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 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.
This thesis is an attempt to apply the tools and techniques developed by quantum information science to gain Insight into strongly-interacting many-body quantum systems. The principle concept we make use of to do this is quantum entanglement. The thesis is divided into two parts. Part I is about quantum entanglement. The second part applies the techniques developed to understand quantum entanglement to complex quantum systems. In Part I we explore the rules governing the manipulation of quantum information. More specifically, we take as our premise the idea that quantum entanglement is the quantity responsible for the difference between classical and quantum information. We discuss the basic principles governing entanglement manipulation, focusing on the problem of quantifying quantum entanglement for quantum states. We prove that the probability that a certain class of quantum algorithms succeeds is an entanglement monotone for the initial state of the register used by these algorithms. We provide an easily-computable formula for a good mixed-state bipartite entanglement measure. We then use this formula to investigate a possible law of entanglement, namely, entanglement cannot be shared arbitrarily. In Part II we focus on using the concepts and tools of quantum information science to study complex quantum systems. The principal tool we use to do this is quantum entanglement. We begin with a definition of what we mean by a complex quantum system. This is followed with a study of the quantum entanglement present in a variety of complex quantum systems. We argue that quantum entanglement constitutes a signature of complex quantum behaviour. We conclude this thesis with a discussion of a well-known technique to understand complex quantum systems, namely, the density matrix renormalization group. We show that the success of this method is due to the way it treats the quantum entanglement present in a system.
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.
Abstract In this work we have performed state-of-the-art configuration-interaction (CI) calculations to determine the linear and von Neumann entanglement entropies for the helium-like systems with varying nuclear charge Z in the range . The focus of the work resides on determining accurate entanglement values for 2-electron systems with the lowest computational cost through compact CI-wave functions. Our entanglement results for the helium atom fully agree with the results obtained with higher quality wave functions of the Kinoshita type (Dehesa [5]). We find that the correlation energy is linearly related to the entanglement measures associated with the linear and von Neumann entropies of the single-particle reduced density matrizes, which sheds new light on the physical implications of entanglement in helium-like systems. Moreover, we report CI-wave-function-based benchmark results for the entanglement values for all members of the helium isoelectronic series with an accuracy similar to that of Kinoshita-type wave functions. Finally, we give parametric expressions of the linear and von Neumann entanglement measures for two-electron systems as Z varies from 1 to 10.
This thesis addresses a number of problems within the emerging field of quantum information science. Quantum information science can be said to encompass the more-established disciplines of quantum computation and quantum information, as well as rather more recent attempts to apply concepts, tools and techniques from these disciplines to gain greater understanding of quantum systems in general. The role of entanglement — non-classical correlation — has been of particular interest to date. Part I contributes to this later goal. In particular, we establish a connection between the energy of a many-body quantum system and the idea of an entanglement witness from the theory of mixed-state entanglement. This connection allows mathematical results about entanglement witnesses to be translated into physical results about many-body quantum systems, specifically energy and temperature thresholds for entanglement. For the case of two qubits we are able to establish fairly detailed results about the behaviour of entanglement with temperature. We also study entanglement in systems of indistinguishable particles, where even the question of which quantum states should be regarded as entangled has been the subject of much controversy. We aim to clarify this issue by applying Wiseman and Vaccaro’s notion of entanglement of particles to a number of wellunderstood model systems. We discuss the advantages of the entanglement of particles approach compared with other methods in common use. Finally, we study the operational meaning of superselection rules in quantum physics, in particular the connection to the existence or not of an appropriate reference frame. We propose an experiment that aims to exhibit a coherent superposition of an atom and a molecule, apparently in violation of the commonly-accepted particle-number superselection rule. This result sheds light on the entanglement of particles approach to entanglement of indistinguishable particles. Part II returns to a fundamental question at the heart of quantum computation and quantum information, namely: how many quantum gates are required to perform a particular quantum computation? In other words, how efficiently can a quantum computer solve a particular computational problem? We establish a connection between this question and the field of Riemannian geometry. Intuitively, optimal quantum circuits correspond to “free-falling” along the shortest path between two points in a curved space. This opens up the possibility of using Riemannian geometry to study quantum computation, a possibility that was previously unknown. We provide explicit calculations of all the basic geometric quantities associated with the space, and give some preliminary results of applying geometric ideas to quantum computing. Finally, we explore more generally the connection between optimal control and quantum circuit complexity, of which the Riemannian metric described above can be viewed as a special case.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.
This thesis investigates questions related with the quantification, creation and application of entanglement. We first show how the notion of entanglement is modified by the presence of superselection rules. Next, we develop protocols for the optimal generation of squeezing and entanglement from noisy operations. The second part of the thesis applies quantum information methods to quantum many-body problems. We start by investigating the relation of Hamiltonian and correlations for lattices of harmonic oscillators. Next, we turn towards the description of many-body systems in terms of Matrix Product States (MPS) and Projected Entangled Pair States (PEPS). We clarify the relation between entropy scaling laws and the approximability by MPS, and determine the computational complexity of creating and simulating PEPS. Finally, we define Gaussian MPS as a generalization of MPS to bosonic systems.