On Generalized Stam Inequalities and Fisher-Rényi Complexity Measures
arXiv:1710.00726 · doi:10.3390/e19090493
Abstract
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas 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. 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.
References in corpus (28)
- A Statistical Measure of Complexity
- Formulation of the uncertainty relations in terms of the Renyi entropies
- Information Entropy, Information Distances and Complexity in Atoms
- Entropic Uncertainty Relations in Quantum Physics
- Statistical complexity, Fisher-Shannon information, and Bohr orbits in the H-atom
- Some extensions of the uncertainty principle
- Universal property of the information entropy in atoms, nuclei and atomic clusters
- Monotone measures of statistical complexity
- Net Fisher information measure versus ionization potential and dipole polarizability in atoms
- General entropy-like uncertainty relations in finite dimensions
- Some results concerning maximum Renyi entropy distributions
- On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles
- Source coding with escort distributions and Renyi entropy bounds
- Classification and Verification of Online Handwritten Signatures with Time Causal Information Theory Quantifiers
- On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
- Position-momentum uncertainty relations based on moments of arbitrary order
- Some properties of generalized Fisher information in the context of nonextensive thermostatistics
- Configuration Complexities of Hydrogenic Atoms
- On generalized Cramér-Rao inequalities, generalized Fisher informations and characterizations of generalized q-Gaussian distributions
- Source Coding for Quasiarithmetic Penalties
- One-parameter class of uncertainty relations based on entropy power
- Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems
- On a (β,q)-generalized Fisher information and inequalities involving q-Gaussian distributions
- One-parameter Fisher-Rényi complexity: Notion and hydrogenic applications
- Statistical measure of complexity for quantum systems with continuous variables
- Rescaling the nonadditivity parameter in Tsallis thermostatistics
- Naudts-like duality and the extreme Fisher information principle
- The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation