Probability space of wave functions

It is well known that in quantum mechanics, when the mean value of an observable is given, entropy maximization (von Neumann, Born, Jaynes) can be successfully applied for constructing a probability distribution on the set of possible values of that observable. In this paper, the entropy maximization technique is extended to the complex domain in order to construct an unbiased probability measure on the set of all wave functions. In particular, a justification and a generalization of the Wiener-Siegel probability distribution of Gaussian type in the differential space of wave functions are given.     

The entropy of observables on quantum logic

The entropy of an abstract observable on quantum logic is defined as an informational property of the corresponding sublogic of a quantum logic associated with the physical system. The main properties of such quantity are stated. It is proved that the entropy is completely characterized by the entropies of the corresponding finite resolutions of the unit (experiments). The connection with the entropy of a state is also mentioned.     

A-entropy for generalized observables

We investigateA-entropy with respect to certain semispectral measures in a given state. It is shown that the entropy with respect to an observable describing simultaneous measurement of position and momentum is greater than the von Neumann entropy. Similar results are obtained for the fuzzy and sharp positions. The continuity properties of this entropy are also examined.     

The Wigner-Yanase Entropy is not Subadditive

Wigner and Yanase introduced in 1963 the Wigner-Yanase entropy defined as minus the skew information of a state with respect to a conserved observable. They proved that the Wigner-Yanase entropy is a concave function in the state and conjectured that it is subadditive with respect to the aggregation of possibly interacting subsystems. While this turned out to be true for the quantum-mechanical entropy, we negate the conjecture for the Wigner-Yanase entropy by providing a counter example.     

Remarks on A-entropy

We study three possibilities of defining the entropy of a density matrix with respect to an observable and show that most of the properties of the usual entropy remain valid cum grano salis.     

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.     

Information and Entropy in Quantum Measurement Processes

Quantum measurement processes of discrete andcontinuous observables are considered from theinformation-theoretic point of view. The informationextracted from the results of quantum measurementperformed on a physical system and the change of theShannon entropy of the measured physical system areinvestigated in detail. It is shown that the amount ofinformation about the intrinsic observable of themeasured physical system can be expressed by the mutualinformation between the physical system and themeasurement apparatus if the intrinsic observablecommutes with the operational observable defined by thequantum measurement process. Furthermore, the conditioncan be obtained under which the amount of informationextracted from the measurement outcomes becomes equal tothe decrease of the entropy of the measured physical system. In addition, the change of theShannon entropy is compared with that of the von Neumannentropy. The general results do not depend on whetherthe readout of the measurement outcome obeys the projection postulate or not. Severalexamples of quantum measurement processes are consideredto examine the general results.     

Primordial fluctuations and non-Gaussianities in multifield Dirac-Born-Infeld inflation

We study Dirac-Born-Infeld inflation models with multiple scalar fields. We show that the adiabatic and entropy modes propagate with a common effective sound speed and are thus amplified at the sound horizon crossing. In the small sound speed limit, we find that the amplitude of the entropy modes is much higher than that of the adiabatic modes. We show that this could strongly affect the observable curvature power spectrum as well as the amplitude of non-Gaussianities, although their shape remains as in the single-field Dirac-Born-Infeld case.     

热场动力学理论中的Husimi分布函数及Wehrl熵的研究

利用量子相空间技术和信息熵理论,研究了热场动力学理论中量子纯态与相应混合态的Husimi分布函数及Wehrl熵的一致性问题.结果表明,热相干态与相应混合态的Husimi分布函数及Wehrl熵完全相同,支持了热场动力学理论.且热相干态的Wehrl熵与平移因子无关,故在热相干态中,量子系统的可观测量的量子涨落及不确定关系也与平移因子无关.     

Continuous QND measurements: no quantum noise

Monitoring an arbitrary observable is analyzed in the framework of Restricted-Path-Integral (RPI) theory of continuous quantum measurements. While in an usual (quantum-demolition) continuous measurement the measurement noise contains both classical and quantum parts, only the classical noise is shown to be present in a quantum nondemolition (QND) continuous measurement. As a result, no absolute restrictions exist on measurability of a QND observable and the measurement output satisfies the classical equation of motion. Monitoring the energy gives an example of a discrete-spectrum observable. Received: 7 April 1996 / Revised version: 7 August 1996     

Effect of Magnetic Field on Internal Energy and Entropy of a Parabolic Cylindrical Quantum Dot

In the present work, the entropy and internal energy of a GaAs cylindrical quantum dot in the presence of an applied magnetic field is studied. For this purpose, the Tsallis formalism is applied to obtain internal energy and entropy. It is found that entropy and internal energy are continuous function and they are zero at special temperatures. Entropy maximum increases with increasing dot radius. Internal energy increases by increasing magnetic field.     

Piecewise spectrally band-pass for compressive coded aperture spectral imaging

Coded aperture snapshot spectral imaging(CASSI) has been discussed in recent years. It has the remarkable advantages of high optical throughput, snapshot imaging, etc. The entire spatial-spectral data-cube can be reconstructed with just a single two-dimensional(2D) compressive sensing measurement. On the other hand, for less spectrally sparse scenes,the insufficiency of sparse sampling and aliasing in spatial-spectral images reduce the accuracy of reconstructed threedimensional(3D) spectral cube. To solve this problem, this paper extends the improved CASSI. A band-pass filter array is mounted on the coded mask, and then the first image plane is divided into some continuous spectral sub-band areas. The entire 3D spectral cube could be captured by the relative movement between the object and the instrument. The principle analysis and imaging simulation are presented. Compared with peak signal-to-noise ratio(PSNR) and the information entropy of the reconstructed images at different numbers of spectral sub-band areas, the reconstructed 3D spectral cube reveals an observable improvement in the reconstruction fidelity, with an increase in the number of the sub-bands and a simultaneous decrease in the number of spectral channels of each sub-band.     

Dissipative hamiltonian systems: A unifying principle

The concept of hamiltonian sysem is generalized to include a wide class of dissipative processes. Evolution of any observable is generated jointly by a hamiltonian, with an entropy-conserving Poisson bracket, and an entropy, with an energy-conserving dissipative bracket. This approach yields many of the standard kinetic equations, such as those representing particle collisions, three-wave interactions, and wave-particle resonances.     

The observer-dependence of cellular space-time structure

It is argued that space-time has a cellular structure, the exact structure being observer-dependent and consistent with the amount of energy he has available for refining his measuring apparatus. The usual concept of a single distance in continuous space is replaced by the concept of a distance set between cells, the elements of each set depending on the cellular structures of both the space and the measuring rod that is used in the measurement. The idea that there are many different ways of measuring the same observable is abandoned: instead, the definition of the original observable becomes split by the different measuring processes used, and the results of a measurement of each new observable defined by this splitting are predicted from the eigenvalues of a common operator by using an observer-dependent construction. Transformations between observers with different cellular structures are considered. The transformation is not as exact as in the continuous case, with at best a cell of one space being associated with a set of cells in the other. This transformation is determined by information being exchanged by the observers concerning the locations in their two spaces of a finite number of common events. The transformation becomes more exact as more information is exchanged.     

Nonequilibrium Linear Response for Markov Dynamics, I: Jump Processes and Overdamped Diffusions

Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory out-of-equilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.     

Curvature in causal BD-type inflationary cosmology

We study a closed model of the universe filled with viscous fluid and quintessence matter components in a Brans-Dicke type cosmological model. The dynamical equations imply that the universe may look like an accelerated flat Friedmann-Robertson-Walker universe at low redshift. We consider here dissipative processes which follow a causal thermodynamics. The theory is applied to viscous fluid inflation, where accepted values for the total entropy in the observable universe are obtained.     

On the Validity of Entropy Production Principles for Linear Electrical Circuits

We explain the (non-)validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principles are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff’s laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows to understand beyond the schemes of irreversible thermodynamics (1) the validity of the least dissipation, the minimum entropy production, and the maximum entropy production principles close to equilibrium; (2) the role of the observables’ parity under time-reversal and, in particular, the origin of Landauer’s counterexample (1975) from the fact that the fluctuating observable there is odd under time-reversal; (3) the critical remark of Jaynes (1980) concerning the apparent inappropriateness of entropy production principles in temperature-inhomogeneous circuits.     

From physics to economics: An econometric example using maximum relative entropy

Econophysics, is based on the premise that some ideas and methods from physics can be applied to economic situations. We intend to show in this paper how a physics concept such as entropy can be applied to an economic problem. In so doing, we demonstrate how information in the form of observable data and moment constraints are introduced into the method of Maximum relative Entropy (MrE). A general example of updating with data and moments is shown. Two specific econometric examples are solved in detail which can then be used as templates for real world problems. A numerical example is compared to a large deviation solution which illustrates some of the advantages of the MrE method.     

An Order Model for Infinite Classical States

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory, 2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and Martin is obtained. The infinite model retains many desirable aspects of the finite model, such as pure states as maximal elements and expected behavior of thermodynamic entropy. But it looses some of the important domain theoretic aspects, such as having a least element and exactness. Shannon entropy is no longer defined over the entire model and both it and thermodynamic entropy cease to be a measurements in the sense of Martin.     

Instabilities of Rényi entropies

We show that for systems with a large number of microstates Rényi entropies do not represent experimentally observable quantities except the Rényi entropy that coincides with the Shannon entropy.Work supported by the DFG (1978); author is recipient of a Feodor-Lynen grant from the Alexander von Humboldt Stiftung.