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

TL;DR

This paper investigates the behavior of various uncertainty measures and complexity measures in high-dimensional hydrogenic and harmonic systems, revealing that many of these measures converge to similar values in the limit as the dimension approaches infinity.

Contribution

It demonstrates that multiple uncertainty and complexity measures exhibit universal first-order behavior in the high-dimensional limit for hydrogenic and oscillator-like systems, and highlights the sensitivity of LMC complexities to system differences.

Findings

Uncertainty products and sums converge to the same first-order values in the high-dimensional limit.

Fisher-information-based and Shannon-entropy-based measures also show similar convergence.

LMC and LMC-Rényi complexities detect differences between hydrogenic and harmonic systems at first order.

Abstract

In this work we find that not only the Heisenberg-like uncertainty products and the R\'enyi-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 ($D \to \infty$) 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 Cr\'amer-Rao and Fisher-Shannon complexities. Moreover, we show that the LMC (L\'opez-Ruiz-Mancini-Calvet) and LMC-R\'enyi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.