Generalized statistical complexity and Fisher-Renyi entropy product in the $H$ -atomGeneralized statistical complexity and Fisher-Renyi entropy product in the $H$-atom

E. Romera, Ricardo López‐Ruiz, Jaime Sañudo A +1 more

By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.By using the Renyi entropy, and following the same scheme that in the Fisher-Renyi entropy product case, a generalized statistical complexity is defined. Several properties of it, including inequalities and lower and upper bounds are derived. The hydrogen atom is used as a test system where to quantify these two different statistical magnitudes, the Fisher-Renyi entropy product and the generalized statistical complexity. For each level of energy, both indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum. Hence, in the same way as happens with the Fisher-Shannon and the statistical complexity, these generalized Renyi-like statistical magnitudes break the energy degeneration in the H-atom.

arXiv:0901.1752Jan 13, 20098 citations

Information Theory and Statistical Entropy

Roger Balian

Springer eBooksJan 1, 19912 citations

Quantum information-entropic measures for exponential-type potential

A. N. Ikot, Gaotsiwe Joel Rampho, P. O. Amadi +3 more

The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.

Results in PhysicsJun 29, 202021 citations

Smooth Entropy and Rényi Entropy

Christian Cachin

Lecture notes in computer scienceJan 1, 199739 citations

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

Nahual Sobrino, David Puertas‐Centeno, I. V. Toranzo +1 more

Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.Abstract In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the D -dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ( D → ∞ ) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

arXiv:1708.07299Journal of Statistical Mechanics Theory and ExperimentAug 1, 201724 citations

Generalized statistical complexity and Fisher-Renyi entropy product in the

H

-atom