly charged cations and 43 anions in their ground state. The values of α?1 (α?1) stress the shell structure for position-space (momentum-space) Rényi information. The relationship between the complexity and Rényi information is also studied.     

Extensive Rényi statistics from non-extensive entropy

We show that starting with either the non-extensive Tsallis entropy in Wang's formalism or the extensive Rényi entropy, it is possible to construct equilibrium non-Gibbs canonical distribution functions which satisfy the fundamental equations of thermodynamics. The statistical mechanics with Tsallis entropy does not satisfy the zeroth law of thermodynamics at dynamical and statistical independence request, whereas the extensive Rényi statistics fulfills all requirements of equilibrium thermodynamics in the microcanonical ensemble. Transformation formulas between Tsallis statistics in Wang representation and Rényi statistics are presented. The one-particle distribution function in Rényi statistics for a classical ideal gas at finite particle number has a power-law tail for large momenta.     

Source coding with escort distributions and Rényi entropy bounds

We discuss the interest of escort distributions and Rényi entropy in the context of source coding. We first recall a source coding theorem by Campbell relating a generalized measure of length to the Rényi-Tsallis entropy. We show that the associated optimal codes can be obtained using considerations on escort-distributions. We propose a new family of measure of length involving escort-distributions and we show that these generalized lengths are also bounded below by the Rényi entropy. Furthermore, we obtain that the standard Shannon codes lengths are optimum for the new generalized lengths measures, whatever the entropic index. Finally, we show that there exists in this setting an interplay between standard and escort distributions.     

Rényi entropy and quantum phase transition in the Dicke model

The Rényi entropies of the Dicke model are presented. This quantum-optical model describes a single-mode bosonic field interacting with an ensemble of N two-level atoms. There is a quantum phase transition in the N→∞ limit. It is shown that there is an abrupt change in the Rényi entropy of order β at the transition point. Around the critical value of the coupling strength λc the Rényi entropy is proportional to the logarithm of the characteristic length and diverges as ln|λc−λ| for any order β. The pseudocapacity defined here in analogy with the heat capacity exhibits the phase transition. The critical exponent for the Dicke model is found to be 1 for any value of the parameter β.     

Tsallis, Rényi and nonextensive Gaussian entropy derived from the respective multinomial coefficients

We explore the generalization of the ordinary multinomial coefficient, based on the deformed q-multiplication and q-division. Aim of this study is to construct the appropriate multinomial coefficients, from which one can obtain the Tsallis, Rényi and nonextensive Gaussian entropy, respectively. We show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient. Its consequence is discussed.     

Physical information entropy and probability Shannon entropy

In a previous work by one of us (R. Urigu) concerning open quantum systems it was remarked that in processes of the type , when evaluating the information entropy of the environment as the Shannon entropy of the outcome probabilities in the channels , the total information entropy may decrease. We remark here that this decrease is easily excluded by requiring a condition of quantum modelizability of the environment even with respect to Shannon entropy (“cybernetic interpretability” of the environment). Further conditions on the quantum model of the environment are defined (“maximal observability” and “Boolean interpretability”), which are proved to be equivalent, and it turns out that, once satisfied in one model, they also are in any model with pure initial state; furthermore, these conditions turn out to be equivalent to the condition that the process consists of pure operations of the first kind. The relevance to the concept of macroscopicity and to the “von Neumann chain” is discussed.     

信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系--兼论玻尔兹曼熵和克劳修斯熵是否等价

论述了信息熵、玻尔兹曼熵以及克劳修斯熵之间的关系;由不涉及具体系统的方法从玻尔兹曼关系、信息熵推导出了克劳修斯熵的表达式;指出玻尔兹曼熵与克劳修斯熵不是等价关系,而是玻尔兹曼熵包含克劳修斯熵,信息熵又包含玻尔兹曼熵。     

Irreversibility, entropy and incomplete information

The open system has been proved to be a system with perfect accessibility represented as a probability space in which is defined a PA-measure. But, the PA-measure is not yet known; consequently, it is difficult to develop the statistical thermodynamics for an irreversible system. Here its integral expression is obtained in order to its use in the statistical thermodynamic analysis of the complex and irreversible systems.     

平面对称黑洞的统计熵

避开求解各种粒子波动方程的困难,直接应用量子统计的方法,计算平面对称黑面背景下玻色场与费米场的配分函数,得到黑面熵的积分表达式.然后应用改进的brickwall方法膜模型,计算黑面视界所对应的统计熵.在所得结果中当所取的积分下限和上限都趋于视界上时,可得到黑面熵与相应黑面视界面积成正比的关系,不存在原brickwall方法中的舍去项与对数发散项.整个计算过程,物理图像清楚,计算简单,为研究黑洞熵提供了一条简捷的新途径 关键词： 量子统计 膜模型 黑面熵     

Stochastic operators,information, and entropy

For a stochastic operatorU on andL ₁-space, i.eU is linear, positive, and norm preserving on the positive cone ofL ₁, it is shown thatU decreases relative information between two nonnegativeL ₁-functions. Furthermore it is shown that the following properties ofU are closely related:U is energy decreasing (energy preserving),U isH-decreasing, whereH is Boltzmann'sH-functional, and the Maxwell distributions are fixed points ofU.     

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).     

On the notions of entropy and information

In this review article we look for some of the historical reasons for the “identification” of the information-theoretical and thermodynamic entropy concepts. We also discuss the Baron-Jauch entropy concept and explicitly show that, for classical systems in thermodynamic equilibrium, there exists a very simple connection between this general definition and the ordinary experimental entropy.     

Continuous-time quantum walks on Erdös-Rényi networks

We study the coherent exciton transport of continuous-time quantum walks (CTQWs) on Erdös-Rényi networks. We numerically investigate the transition probability between two nodes of the networks, and compare the classical and quantum transport efficiency on networks of different connectivity. In the long time limiting, we find that there is a high probability to find the exciton at the initial node. We also study how the network parameters affect such high return probability.     

The Tsallis entropy of natural information

Estimating the information contained in natural data, such as electroencephalography data, is unusually difficult because the relationship between the physical data and the information that it encodes is unknown. This unknown relationship is often called the encoding problem. The present work provides a solution to this problem by deriving a method to estimate the Tsallis entropy in natural data. The method is based on two findings. The first finding is that the physical instantiation of any information event, that is, the physical occurrence of a symbol of information, must begin and end at a discontinuity or critical point (maximum, minimum, or saddle point) in the data. The second finding is that, in certain data types such as the encephalogram (EEG), the variance within of an EEG waveform event is directly proportional to its probability of occurrence.These two outcomes yield two results. The first is the easy binning of data into separate information events. The second is the ability to estimate probabilities in two ways: frequency counting and computing the variance within of an EEG waveform. These results are used to derive a linear estimator of the Tsallis entropy functional, allowing it to be estimated without deducing the encoding.This method for estimating the Tsallis entropy is first used to estimate the information in simple signals. The amount of information estimated is highly accurate. The method is then applied to two problems in electroencephalography. The first is distinguishing normal aging from very early Alzheimer's disease (mild cognitive impairment), and the second is medication monitoring of Alzheimer's disease treatment. The former is done with an accuracy of 92% and the latter with an accuracy of 91%. This detection accuracy is the highest published accuracy in the literature, which suggests that this method for Tsallis entropy estimation is both accurate and useful.     

Von neumann entropy as information rate

Recently it has been shown that quantum theory can be viewed as a classical probability theory by treating Hilbert space as a measure space (H, B(H)) of events or hidden states. Each density operator defines a set of probability measures such that(E _n )=w _n (alln). Coding elements H by subspacesE _n entails distortion. We show that the von Neumann entropyS() = -trInequals the effective rate at which the Hilbert space produces information with zero expected distortion, and comment on the meaning of this.     

Maxwell's demon and the entropy cost of information

We present an analysis of Szilard's one-molecule Maxwell's demon, including a detailed entropy accounting, that suggests a general theory of the entropy cost of information. It is shown that the entropy of the demon increases during the expansion step, due to the decoupling of the molecule from the measurement information. It is also shown that there is an entropy symmetry between the measurement and erasure steps, whereby the two steps additivelv share a constant entropy change, but the proportion that occurs during each of the two steps is arbitrary. Therefore the measurement step may be accompanied by an entropy increase, a decrease, or no change at all, and likewise for the erasure step. Generalizing beyond the demon, decorrelation between a physical system and information about that system always causes an entropy increase in the joint system comprised of both the original system and the information. Decorrelation causes a net entropy increase in the universe unless, as in the Szilard demon, the information is used to decrease entropy elsewhere before the correlation is lost. Thus, information is thermodynamically costly precisely to the extent that it is not used to obtain work from the measured system.     

Rényi Entropy and Free Energy

The Rényi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, Rényi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can perform as it moves to equilibrium at the new temperature divided by the change in temperature equals the system’s Rényi entropy in its original state. This result applies to both classical and quantum systems. Mathematically, we can express this result as follows: the Rényi entropy of a system in thermal equilibrium is without the ‘q−1-derivative’ of its free energy with respect to the temperature. This shows that Rényi entropy is a q-deformation of the usual concept of entropy.