Local wave-vector, Shannon and Fisher information

Ágnes Nagy, E. Romera, Shubin Liu

Ágnes Nagy

Abstract Relationship of Fisher information and density functional theory is reviewed. Links between the Fisher and Shannon information, the local wave‐vector and the relative information are displayed. Euler equations for the Fisher and relative Fisher information are presented. A combined information theoretical and thermodynamic view of density functional theory is analyzed. The extremum of the Shannon and Fisher information results a constant temperature and for Coulomb systems a simple relation between the total energy and phase‐space Fisher information. Relations for phase‐spase fidelity, fidelity susceptibility and relative information are also presented.

István Hornyák, Ágnes Nagy

Mojtaba Alipour

In the context of informational analysis of atomic and molecular systems, the connection between local quantum observables and information measures is of interest. In this paper, analytical relationships for the imaginary part of the total local momentum (coordinate) in terms of information theoretic measures have been established. Moreover, on the basis of another scheme in which the relationship among densities of information energy, Shannon entropy and Fisher information has been proposed [M. Alipour and A. Mohajeri, Mol. Phys. 110, 403 (2012)], the general formulae for the imaginary part of the total local momentum and the corresponding variance are expressed. The presented proof may be viewed in light of the relation between the local wave vector and the information energy density. Through this study, another noteworthy application of information theory is highlighted.

Sudin Singh, A. Saha

The article intertwines the study of wave packet dynamics with information-theoretic measurements in one dimensional (D =1) system. A localized wave packet at time t=0 has been considered here and its change at later instant of time t is calculated for the position space wave function. The momentum space wave function is obtained by taking the Fourier transform of the position space wave function at time t= 0 . These wave functions are then employed to construct the wave packet's respective probability densities in position and momentum space, and later have been used to compute the corresponding position and momentum space Shannon (S) and Fisher information (I) entropies. It has been observed that although the Shannon and Fisher information entropies explicitly depend on the standard deviation, neither the Shannon entropy sum nor the product of Fisher information entropies is. Moreover, for D dimensional systems, the computed values of the Shannon and Fisher entropies are found to satisfy the lower bounds of the Bialynicki-Birula and Myceilski (BBM) inequality relation and the Stam-Cramer-Rao inequalities better known as the Fisher based uncertainty relation. Thus, our theoretical study explores the validation of information-theoretic measurements for wave packet dynamics using the basic formulations of information theory.