Abstract
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 ( p , β , λ ) -Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function ρ characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of ρ , 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 ( p , β , λ ) -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.
Highlights
The definition of complexity measures to quantify the internal disorder of physical systems is an important and challenging task in science, basically because of the many facets of the notion of disorder [1,2,3,4,5,6,7,8,9,10,11,12]
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
Let us begin with the definitions of the following information-theoretic quantities of the probability density ρ: the Rényi entropy power Nλ [ρ], the ( p, β)-Fisher information Fp,β [ρ], and the ( p, β, λ)-Fisher–Rényi complexity C p,β,λ [ρ]
Summary
The definition of complexity measures to quantify the internal disorder of physical systems is an important and challenging task in science, basically because of the many facets of the notion of disorder [1,2,3,4,5,6,7,8,9,10,11,12]. Tsallis distributions [48,49] This extended inequality allows to define again a complexity measure, based on this generalized Fisher information and the Rényi entropy power [27]. We firstly review the extension of the Stam inequality based on the efforts of Lutwak et al and Bercher [34,35,36], or more generally, based on that of Agueh [51,52] To this aim, we introduce a three-parametric Fisher–Rényi complexity, showing its scaling and translation invariance and non-trivial bounding from below. Let us begin with the definitions of the following information-theoretic quantities of the probability density ρ: the Rényi entropy power Nλ [ρ], the ( p, β)-Fisher information Fp,β [ρ], and the ( p, β, λ)-Fisher–Rényi complexity C p,β,λ [ρ]. Exhibits several situations where the solution is known explicitly (and the optimal bound as well), as summarized in the subsection
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