Exact R\'enyi entropies of $D$-dimensional harmonic systems

TL;DR

This paper derives exact Rénnyi entropy measures for all quantum states of a multidimensional harmonic oscillator, providing precise uncertainty quantification in quantum systems.

Contribution

It presents the first exact formulas for Rénnyi entropies of all states in D-dimensional harmonic systems, extending beyond high-energy and high-dimensional approximations.

Findings

Exact Rénnyi entropies for all states derived

Formulas expressed in terms of D, potential strength, and quantum numbers

Provides comprehensive uncertainty measures for multidimensional quantum systems

Abstract

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue \textit{per se} and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and R\'enyi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a $D$-dimensional quadratic potential, $D > 1$) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the R\'enyi uncertainty measures of the $D$-dimensional harmonic system for all ground and excited quantum states directly in terms of $D$, the potential strength and the hyperquantum numbers.