In this Thesis we use the notions of information entropy, statistical complexity and\nquantum entanglement, together with the methods of Classical and Quantum Information\nand 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\nthe charge and matter distributions of the multidimensional quantum systems of bosonic\nand fermionic character. These facets, which determine both the uncertainty measures of\nthe systems and their physical and chemical properties, are manifested in the enormous\ndiversity of multidimensional geometries of the single-particle densities which characterize\nthe non-relativistic mechano-quantically allowed states of such systems, according to\nthe Density Functional Theories based on the Hohenberg-Kohn theorem and generalizations.\nThe aim of this Thesis is to contribute to the emergent informational representation\nof the quantum systems, which complements the standard representation\nbased on energetic concepts. The analytical determination of the entropy-,\ncomplexity- and entanglement-like measures and their associated uncertainty relations is a fundamental task (not yet solved except for the ground state and\na few low-lying excited states) even for the very small number of elementary quantummechanical\npotentials which are utilized to approach and model the self-consistent mean-\n eld of the physical systems (atoms, molecules,...), as this work shows. Basically, this\nis because the quantum manifestations of the multidimensional physical systems are\ngenerally inaccessible as the associated Schr odinger equation cannot be exactly solved\nexcept for a few cases where the quantum-mechanical potential has a known symmetry.
In this Thesis we use the notions of information entropy, statistical complexity and\nquantum entanglement, together with the methods of Classical and Quantum Information\nand 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\nthe charge and matter distributions of the multidimensional quantum systems of bosonic\nand fermionic character. These facets, which determine both the uncertainty measures of\nthe systems and their physical and chemical properties, are manifested in the enormous\ndiversity of multidimensional geometries of the single-particle densities which characterize\nthe non-relativistic mechano-quantically allowed states of such systems, according to\nthe Density Functional Theories based on the Hohenberg-Kohn theorem and generalizations.\nThe aim of this Thesis is to contribute to the emergent informational representation\nof the quantum systems, which complements the standard representation\nbased on energetic concepts. The analytical determination of the entropy-,\ncomplexity- and entanglement-like measures and their associated uncertainty relations is a fundamental task (not yet solved except for the ground state and\na few low-lying excited states) even for the very small number of elementary quantummechanical\npotentials which are utilized to approach and model the self-consistent mean-\n eld of the physical systems (atoms, molecules,...), as this work shows. Basically, this\nis because the quantum manifestations of the multidimensional physical systems are\ngenerally inaccessible as the associated Schr odinger equation cannot be exactly solved\nexcept for a few cases where the quantum-mechanical potential has a known symmetry.
Due to their great importance, both from the fundamental and from the practical points of view, it is imperative that the various facets of the concepts of information and entanglement are explored systematically in connection with diverse physical systems and processes. These concepts are at the core of the emerging field of the Physics of Information. In this Thesis I investigate some aspects of the dynamics of information in both classical and quantum mechanical systems and then move on to explore entanglement in fermion systems by searching for novel ways to classify and quantify entanglement in fermionic systems. In Chapter 1 a brief review of the different information and entropic measures as well as of the main evolution equations of classical dynamical and quantum mechanical systems is given. The conservation of information as a fundamental principle both at the classical and quantum levels, and the implications of Landauer's theorem are discussed in brief. An alternative and more intuitive proof of the no-broadcasting theorem is also provided. Chapter 2 is a background chapter on quantum entanglement, where the differences between the concept of entanglement in systems consisting of distinguishable subsystems and the corresponding concept in systems of identical fermions are emphasized. Different measures of entanglement and relevant techniques such as majorization, are introduced. To illustrate some of the concepts reviewed here I discuss the entanglement properties of an exactly soluble many-body model which was studied in paper (E) of the publication list corresponding to the present Thesis. An alternative approach to the characterization of quantum correlations, based on perturbations under local measurements, is also briefly reviewed. The use of uncertainty relations as entanglement indicators in composite systems having distinguishable subsystems is then examined in some detail. Chapter 3 is based on papers (A) and (B) of the list of publications. Extended Landauer-like principles are developed, based amongst others on the conservation of information of divergenceless dynamical systems. Conservation of information within the framework of general probabilistic theories, which include the classical and quantum mechanical probabilities as particular instances, is explored. Furthermore, Zurek's information transfer theorem and the no-deleting theorem are generalized. Chapter 4 is based on articles (C) and (D) mentioned in the publication list, and investigates several separability criteria for fermions. Criteria for the detection of entanglement are developed based either on the violation of appropriate uncertainty relations or on inequalities involving entropic measures. Chapter 5 introduces an approach for the characterization of quantum correlations (going beyond entanglement) in fermion systems based upon the state disturbances generated by the measurement of local observables. Chapter 6 summarizes the conclusions drawn in the previous chapters. The work leading up to this Thesis has resulted in five publications in peer reviewed science research journals.
The thesis aims to study the applicability of techniques of Quantum Information theory to complex systems, like molecules and biomolecules. Recently, concepts of Quantum Information theory have been applied to the study of phenomena involving molecules, for instance the interaction light-biomolecules. In particular, the entanglement has been proposed to quantify the non-classicality of interactions. The computation of molecular structures and interactions, on the other hand, has been studied for decades mainly to understand reactions? kinetic and energetic characteristics. It is not straightforward to apply these methods to the computation of quantum correlations. We have therefore explored the computability of such effects using Quantum Information techniques. We started with the study of the electronic correlations in the Helium atom, that is the simplest non trivial case, in order to evaluate the computational problems and to develop approximate methods suitable to more complex systems. We start recalling the definition and basic properties of entanglement, as it is considered in Quantum Information theory. We illustrate how it can be measured and the difficulties one encounters when identical particles are involved, then we concentrate on the case of Helium in particular. In order to study the entanglement in Helium, we treat separately the singlet and the triplet configurations. The method we use consists in computing the reduced density matrix and then the entropy. The possible choices of coordinates, basis functions and computation strategies are discussed, together with their impact on the algorithms. The reduced, single-electron von Neumann and linear entropy for several low-energy eigenstates of Helium are computed by means of a simple configuration-interaction variational method.
The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
Entanglement and circuit complexity are fundamental concepts in quantum information theory which turned out to have considerable applications in various contexts ranging from many-body systems and condensed matter theory to quantum gravity and black hole physics. In order to find connections between these aspects, an insightful approach consists in comparing the complexity with some entanglement measures of a given bipartition, as for instance the Von Neumann entanglement entropy. Since, for a system in a given quantum state, the bipartite entanglement can be determined once a spatial subsystem and its complement are chosen, this comparison has to be made considering the complexity of reduced density matrices. This motivates the development of techniques for determining the circuit complexity of mixed states. In this thesis, focussing on free bosonic systems, we study the circuit complexity of mixed Gaussian states by employing the Fisher information geometry for the covariance matrices. This approach allows computing the complexity between reduced density matrices (subsystem complexity), as well as the complexity between thermal states. It also leads to a precise prescription for computing the spectrum and the basis complexity and the temporal evolution of the subsystem complexity after global and local quantum quenches. The study of the subsystem complexity of a given bipartition and a complete characterisation of its entanglement requires the detailed knowledge of the corresponding reduced density matrix, or, equivalently, of the entanglement Hamiltonian. A part of this thesis is devoted to determining the entanglement Hamiltonians in free theories. At equilibrium, we consider gapless systems, with the aim of recovering the underlying CFT results, and gapped lattice models. The evolution of entanglement Hamiltonians in quantum chains after a global quench is also studied. In addition, other entanglement quantifiers, as the entanglement spectrum and the contour for the entanglement entropies, are discussed in some of these settings. Finally, motivated by theoretical and experimental advances, we also address the issue of how the entanglement splits into the different charge sectors of a system endowed with a U(1) symmetry. We focus in particular on massive theories, either continuous or on the lattice, and we try to understand whether the entanglement equipartition, which is known to characterise gapless systems, survives away from criticality.
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 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 leading term of the Rényi entropies of the D-dimensional 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.