A Generalized Statistical Complexity Measure: Applications to Quantum Systems

TL;DR

This paper introduces a two-parameter family of complexity measures based on Rényei entropies, generalizing existing measures and applying them to analyze quantum systems like the hydrogen atom, harmonic oscillator, and square well.

Contribution

It develops a generalized complexity measure using Rényei entropies and explores its mathematical properties and applications to quantum systems.

Findings

Complexity measures reduce to known forms for specific parameters.

Global and local factors of complexity are characterized.

Complexity calculated for various quantum systems.

Abstract

A two-parameter family of complexity measures $\tilde{C}^{(\alpha,\beta)}$ based on the R\'enyi entropies is introduced and characterized by a detailed study of its mathematical properties. This family is the generalization of a continuous version of the LMC complexity, which is recovered for $\alpha=1$ and $\beta=2$. These complexity measures are obtained by multiplying two quantities bringing global information on the probability distribution defining the system. When one of the parameters, $\alpha$ or $\beta$, goes to infinity, one of the global factors becomes a local factor. For this special case, the complexity is calculated on different quantum systems: H-atom, harmonic oscillator and square well.