Geometric Rényi divergence: A comparative measure with applications to atomic densities

Antolín, J.; Bouvrie, P. Alexander; Angulo, J. C.

doi:10.1103/physreva.84.032504

Cited by 5 publications

(3 citation statements)

References 46 publications

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“…Let us mention the relevant role played by the exchange and kinetic terms in the expansion of the functional for the total energy, 1 the Shannon entropy 25 (as a limiting case having its roots in information theory, statistical mechanics, and thermodynamics), or the disequilibrium 33 in the study of organization patterns in statistical mechanics, to mention a few of them. Additionally, let us remark that some of the previous functionals are essential ingredients in defining a variety of further information-theoretic concepts (e.g., complexity, 34 similarity, 37 and divergence [38][39][40] ).…”

Section: Upper Bounds On Frequency Moments In Terms Of Radial Exmentioning

confidence: 99%

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Angulo

Bouvrie

Antolín

2012

Journal of Mathematical Physics

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Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others. C 2012 American Institute of Physics. [http://dx.are also known. Although these relationships are usually applied in three-dimensional systems (i.e., with vectors of three components r and p), all of them are valid for arbitrary dimensionality. 16,17 Such uncertainty relations are physically relevant, not only because of their importance in a theoretical quantum-mechanical framework, 18-20 but also in the development of quantum information and computation. 21,22 In this sense, the studies of entropic uncertainty relations 23 and their connection with entanglement 24 are also remarkable.The aim of this study is to present uncertainty inequalities, in the form of bounds on a frequency moment (quantity defined in Sec. II) in a given space (position or momentum) in terms of two radial expectation values in the conjugate space. Such inequalities can handle radial expectation values of positive or negative orders. The bounds provided here are of universal validity (i.e., for any d-dimensional quantum mechanical system). By way of example, we carry out a numerical study for selected inequalities of physical interest in atomic systems, and the results are interpreted taking into account that some radial expectation values for atomic densities, in both position and momentum spaces, are physically relevant and/or experimentally accessible.The paper is structured as follows: Section II is devoted to the definition of the one-particle densities from the wave function, as well as the main quantities we will deal with (radial expectation values, frequency moments, Rényi entropies). In Sec. III, the uncertainty relations associated to those quantities are provided, as universal bounds on frequency moments in terms of radial expectation values in the conjugate space. Particular cases of physical interest are detailed in Se...

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Section: Upper Bounds On Frequency Moments In Terms Of Radial Exmentioning

confidence: 99%

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Angulo

Bouvrie

Antolín

2012

Journal of Mathematical Physics

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“…Therefore, order 2 JRD is not a preferable measure for alignment of two distributions. To resolve the negativity issue, Antolín et al [1] proposed geometric Rényi divergence (GRD):…”

Section: Geometric Rényi Divergencementioning

confidence: 99%

Shape from semantic segmentation via the geometric Rényi divergence

Koizumi

Smith

2021

2021 IEEE Winter Conference on Applications of Computer Vision (WACV)

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“…KL has also been used in molecular dynamics simulations . Relative information, based on other measures such as Fisher information, or Rényi entropy, has also been examined in varied contexts.…”

Section: Introductionmentioning

confidence: 99%

Entropic Kullback‐Leibler type distance measures for quantum distributions

Laguna

Salazar

Sagar

2019

Int J of Quantum Chemistry

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Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback-Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback-Leibler (CRKL) distance.The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two-variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure.The use of CRKL in measuring distances between orbitals is also discussed. K E Y W O R D S cumulative residual Kullback-Leibler entropy, Kullback-Leibler, quantum distributions, relative entropy

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Geometric Rényi divergence: A comparative measure with applications to atomic densities

Cited by 5 publications

References 46 publications

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Uncertainty inequalities among frequency moments and radial expectation values: Applications to atomic systems

Shape from semantic segmentation via the geometric Rényi divergence

Entropic Kullback‐Leibler type distance measures for quantum distributions

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