Information and Entanglement Measures in Quantum Systems With Applications to Atomic Physics

This thesis is a multidisciplinary contribution to the information theory of single-particle Coulomb systems in their relativistic and not relativistic description, to the theory of special functions of mathematical physics with the proposal and analysis of a new set of measures of spreading for orthogonal polynomials, to quantum computation and learning devices and to the analysis of entanglement in systems of identical fermions, in this field we propose a separability criteria for pure states of N identical fermions and the entanglement of two-electron atoms is studied, a new separability criteria for continuous variable systems is also analyzed. The notions of information, complexity and entanglement play a central role.

💡 Research Summary

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This dissertation presents a multidisciplinary study that intertwines information theory, complexity measures, special‑function analysis, quantum learning, and quantum entanglement, with a focus on single‑particle Coulomb systems and their extensions. The work is organized into seven self‑contained chapters, each of which can be read independently and corresponds to one or two peer‑reviewed publications.

Chapter 1 – Information‑theoretic measures for non‑relativistic hydrogenic atoms
The author investigates the internal disorder of hydrogenic orbitals by analysing the position‑space density ρ(r) and the momentum‑space density γ(p). Beyond the usual variance and disequilibrium, three composite information‑theoretic quantities are evaluated: the Fisher‑Shannon product, the Cramér‑Rao complexity and the LMC (Lopez‑Ruiz‑Mancini‑Calbet) shape complexity. A systematic study of their dependence on the nuclear charge Z and the quantum numbers (n, l, m) shows that all three composite measures are Z‑independent, while the Fisher‑Shannon complexity grows quadratically with the principal quantum number n. Exact analytical expressions for the LMC shape complexity are derived, and sharp upper bounds for the Fisher‑Shannon product are obtained, improving on earlier results.

Chapter 2 – D‑dimensional hydrogenic systems
The analysis is generalized to D‑dimensional Coulombic atoms. The author expresses the LMC shape complexity of arbitrary stationary states in terms of entropic functionals of Laguerre and Gegenbauer (ultraspherical) polynomials. Special attention is given to ground and circular states, for which closed‑form expressions are obtained. The dimensional dependence of the complexity is discussed, and the corresponding uncertainty products are evaluated. In the limits of large D (dimensional scaling) and large principal quantum number (Rydberg limit) the author provides explicit asymptotic formulas, revealing how complexity interpolates between classical and quantum regimes.

Chapter 3 – Relativistic effects at the Klein‑Gordon level
The thesis extends the previous non‑relativistic treatment to spin‑zero particles described by the Klein‑Gordon equation in a Coulomb field. Both single‑parameter measures (variance, Shannon entropy, Fisher information) and composite measures (Fisher‑Shannon and LMC complexities) are computed for pionic atoms. The results demonstrate that relativistic charge compression is more pronounced for low‑lying states and for larger nuclear charge Z. While the variance and Shannon entropy increase with Z, the Fisher information shows the opposite trend. Importantly, after accounting for Lorentz invariance, the Fisher‑Shannon and LMC complexities increase with Z, contrary to the non‑relativistic case, highlighting a genuine relativistic signature in information‑theoretic descriptors.

Chapter 4 – Spreading lengths of orthogonal polynomials
A new family of “spreading lengths” is introduced for classical orthogonal polynomials, namely Hermite and Laguerre families. Four distinct lengths are defined: the standard deviation (ordinary moment), the Rényi length of order q, the Shannon length, and the Fisher length. The Rényi lengths are obtained analytically using multivariate Bell polynomials (for Hermite) and the linearization technique of Srivastava‑Nikkunen (for Laguerre). For the Shannon length, which involves a logarithmic functional, only asymptotic behaviours and rigorous upper bounds are derived. The Fisher length admits a closed expression. These spreading measures quantify the “width” of the associated Rakhmanov probability density on the orthogonality interval and provide a novel toolbox for assessing the localisation properties of wavefunctions expressed in polynomial bases.

Chapter 5 – Quantum learning automaton
A quantum learning model is proposed in which a single‑qubit device can implement any unitary operation. The learning task consists of teaching the device to perform the k‑th root of the classical NOT gate (k = 2^m). Analytical results show that the quantum learning time is independent of k, whereas a classical learning algorithm would require a time scaling as k^2. This demonstrates a quadratic speed‑up for the quantum learner and illustrates how quantum systems can efficiently acquire classical logical operations.

Chapter 6 – Overview of quantum entanglement for fermionic systems
The author reviews the state‑of‑the‑art in entanglement theory, emphasizing the distinction between systems with distinguishable subsystems and those composed of identical fermions. Existing separability criteria and entanglement measures are discussed, and the conceptual difficulties of defining entanglement for indistinguishable particles are highlighted.

Chapter 7 – Entanglement of identical fermions and two‑electron atoms
Building on the previous review, new separability criteria for pure states of N identical fermions are derived from the linear entropy and von Neumann entropy of the one‑particle reduced density matrix. These criteria are considerably simpler than those previously reported. Inequalities between the two entropies lead to natural entanglement measures for N‑fermion pure states, and connections with Hartree‑Fock theory are elucidated. The measures are applied to exactly solvable two‑electron models (the Crandall and Hooke atoms) and to realistic helium‑like atoms using high‑precision Kinoshita‑type wavefunctions. In all cases, entanglement grows with increasing energy, with stronger confining potentials, and with larger nuclear charge. Finally, a continuous‑variable separability criterion introduced by Walborn et al. (2009) is examined numerically for both pure and mixed states, confirming its effectiveness in detecting entanglement in continuous‑variable systems.

Overall impact
The dissertation establishes a coherent framework where information‑theoretic complexity and quantum entanglement are used as complementary lenses to probe atomic and molecular systems, high‑dimensional quantum models, relativistic effects, and quantum learning processes. The newly introduced spreading lengths for orthogonal polynomials, the refined bounds for Fisher‑Shannon complexity, and the simplified separability criteria for identical fermions constitute methodological advances that can be directly employed in quantum chemistry, quantum information processing, and the study of relativistic quantum systems. The work therefore bridges fundamental theoretical physics with practical tools for emerging quantum technologies.