I. V. Toranzo

Monotone measures of statistical complexity

Abstract 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.

Entanglement in N-harmonium: bosons and fermions

The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely integrable model of N particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of N and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. A critical number, Nc, of particles exists, and below it, for positive values of the coupling constant, the entanglement grows when the number of particles increases; the opposite occurs for . For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when N grows. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increases values of the coupling constant, and (b) to grow when the energy increases. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.

Entropy and complexity analysis of hydrogenic Rydberg atoms

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of Cramer-Rao, Fisher-Shannon, and Lopez Ruiz-Mancini-Calvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.

Entropic properties of D-dimensional Rydberg systems

The fundamental information-theoretic measures (the Renyi Rp[ρ] and Tsallis Tp[ρ] entropies, p>0) of the highly-excited (Rydberg) quantum states of the D-dimensional (D>1) hydrogenic systems, which include the Shannon entropy (p→1) and the disequilibrium (p=2), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability ρ(r→) densities associated to the Rydberg hydrogenic wavefunctions Ψn,l,{μ}(r→), which are closely connected to the Lp-norms of the associated Laguerre polynomials. Then, we determine the (n→∞)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality D, the nuclear charge and the hyperquantum numbers (n,l,{μ}) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.

Pauli effects in uncertainty relations

Abstract In this Letter we analyze the effect of the spin dimensionality of a physical system in two mathematical formulations of the uncertainty principle: a generalized Heisenberg uncertainty relation valid for all antisymmetric N-fermion wavefunctions, and the Fisher-information-based uncertainty relation valid for all antisymmetric N-fermion wavefunctions of central potentials. The accuracy of these spin-modified uncertainty relations is examined for all atoms from Hydrogen to Lawrencium in a self-consistent framework.

Entropic uncertainty measures for large dimensional hydrogenic systems

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 Renyi 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 non-standard 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 lea...

Quantum entanglement in (d−1)-spherium

There are very few systems of interacting particles (with continuous variables) for which the entanglement of the concomitant eigenfunctions can be computed in an exact, analytical way. Here we present analytical calculations of the amount of entanglement exhibited by s-states of spherium. This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a (d−1)-sphere (that is, to the surface of a d-dimensional ball). We investigate the dependence of entanglement on the radius R of the system, on the spatial dimensionality d, and on energy. We find that entanglement increases monotonically with R, decreases with d, and also tends to increase with the energy of the eigenstates. These trends are discussed and compared with those observed in other two-electron atomic-like models where entanglement has been investigated.

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

textabstractThe determination of the physical entropies (Renyi, 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 () and Gegenbauer () polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.

One-Parameter Fisher–Rényi Complexity: Notion and Hydrogenic Applications

In this work, the one-parameter Fisher–Renyi 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.

Exact Rényi entropies of D -dimensional harmonic systems

Abstract The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue per se and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a D-dimensional quadratic potential, D > 1) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the Rényi uncertainty measures of the D-dimensional harmonic system for all ground and excited quantum states directly in terms of D, the potential strength and the hyperquantum numbers.