This thesis presents some results about two important subjects of the quantum information theory: quantum entanglement and quantum algorithms. Regarding the entanglement, a relationship between negativity and minors principals of Hermitian matrices was stablished, allowing the analysis of the entanglement of two qubits states, three and four qubits pure states, using the minors principals. It was also proposed a new measure, using negativity, for calculating the entanglement of pure states of six qubits. Finally, with respect to quantum algorithms it was proposed a quantum search algorithm able to finding the minimum of a function by performing only one measurement. The algorithm is described and an example of its usage in the calculation of the period of a periodic function is presented.
This thesis presents some results about two important subjects of the quantum information theory: quantum entanglement and quantum algorithms. Regarding the entanglement, a relationship between negativity and minors principals of Hermitian matrices was stablished, allowing the analysis of the entanglement of two qubits states, three and four qubits pure states, using the minors principals. It was also proposed a new measure, using negativity, for calculating the entanglement of pure states of six qubits. Finally, with respect to quantum algorithms it was proposed a quantum search algorithm able to finding the minimum of a function by performing only one measurement. The algorithm is described and an example of its usage in the calculation of the period of a periodic function is presented.
Entanglement is the resource that most sharply distinguishes quantum information processing from its classical counterpart, and this thesis examines it from two complementary directions: how it can be detected and quantified, and how it, together with the broader machinery of quantum correlations, interference, and measurement, can be put to use. We first study two simple, measurable conditions that detect bipartite entanglement, and show that the extent to which either is violated yields a lower bound on the negativity of a state. The bound is established for two qubits, extended to systems of arbitrary finite dimension, and sharpened when partial knowledge of the Schmidt vectors is available. We then turn to discrete-outcome quantum sensor networks, in which one seeks only to determine which detectors, or which group or spatial pattern of detectors, has responded to a signal. Working in the weak-interaction regime, where the output states are nonorthogonal, we use minimum-error and unambiguous state discrimination to show that entangled initial states and collective measurements confer an advantage when detectors are grouped, and we develop a shift-operator formalism that gives closed-form success probabilities for the detection of shifted patterns in one- and two-dimensional arrays. Next we consider the problem of learning two unknown pure qubit states, together with their probabilities of occurrence, from a scrambled stream; we show that receiving the systems in identically prepared pairs supplies exactly the extra quantum information needed, since the resulting two-qubit state in the symmetric subspace has a unique decomposition into pure product states, which we recover by tomography. Finally, we show how interference and measurement can convert information encoded in the phases of quantum amplitudes into information encoded in their magnitudes, amplifying the amplitudes that carry useful information and suppressing those that do not, and we apply this technique to states produced by the quantum approximate optimization algorithm for the MaxCut and minimum-vertex-cover problems. Throughout, a single theme recurs: the correlations and superpositions permitted by quantum mechanics are at once something to be measured and something to be exploited.
This thesis is divided in three parts. The first one brings a brief review of the basic concepts about entanglement of quantum states with two and multiple qubits, including the entanglement monotones and the Groverian entanglement measure, which is calculated by a genetic algorithm. The second part is focused on the generation of a entanglement measures that identify only four qubit states with genuine entanglement. Some examples of the calculus of the entanglement of four qubit states, using the proposed measure, when these states are processed by quantum circuits, are presented. In the third part, the proposed measure is extended, to calculate the genuine entanglement of pure six qubits quantum states. Numerical simulations were realized aiming to show the variation, creation and annihilation of the entanglement of pure six qubits states.
Quantum entanglement is a key property of quantum information theory, that is at the heart of numerous promising applications in fields such as quantum cryptography, quantum computing or quantum sensing. In the past decades, the advent of such innovative technologies has reinforced the need for a better understanding of entanglement. The aim of this thesis is to contribute to this effort through the development of new tools targeting the characterization of several features of entanglement. Concerning the issue of entanglement detection, we present an optimization of the approach that exploits the concept of generalized concurrences to solve the separability problem for pure states. We then reformulate the separability question of mixed states into a matrix analysis problem, from which we obtain general separability criteria for multipartite states of ranks two and three. We also briefly discuss some properties of separable states. In particular, we characterize optimal separable decompositions of symmetric (i.e. permutation invariant) states of two and three qubits with maximal rank properties. Regarding the quantification of entanglement, we propose a function to quantify the entanglement of symmetric multiqubit states within classes of entangled states gathering states that are stochastically equivalent through local operations assisted with classical communication. This function establishes a link between the amount of entanglement of a symmetric state and the distribution of its Majorana points on the Bloch sphere. We finally investigate the robustness of entanglement with respect to particle loss and provide a full description of all multiqubit states that are fragile for the loss of one of their qubits. For symmetric states, the fragility for the loss of one qubit is shown to be related to a particular symmetry of the Majorana points.
This thesis studies four related projects: tuning entanglement for spin systems, de-coherence and dynamics of entanglement, entanglement as measure of electron correlation in chemical systems and quantum algorithms to search for global minima. Chapter 1 presents a general introduction and the motivation for these research areas. It is composed of a brief review of the history and the future of modern computers and the state-of-art areas in quantum computation and information. Chapter 2 will be dedicated to introducing the concept of entanglement, its importance, quantification and applications. Then in Chapter 3, the properties of a set of localized spins, which are coupled to an external magnetic field through exchange interaction, are investigated. It is shown that entanglement can be controlled and tuned by varying the anisotropy parameter in the Hamiltonian, and by introducing impurities into the systems. Furthermore, the scheme for amplifying the internal entanglement by external interaction is also studied. In Chapter 4, the applications of the entanglement concept for de-coherence and dynamics is considered. In particular, the dynamics of entanglement for one-dimensional spin systems, which are coupled through nearest neighbor exchange interaction and subject to an external time-dependent magnetic field, is investigated. Using the two-site density matrix, the time-dependent entanglement of formation between nearest neighbor qubits is calculated. It is found that the entanglement can be localized between nearest neighbor qubits for certain values of the external time-dependent magnetic field. In quantum chemistry calculations, the correlation energy is defined as the difference between the Hartree-Fock limit energy and the exact solution of the non-relativistic Schrödinger equation. In Chapter 5, the application of entanglement as an alternative measure of the electron correlation in quantum chemistry calculations is proposed. Entanglement is directly observable and it is one of the most striking properties of quantum mechanics. Finally, Chapter 6 presents the latest research results in developing quantum algorithms for searching the global minima. In this Chapter, the modified Grover’s quantum algorithm using modest numbers of quantum bits to find a global minimum for real problems is demonstrated.
Quantum Computers are provisioned as very high performance computers using quantum mechanical aspects for information processing. Quantum Information Theory and Quantum Computing topics are really popular topics in academia aiming the construction of theoretical background of Quantum Computers. The information processing units for Quantum Computers are defined as qubits but for some problems three level (trinary) systems may be applied as well. We call these three level systems as qutrits. The scope of this work is the analysis of well-defined entanglement measures Negativity and Relative Entropy of Entanglement (REE) [1-5] for qubit-qutrit quantum systems. In this manner, for randomly generated 1000 qubit-qutrit and 1000 two qubit states the mentioned measures are calculated and these values are compared. These comparisons are analyzed and for quantum state ordering problem, very interesting results are reported.
The focal point of this thesis is strongly based on quantum entanglement within Quantum Information Science. We provide an indepth analysis of entanglement quantification in both bipartite and multipartite quantum systems, with an emphasis on many-body entanglement. We examine the definitions of entanglement, introduce several entanglement measures, and examine the challenges in expanding these measures to multipartite scenarios. For bipartite systems, we detail entropy-based measures such as von Neumann and Rényi entropies, the entanglement spectrum and important algebraic metrics such as negativity and concurrence. We then elaborate on multipartite entanglement, introducing modern measures and detection techniques including geometric and global entanglement measurements, generalised multipartite concurrence, multipartite negativity, fidelity-based entanglement benchmarks, and entanglement witnesses. Some examples include Greenberger-Horne-Zeilinger (GHZ), and the W states to illustrate the distinction between biseparable and truly multi-partite-entangled states, as well as the SLOCC classification of families of entangled states. We also briefly introduce practical ways of measuring entanglement, such as the SWAP test, quantum state tomography for experimental estimation of entanglement and consequently randomised measurements. We apply these measures to specific quantum states (GHZ, W , graph states, and random states), analysing how entanglement evolves under operations and how different metrics capture distinct features of multi-qubit entanglement. Finally, we conclude with a summary of findings, a discussion of current limitations such as scalability of measures. The thesis is organised into five chapters: Introduction, Entanglement, Multipartite Entanglement Measures, Negativity vs. Three-Tangle, and Conclusion.