P. Sánchez-Moreno

Uncertainty relation for Fisher information of D-dimensional single-particle systems with central potentials

An uncertainty Fisher information relation in quantum mechanics is derived for multidimensional single-particle systems with central potentials. It is based on the concept of Fisher information in the two complementary position and momentum spaces, which is a gradient functional of the corresponding probability distributions. The lower bound of the product of position and momentum Fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality. Applications to various elementary systems is discussed.

The Fisher-information-based uncertainty relation, Cramer-Rao inequality and kinetic energy for the D-dimensional central problem

The inequality , with L being the grand orbital quantum number, and its conjugate relation for (r2, p−2) are shown to be fulfilled in the D-dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation (IρIγ≥ const) and the Cramer–Rao inequalities (r2Iρ ≥ D2; p2Iγ ≥ D2). In addition, the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by Iρ and Iγ, respectively.

Improvement of the Heisenberg and Fisher-information-based uncertainty relations for D-dimensional central potentials

The Heisenberg and Fisher-information-based uncertainty relations are improved for stationary states of single-particle systems in a D-dimensional central potential. The improvement increases with the squared orbital hyper- angular quantum number. The new uncertainty relations saturate for the isotropic harmonic oscillator wavefunction.

A quantum uncertainty relation based on Fisher's information

We explore quantum uncertainty relations involving the Fisher information functionals Ix and Ip evaluated, respectively, on a wavefunction Ψ(x) defined on a D-dimensional configuration space and the concomitant wavefunction on the conjugate momentum space. We prove that the associated Fisher functionals obey the uncertainty relation IxIp ≥ 4D2 when either Ψ(x) or is real. On the other hand, there is no lower bound to the above product for arbitrary complex wavefunctions. We give explicit examples of complex wavefunctions not obeying the above bound. In particular, we provide a parametrized wavefunction for which the product IxIp can be made arbitrarily small.

Quantum entanglement in exactly soluble atomic models: the Moshinsky model with three electrons, and with two electrons in a uniform magnetic field

AbstractWe investigate the entanglement-related features of the eigenstates of two exactly soluble atomic models: a one-dimensional three-electron Moshinsky model, and a three-dimensional two-electron Moshinsky system in an external uniform magnetic field. We analytically compute the amount of entanglement exhibited by the wavefunctions corresponding to the ground, first and second excited states of the three-electron model. We found that the amount of entanglement of the system tends to increase with energy, and in the case of excited states we found a finite amount of entanglement in the limit of vanishing interaction. We also analyze the entanglement properties of the ground and first few excited states of the two-electron Moshinsky model in the presence of a magnetic field. The dependence of the eigenstates’ entanglement on the energy, as well as its behaviour in the regime of vanishing interaction, are similar to those observed in the three-electron system. On the other hand, the entanglement exhibits a monotonically decreasing behavior with the strength of the external magnetic field. For strong magnetic fields the entanglement approaches a finite asymptotic value that depends on the interaction strength. For both systems studied here we consider a perturbative approach in order to shed some light on the entanglement’s dependence on energy and also to clarify the finite entanglement exhibited by excited states in the limit of weak interactions. As far as we know, this is the first work that provides analytical and exact results for the entanglement properties of a three-electron model.

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.

Fisher information of special functions and second-order differential equations

We investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a non-negative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic D-dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometric-type functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others.

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.

Jensen divergence based on Fisher?s information

The measure of Jensen–Fisher divergence between probability distributions is introduced and its theoretical grounds set up. This quantity, in contrast to the remaining Jensen divergences, grasps the fluctuations of the probability distributions because it is controlled by the (local) Fisher information, which is a gradient functional of the distribution. So it is appropriate and informative when studying the similarity of distributions, mainly for those having oscillatory character. The new Jensen–Fisher divergence shares with the Jensen–Shannon divergence the following properties: non-negativity, additivity when applied to an arbitrary number of probability densities, symmetry under exchange of these densities, vanishing under certain conditions and definiteness even when these densities present non-common zeros. Moreover, the Jensen–Fisher divergence is shown to be expressed in terms of the relative Fisher information as the Jensen–Shannon divergence does in terms of the Kullback–Leibler or relative Shannon entropy. Finally, the Jensen–Shannon and Jensen–Fisher divergences are compared for the following three large, non-trivial and qualitatively different families of probability distributions: the sinusoidal, generalized gamma-like and Rakhmanov–Hermite distributions, which are closely related to the quantum-mechanical probability densities of numerous physical systems.

The Shannon-entropy-based uncertainty relation for D-dimensional central potentials

The uncertainty relation based on the Shannon entropies of the probability densities in position and momentum spaces is improved for quantum systems in arbitrary D-dimensional spherically symmetric potentials. To find this, we have used the Lp – Lq norm inequality of De Carli and the logarithmic uncertainty relation for the Hankel transform of Omri. Applications to some relevant three-dimensional central potentials are shown.