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arXiv:1305.5334 · AIP conference proceedings

Rigorous bounds for Renyi entropies of spherically symmetric potentials

Pablo Sánchez-Moreno, Steeve Zozor, Jesús S. Dehesa, Ali Mohammad-djafari A, Jean‐François Bercher, Pierre Bessìère

Published Jan 1, 2011• 2 citations
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Abstract

The Rényi and Shannon entropies are information‐theoretic measures which have enabled to formulate the position‐momentum uncertainty principle in a much more adequate and stringent way than the (variance‐based) Heisenberg‐like relation. Moreover, they are closely related to various energetic density‐functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment 〈r2〉 for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.

DOI
10.1063/1.3573616

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The Rényi and Shannon entropies are information‐theoretic measures which have enabled to formulate the position‐momentum uncertainty principle in a much more adequate and stringent way than the (variance‐based) Heisenberg‐like relation. Moreover, they are closely related to various energetic density‐functionals of quantum systems. Here we find sharp upper bounds to these quantities in terms of the second order moment 〈r2〉 for general spherically symmetric potentials, which substantially improve previous results of this type, by means of the Rényi maximization procedure with a covariance constraint due to Costa, Hero and Vignat [1]. The contributions to these bounds coming from the radial and angular parts of the physical wavefunctions are explicitly given.
The Rényi and Shannon entropies are information-theoretic measures, which have enabled to formulate the position–momentum uncertainty principle in a much more adequate and stringent way than the (variance-based) Heisenberg-like relation. Moreover, they are closely related to various energetic density functionals of quantum systems. Here we derive upper bounds on these quantities in terms of the second-order moment 〈r2〉 for general central potentials. This improves previous results of this type. The proof uses the Rényi maximization procedure with a covariance constraint due to Costa et al. [in Proceedings of the Fourth International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), edited by A. Rangarajan, M. A. T. Figueiredo, and J. Zerubia (Springer-Verlag, Lisbon, 2003), [Lect. Notes Comput. Sci. 52, 211 (2003).]] The contributions to these bounds coming from the radial and angular parts of the physical wave functions are taken into account. Finally, the application to the d-dimensional (d ⩾ 3) hydrogenic and oscillator-like systems is provided.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1‐dimensional) quantum particle in terms of variances: 〈‖x‖2〉〈‖u‖2〉⩾1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon’s or Rényi’s entropies). As another alternative, we consider moments‐based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy under statistical moments’ constraints (〈‖x‖a〉 and 〈‖u‖b〉). Our result improves a relation proposed very recently by Dehesa et al.. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a =b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance‐based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d‐dimensional hydrogenic systems.
Abstract The Shannon entropy, the desequilibrium and their generalizations (Rényi and Tsallis entropies) of the three‐dimensional single‐particle systems in a spherically symmetric potential V(r) can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this article, we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator‐like systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case, we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator‐like wavefunctions.
The position-momentum Shannon and R\'enyi uncertainty products of general quantum systems are shown to be bounded not only from below (through the known uncertainty relations), but also from above in terms of the Heisenberg-Kennard product $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}$. Moreover, the Cram\'er-Rao, Fisher-Shannon, and L\'opez-Ruiz, Mancini, and Calbet shape measures of complexity (whose lower bounds have been recently found) are also bounded from above. The improvement of these bounds for systems subject to spherically symmetric potentials is also explicitly given. Finally, applications to hydrogenic and oscillator-like systems are done.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.
The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg’s uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in 1/D in similar systems with a non-standard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-D limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The D-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the D-dimensional hydrogenic atom at the limit of large D. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.