Entanglement, complexity and entropic properties of quantum systems

Resumen

This Thesis aims to contribute to the emergent informational representation of the quantum systems, which complements the standard representation based on energetic concepts. Thus, we use the notions of information entropy, statistical complexity and quantum entanglement, together with the methods of Classical and Quantum Information and the algebraic and asymptotic techniques of the theory of orthogonal polynomials and special functions of Applied Mathematics and Mathematical Physics, to study and quantify the multiple facets of the spatial delocalization of the charge and matter distributions of the multidimensional quantum systems of bosonic and fermionic character. These facets, which determine both the uncertainty measures of the systems and their physical and chemical properties, are manifested in the enormous diversity of multidimensional geometries of the single-particle densities which characterize the non-relativistic mechano-quantically allowed states of such systems, ac- cording to the Density Functional Theories based on the Hohenberg-Kohn theorem and generalizations. The Thesis is composed by three chapters (Introduction, Methodology, Applications) followed by some Conclusions and open problems, and Bibliography. Chapter 1, Introduction, is devoted to the presentation and brief explanation of the basic notions (uncertainty, complexity and entanglement) that have been used in the research work about the quantum systems considered in this Thesis (multidimensional blackbody, harmonic and Coulombian systems). In addition, it contains the physico-mathematical motivation about the interest and relevance of the selected systems. Chapter 2, Methodology, includes (i) the mathematical techniques based on orthogonal polynomials that we have derived to determine analytically the informational measures of the quantum states of the harmonic and Coulombian systems including the extreme high-energy (Rydberg) and high-dimensional (pseudoclassical) cases, (ii) a number of inequality-based physico-mathematical methods and informational extremization techniques used to improve the uncertainty relations of the quantum multidimensional systems, and (iii) a brief description of the computational methods used in this work to calculate the wavefunctions and physical observables of the atomic and molecular systems. The results, as shown below, of the Thesis are given in sections 2.2, 2.3 and 2.4, as well as in the eleven sections of chapter 3. These sections correspond to the different methodological and quantum-physical issues considered in this work which gave rise to the scientific articles gathered in the paragraph Author’s Publications. Sections 2.2, 2.3 and 2.4 contain the various mathematical theorems and propositions we have found relative to the various entropic functionals of orthogonal hypergeometric polynomials, which are closely related to the weighted analytical norms of Lq type of these functions. The corresponding results allow, among other issues, for the determination of the asymptotics of various entropic integrals of these polynomials when their degree or the parameters of their associated weight function tend towards infinity. In Sections 3.1, 3.2 and 3.3 we use various physico-mathematical methods (the Lieb-Thirring and Daubechies-Thakkar inequalities and the informational-extremization method) to determine the combined spatial and spin dimensionality effects on the following two mathematical formulations of the Uncertainty Principle of quantum physics: the Heisenberg-like uncertainty relations and the Fisher-information-based uncertainty relation of the multidimensional fermionic systems. Moreover, we study the accuracy of the subsequent uncertainty relations for a large number of neutral atoms, singly-ionized ions and light and heavy molecular systems. In Section 3.4 we determine in an analytical way the generalized Heisenberg-like measures based on the radial expectation values of arbitrary order for the bound non-relativistic stationary states of the high-dimensional hydrogenic systems in the position and momentum spaces; this is done by use of a wide spectrum of mathematical techniques which include the parametric asymptotics of the generalized hypergeometric functions of p+1Fp type and the asymptotics of the Laguerre and Gegenbauer polynomials. In Section 3.5 we investigate the conditions that an informational quantity must fulfill to be considered as a true statistical complexity measure of a complex physical system. Moreover, the notion of monotonicity of a complexity measure of a probability distribution is discussed and applied to the most popular measures of complexity and some of their generalizations. In Section 3.6 we propose a novel monoparametric complexity measure of Fisher-Rényi type and we study their analytical properties; moreover, we calculate explicitly the values of this measure for the quantum states of the hydrogen atom. In Section 3.7 we determine analytically the basic quantities of entropy (Shannon, Rényi,...) and complexity (LMC, Fisher-Shannon and Crámer-Rao) types for the frequency distribution of the multidimensional blackbody radiation, and we discuss their potential relevance on the cosmic microwave background (CMB in short) which baths our universe. In Sections 3.8 y 3.9 we explicitly calculate the Rényi entropies of arbitrary order for the highly-excited (Rydberg) states of the multidimensional systems of harmonic and Coulomb types in position space in terms of the hyperquantum numbers and the potential strength; to this aim we use the strong asymptotics of the Laguerre polynomials when the polynomial degree becomes very high. In Sections 3.10 y 3.11 we study the quantum entanglement features of the bosonic and fermionic multidimensional systems of Harmonium and Spherium types, respectively; the resulting values are later compared with the corresponding ones in other realistic (helium) and model (Moshinsky, Hooke, Crandall,...) systems analyzed by other authors in recent years. Below is shown a scheme of the main contents of this Thesis: Author's publications 1 Introduction 2 Methodology 2.1 Computational approach and inequality-based physico-mathematical approaches 2.2 Laguerre entropic integrals: Strong asymptotics 2.3 Laguerre and Gegenbauer entropic integrals: Parameter asymptotics 2.4 Entropic functionals of general hypergeometric polynomials 3 Applications 3.1 Pauli effects in uncertainty relations 3.2 Heisenberg-like and Fisher-information-based uncertainty relations for multidimensional electronic systems 3.3 Semiclassical Heisenberg-like uncertainty relations of multidimensional fermionic systems 3.4 Heisenberg- and entropy-like uncertainty measures of high-dimensional hydrogenic systems 3.5 Monotonicity property of complexity measures 3.6 Monoparametric complexities for Coulomb systems 3.7 Entropy and complexity properties of the d-dimensional blackbody radiation 3.8 Renyi entropies of the highly-excited states of multidimensional harmonic oscillators 3.9 Renyi, Shannon and Tsallis entropies of Rydberg hydrogenic systems 3.10 Quantum entanglement of Harmonium-type bosonic and fermionic systems 3.11 Quantum entanglement of Spherium-like two-electron systems Conclusions and open problems Bibliography