One-parameter Fisher-R\'enyi complexity: Notion and hydrogenic applications

In this work the one-parameter Fisher-R\'enyi measure of complexity for general $d$-dimensional probability distributions is introduced and its main analytic properties are discussed. Then, this quantity is determined for the hydrogenic systems in terms of the quantum numbers of the quantum states and the nuclear charge.


I. INTRODUCTION
We all have an intuitive sense of what complexity means. In the last two decades an increasing number of efforts have been published [1][2][3][4][5][6][7][8][9][10][11][12] to refine our intuitions about complexity into precise, scientific concepts, pointing out a large amount of open problems. Nevertheless there is not a consensus for the term complexity nor whether there is a simple core to complexity.
The latter suggests that complexity should be minimal at either end of the scale. However, a complexity quantifier to take into account the completely ordered and completely disordered limits (i.e., perfect order and maximal randomness, respectively) and to describe/explain the maximum between them is not known up until now.
Recently, keeping in mind the fundamental principles of the density functional theory, some statistical measures of complexity have been proposed to quantify the degree of structure or pattern of finite many-particle systems in terms of their single-particle density, such as the Crámer-Rao [23,26], Fisher-Shannon [18,21,24] and LMC (López-ruiz, Mancini and Calvet) [12,17] complexities and some modifications of them [13,22,25,[27][28][29]. They are composed by a two-factor product of entropic measures of Shannon [31], Fisher [6,32] and Rényi [33] types. Most interesting for quantum systems are those which involve the Fisher information (namely, the Crámer-Rao and the Fisher-Shannon complexities, and their modifications [25,27,34]), mainly because this is by far the best entropy-like quantity to take into account the inherent fluctuations of the quantum wave functions by quantifying the gradient content of the single-particle density of the systems.
The objetive of this article is to extend and generalize these Fisher-information-based measures of complexity by introducing a new complexity quantifier, the one-parameter Fisher-Rényi complexity, to discuss its properties and to apply it to the main prototype of Coulombian systems, the hydrogenic system. This notion is composed by two factors: a λ-dependent Fisher information (which quantifies various aspects of the quantum fluctuations of the physical wave functions beyond the density gradient, since it reduces to the standard Fisher information for λ = 1) and the Rényi entropy of order λ (which measures various facets of the spreading or spatial extension of the density beyond the celebrated Shannon entropy which corresponds to the limiting case λ → 1).
The article is structured as follows. In Section I we introduce the notion of one-parameter Fisher-Rényi measure of complexity. In Section II we discuss the main analytical properties of this complexity, showing that it is bounded from below, invariant under scaling transformations and monotone. In addition the near-continuity and the invariance under replications are also discussed. In Section III, we apply the new complexity measure to the hydrogenic systems. Finally some concluding remarks are given.

II. ONE-PARAMETER FISHER-RÉNYI COMPLEXITY MEASURE
In this section the notion of one-parameter Fisher-Rényi complexity C (λ) F R [ρ] of a d-dimensional probability density is introduced and its main analytic properties are discussed. This quantity is composed by two entropy-like factors of local (the one-parameter Fisher information of Johnson and Vignat [35],F λ [ρ]) and global (the λ-order Rényi entropy power [36], N λ [ρ]) characters.

A. The notion
The one-parameter Fisher-Rényi complexity measure C where D λ is the normalization factor given as This purely numerical factor is necessary to let the minimal value of the complexity be equal to unity, as explained below in paragraph 2.2.1. TheF λ [ρ] denotes the (scarcely known) λ-weighted Fisher information [35] defined bỹ ρ dx), being dx the d-dimensional volume element. Finally, the symbol N λ [ρ] denotes the λ-Rényi entropy power (see e.g., [36]) given as The complexity measure C (λ) F R [ρ] has a number of conceptual advantages with respect to the Fisher-information-based measures of complexity previously defined; namely, the Crámer-Rao and Fisher-Shannon complexity and their modifications. Indeed, it quantifies the combined balance of different (λ-dependent) aspects of both the fluctuations and the spreading or spatial extension of the single-particle density ρ in such a way that there is no dependence on any specific point of the system's region. The Crámer-Rao complexity [23,26] (which is the product of the standard Fisher information F [ρ] mentioned above and the variance V [ρ] = r 2 − r 2 ) measures a single aspect of the fluctuations (namely, the density gradient) together with the concentration of the probability density around the centroid r . The Fisher-Shannon complexity [18,21,24], defined by , quantifies the density gradient jointly with a single aspect of the spreading given by the Shannon entropy S[ρ] mentioned above. A modification of the previous measure by use of the Rényi entropy R λ [ρ] = 1 1−d ln R d ρ λ (x) dx instead of the Shannon entropy, the Fisher-Rényi product of complexity-type, has been recently introduced [25,27,34]; it measures the gradient together with various aspects of the spreading of the density.

B. The properties
Let us now discuss some properties of this notion: bounding from below, invariance under scaling transformations, monotonicity, behavior under replications and near continuity.
, with λ = 1), and the minimal complexity occurs, as implicitly proved by Savaré and Toscani [36], if and only if the density has the following generalized Gaussian form where (x) + = max{x, 0} and C λ is the normalization constant given by with Thus, the complexity measure C (λ) F R (ρ) has a universal lower bound of minimal complexity, that is achieved for the family of densities B λ (x).

2.
Invariance under scaling and translation transformations. The complexity measure F R (ρ) are scaling and translation invariant in the sense that where To prove this property we follow the lines of Savaré and Toscani [36]. First we calculate the generalized Fisher information of the transformed density, obtaining Note that in writing the first equality we have used that Then, we determine the value of the λ-entropy power of the density ρ a,b (x) which turns out to be equal to In particular, we have Two of the main properties of rearrangements is that they preserve the L p norms, which implies that the rearrangements of a probability density give rise to another probability density, and that they make everything spherically symmetric. The second feature makes the rearrangement operation relevant for quantification of statistical complexity [11], since a spherically symmetric variant of a probability density can in an atomic context be viewed as less complex. Then, we introduce the definition of this operation as well as its effects over the entropic quantities that make up our complexity measure.
Let f be a real-valued function, f : decreasing rearrangement of f is defined as with χ {x∈A * t } = 1 if x ∈ A * t and 0 otherwise. A t represents the super-level set of the function f and A * (which denotes the symmetric rearrangement of a set A ⊂ R n ) is the Euclidean ball centered at 0 such as V ol(A * ) = V ol(A).
The central idea of this transformation is to build up f * from the rearranged super-level sets in the same manner that f is built from its super-level sets. As a by-product from its construction, f * turns out to be a spherically symmetric decreasing function (i.e. f * (x) = f * (|x|) and moreover f * (b) < f * (a) ∀b > a, where a, b ∈ A * t ) which means that for any function f : R n → [0, ∞) and all t ≥ 0 or in other words, that for any measurable subset B ⊂ [0, ∞), the volume of the sets It is known [37] that under this transformation and for any p ∈ [0, 1) ∪ (1, ∞] the Rényi and Shannon entropies remain unchanged, i.e. Let us now consider the biparametric Fisher-like information, I β,q [f ], of a probability density function f (x) which is defined [38] by with q ≥ 0, β > 1. Then one notes that the one-parameter Fisher information,F λ [ρ], given by (3) can be expressed in terms of the previous quantity with β = 2 and q ≡ λ as On the other hand, considering the transformation ρ = u(x) k with k = β β(q−1)+1 , the biparametric Fisher information becomes also known as the β-Dirichlet energy of u(x). If k = 2, note that the function u(x) corresponds to a quantum-mechanical wave function. By using the symmetric decreasing rearrangement to the density function ρ, the well-known Pólya-Szegö inequality states that which implies that the minimizer of the left side is necessarily radially symmetric and decreasing, so the extremal function belongs to the subset of radially symmetric probability densities, and is represented by the generalized Gaussian given in (6). Now by taking into account (14) and the invariance of the Rényi entropy (and therefore the Rényi entropy power, ) upon rearrangements one obtains the monotonic behavior ofF λ [ρ] as Finally, this observation together with (1) allows us to obtain the monotonic behavior of this complexity measure C (λ) F R (ρ) proved by rearrangements, i.e.
where the inequality is saturated for the generalized Gaussian, ρ(x) = B λ (x), which also means that the symmetric rearrangement of a generalized Gaussian gives another generalized Gaussian, i.e. rearrangements preserve this subset of radially symmetric probability densities

4.
Behavior under replications. Let us now study the behavior of the Fisher-Rényi complexity C (λ) F R (ρ) under n replications. We have found that for one-dimensional densities ρ(x), x ∈ R with bounded support, this complexity measure behaves as follows: where the densityρ representing n replications of ρ is given bỹ where the points b m are chosen such that the supports Λ m of each density ρ m are disjoints.
Then, the integrals where the change of variable y = n F R (ρ) gets modified as elsewhere.
Due to the increasing oscillatory behaviour ofρ for x ∈ (0, δ 5 π) as δ tends to zero, the generalized Fisher informationF grows rapidly as δ decreases, while the Rényi entropy power tends to a constant value. Then, the more similar ρ andρ are, the more different are their values of C (λ) F R . Therefore, the Fisher-Rényi complexity measure is not near continuous.

III. THE HYDROGENIC APPLICATION
In this section we determine the one-parameter Fisher-Rényi complexity measure C wherer = 2Z n r, and the symbols ρ n,l (r) and Θ l,m (θ, φ) are the radial and angular parts of the density, which are given by and respectively. In addition, L α n (x) denotes the orthonormal Laguerre polynomials [39] with respect to the weight function ω α = x α e −x on the interval [0, ∞), and Y l,m (θ, φ) are the well-known spherical harmonics which can be expressed in terms of the Gegenbauer polynomials, C m n (x) via where 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π. Let us now compute the complexity measure C (λ) F R [ρ n,l,m ] of the hydrogenic probability density which, according to (1), is given by where D λ is the normalization constant given by (2) and the symbols I 1 and I 2 denote the integrals which can be solved by following the lines indicated in Appendix A.
In the following, for simplicity and illustration purposes, we focus our attention on the computation of the complexity measure for two large, relevant classes of hydrogenic states: the (ns) and the circular (l = m = n − 1) states.

Generalized Fisher-Rényi complexity of hydrogenic (ns) states.
In this case, Θ 0,0 (θ, φ) = |Y 0,0 (θ, φ)| 2 = 1 4π so that the three angular integrals can be trivially determined, and the radial integrals simplify as Thus, finally, the one-parameter (λ) Fisher-Rényi complexity measure C (λ) F R [ρ ns ] for the (ns)like hydrogenic states is given by where In particular, for the ground state (i.e., when n = 1, l = m = 0) we have shown in Appendix B that which allows us to find the following value for the one-parameter Fisher-Rényi complexity measure of the hydrogenic ground state, keeping in mind the value (2) for the normalization factor D λ . We have done this calculation in detail to check our methodology; we are aware that in this concrete example it would have been simpler to start directly from the explicit expression of the wave function of the orbital 1s. Operating in a similar way we can obtain the complexity values for the rest of ns-orbitals.
Then, we obtain the following analytical expressions for the radial integrals in terms of the parameters {Z, λ, n, l} of the system: where one should keep in mind that the Φ 0 functions are given as in (A13).
Similarly we can obtain the angular integrals by means of linerization-like formulas of the Gegenbauer polynomials or the associated Legendre polynomials of the first kind.