On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures
GIPSA-Lab, Université Grenoble Alpes, 11 rue des Mathématiques, 38420 Grenoble, France
Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain
Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain
Author to whom correspondence should be addressed.
Received: 21 August 2017 / Revised: 8 September 2017 / Accepted: 12 September 2017 / Published: 14 September 2017
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named
-Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function
characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of
, and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the
-Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
Scifeed alert for new publications
Never miss any articles
matching your research from any publisher
- Get alerts for new papers matching your research
- Find out the new papers from selected authors
- Updated daily for 49'000+ journals and 6000+ publishers
Define your Scifeed now
Share & Cite This Article
MDPI and ACS Style
Zozor, S.; Puertas-Centeno, D.; Dehesa, J.S. On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. Entropy 2017, 19, 493.
Zozor S, Puertas-Centeno D, Dehesa JS. On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. Entropy. 2017; 19(9):493.
Zozor, Steeve; Puertas-Centeno, David; Dehesa, Jesús S. 2017. "On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures." Entropy 19, no. 9: 493.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.