This bachelor thesis on entanglement in nano systems includes a short presen-tation of the history of quantum mechanics before entanglement takes the spot light. A definition of entanglement will be stated as well as a prescription to quantify it for a pure system. The knowledge will then be used to quantify the entanglement for the ground state of few particle systems containing 2, 3 and 4 particles. The model that will be used is chosen to be as simple as possible, but it can still give a feeling of how the entropy behaves in these small systems. The few-particle systems can be compared to atoms in a molecule or quantum dots in a nano system.
This bachelor thesis on entanglement in nano systems includes a short presen-tation of the history of quantum mechanics before entanglement takes the spot light. A definition of entanglement will be stated as well as a prescription to quantify it for a pure system. The knowledge will then be used to quantify the entanglement for the ground state of few particle systems containing 2, 3 and 4 particles. The model that will be used is chosen to be as simple as possible, but it can still give a feeling of how the entropy behaves in these small systems. The few-particle systems can be compared to atoms in a molecule or quantum dots in a nano system.
THESIS SUMMARYTEXT:This thesis is made of two parts. In the first one, the issue of entanglement in many body systems is addressed. The concept of entanglement and some of the recent progress on the study of entropy of entanglement in many body quantum systems are reviewed. Emphasis is placed on the scaling properties of entropy for one-dimensional models at quantum phase transitions. Then, we focus on the area-law scaling of the entanglement entropy. An explicit computation in arbitrary dimensions of the entanglement entropy of the ground state of a discretized scalar free field theory that shows the expected area law result is also presented. For this system, it is shown that area law scaling is a manifestation of a deeper reordering of the vacuum produced by majorization relations.To finish this first part, the issue of how simple can a quantum system be such as to give a highly entangled ground state is addressed. In particular, we propose a Hamiltonian of a XX model with a ground state whose entropy scales linearly with the size of the block. It provides a simple example of a one dimensional system of spin-1/2 particles with nearest neighbour interactions that violates area-law for the entanglement entropy.The second part of this thesis deals with the problem of simulating quantum mechanics for highly entangled systems. Two different approaches to this issue are considered. One consists of using ultra-cold atoms systems as quantum simulators. With this aim, some experimental techniques related to cold atoms that allow to simulate strongly correlated many body quantum systems are reviewed an explicit example of simulation is presented. In particular, we analyze how to achieve a Mott state of Laughlin wave functions in an optical lattice and study the consequences of considering anharmonic corrections to each single site potential expansion that were not taken into account until now.Finally, a different approach to simulate strongly correlated systems is considered: to use small quantum computers to simulate them. An explicit quantum algorithm that creates the Laughlin state for an arbitrary number of particles n in the case of falling fraction equal to one is presented. We further prove the optimality of the circuit using permutation theory arguments and we compute exactly how entanglement develops along the action of each gate. We also discuss its experimental feasibility decomposing the qudits and the gates in terms of qubits and two qubit-gates as well as the generalization to arbitrary falling fraction.KEYWORDS: Entanglement, Many body quantum systems, Quantum Information Condensed Matter, Cold atoms, Spin chains, Quantum simulator, Quantum computation.
The goal of this thesis is to investigate theoretically the generation and behaviour of multipartite entanglement for solid-state nanosystems, in particular electron spin quantum bits (so-called 'qubits') in quantum dots. A quantum dot is a tiny potential well where a single electron can be trapped. A quantum bit can be implemented in this system by applying a magnetic field, and thereby lifting the degeneracy of the spin states of the electron. These spins can then be used as single qubits, and engineering many of these quantum dots next to each other gives as a register of qubits. In this scheme, the so-called Loss-DiVincenzo quantum computer, the single spins can be rotated e.g. by applying a time dependent magnetic field, and two spins can interact through controlling the potential barrier between them. A qubit cannot only be in a superposition of the two computational states 0 and 1 at the same time, but an even stranger characteristic arises for multiple qubits: this phenomenon is called entanglement and refers to a strong correlation between two or more qubits, which can not be achieved within the framework of classical physics, and exponentially enlarges the possible states for a N-qubit system as compared to a classical N-bit system. In this thesis we devise algorithms how to generate multipartite entangled states in electron spin qubits in quantum dots. We compare which classes of entangled states can be generated efficiently in this system. Once the states are created, they decay due to a process called decoherence. We compare how entangled states can be generated and detected in a realistic experiment, and which classes of states are the most suitable. Furthermore, we compare which classes conserve the entanglement, and quantify the robustness of various classes of entangled states. In the last chapter, we devise a scheme of how to execute a simple quantum algorithm, the Deutsch-Jozsa algorithm, in a system containing another type of solid-state qubit, the so-called flux qubit.
Quantum entanglement is at the heart of many tasks in quantum information. Apart from simple cases (low dimensions, few particles, pure states), however, the mathematical structure of entanglement is not yet fully understood. This tutorial is an introduction to our present knowledge about how to decide whether a given state is separable or entangled, how to characterize entanglement via witness operators, how to classify entangled states according to their usefulness (i.e. distillability), and how to quantify entanglement with appropriate measures.
Quantum entanglement is at the heart of many tasks in quantum information. Apart from simple cases (low dimensions, few particles, pure states), however, the mathematical structure of entanglement is not yet fully understood. This tutorial is an introduction to our present knowledge about how to decide whether a given state is separable or entangled, how to characterize entanglement via witness operators, how to classify entangled states according to their usefulness (i.e., distillability), and how to quantify entanglement with appropriate measures.
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.