We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the three main intrinsic measures of complexity (Cramer-Rao, Fisher-Shannon, LMC) and some of their generalizations.Comment: Version accepted in Physics Letters

The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system but also the entropic uncertainty measures of Rényi type, which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in 1/D in similar systems with a nonstandard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-D limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The D-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work, we rigorously determine the leading term of the Rényi entropies of the Ddimensional hydrogenic atom at the limit of large D. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations. Published by AIP Publishing. https://doi

Abstract. The Rényi entropies R p [ρ], p > 0, 1 of the highly-excited quantum states of the D-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability ρ n (r) density of the Rydberg wavefunctions Ψ n,l,{µ} (r); so, to the L p -norms of the associated Laguerre polynomials. Then, we determine the asymptotics n → ∞ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the Rényi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers (n, l), the parameter order p and the universe dimensionality D for all possible cases D ≥ 1. We find that (a) the Rényi entropy power decreases monotonically as the order p is increasing and (b) the disequilibrium (closely related to the second order Rényi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of D.

The Rényi entropies R p [ρ], 0 < p < ∞ of the probability density ρ n,l,m ( r) of a physical system completely characterize the chemical and physical properties of the quantum state described by the three integer quantum numbers (n, l, m). The analytical determination of these quantities is practically impossible up until now, even for the very few systems where their Schrödinger equation is exactly solved. In this work, the Rényi entropies of Rydberg (highly-excited) hydrogenic states are explicitly calculated in terms of the quantum numbers and the parameter p. To do that we use a methodology which first connects these quantities to the L p -norms N n,l (p) of the Laguerre polynomials which characterize the state's wavefunction. Then, the Rényi, Shannon and Tsallis entropies of the Rydberg states are determined by calculating the asymptotics (n → ∞) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge. PACS numbers: 89.70.Cf, 89.70.-a, 32.80.Ee, 31.15.-p Keywords: Information theory of quantum systems, Rényi entropy of quantum systems, Rydberg states, hydrogenic atoms. I. INTRODUCTIONRecent years have witnessed a growing interest in the analytical information theory of finite quantum systems. A major goal of this theory is the explicit determination of the entropic measures (Fisher information and Shannon, Rényi and Tsallis entropies,...) in terms of the quantum numbers which characterize the state's wavefunction of the system. These quantities, which quantify the spatial delocalization of the single-particle density of the systems in various complementary ways, are most appropriate uncertainty measures because they do not make any reference to some specific point of the corresponding Hilbert space, in contrast to the variance * ivtoranzo@ugr.es † dehesa@ugr.es arXiv:1603.09494v1 [quant-ph]

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre (L (α) m (x)) and Gegenbauer (C (α) m (x)) polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as α → ∞ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.Keywords: asymptotic analysis of integrals, information theory of orthogonal polynomials, entropic functionals of Laguerre and Gegenbauer polynomials.