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Estudo do emaranhamento quantico com base na teoria da codificação cloassica

Wanessa Carla Gazzoni

Published Aug 15, 2008• 0 citations
Publisher

Abstract

In this thesis we present some contributions to a better understanding of quantum entanglement and its applications.With the purpose of obtaining a classification of the arbitrary pure quantum states as separable or entangled, a separability criterion is presented.This criterion is based on an homologic-geometric interpretation which allowed us to formalize some conclusions on the entanglement quantification of arbitrary pure states with three qubits.From this interpretation, it was possible to associate a description of the kets' content of an arbitrary pure state with the concepts of the classical coding theory.Based on this association, we propose a simplified form to determine a mathematical description of arbitrary quantum states satisfying the maximum global entanglement.From the concepts of coding theory we considered the states of maximum global entanglement with respect to its inherent error protection.In this context, we present a new class of states satisfying all the previous properties and which were not known in the open literature.

DOI
10.47749/t/unicamp.2008.432006

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In this thesis we present some contributions to a better understanding of quantum entanglement and its applications.With the purpose of obtaining a classification of the arbitrary pure quantum states as separable or entangled, a separability criterion is presented.This criterion is based on an homologic-geometric interpretation which allowed us to formalize some conclusions on the entanglement quantification of arbitrary pure states with three qubits.From this interpretation, it was possible to associate a description of the kets' content of an arbitrary pure state with the concepts of the classical coding theory.Based on this association, we propose a simplified form to determine a mathematical description of arbitrary quantum states satisfying the maximum global entanglement.From the concepts of coding theory we considered the states of maximum global entanglement with respect to its inherent error protection.In this context, we present a new class of states satisfying all the previous properties and which were not known in the open literature.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
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 is an attempt to enhance understanding of the following questions A- Given a multipartite quantum state (possibly mixed), how to find out whether it is entangled or separable? (Detection of entanglement.) B- Given an entangled state, how to decide how much entangled it is? (Measure of entanglement.), in the context of multipartite quantum states. We have explored two approaches. In the first approach, we assign a weighted graph with multipartite quantum state and address the question of separability in terms of these graphs and various operations involving them. In the second approach we use the so called Bloch representation of multipartite quantum states to establish new criteria for detection of multipartite entangled states. We further give a new measure for entanglement in NNN-qubit entangled pure state and formally extend it to cover NNN-qubit mixed states.
In this thesis, we discuss several aspects of the characterization of entanglement in multipartite quantum systems, including detection, classification and quantification of entanglement. First, we discuss triqubit pure entanglement and propose a special true tripartite entanglement, the mixed entanglement, besides the Greenberger-Horne-Zeilinger (GHZ) entanglement and the W entanglement. Then, based on quantitative complementarity relations, we draw entanglement Venn diagrams for triqubit pure states with different entanglements and introduce the total tangle Tau sup(T) to quantify total entanglement of triqubit pure states by defining the union I that is equivalent to the total tangle Tau sup(T) from the mathematical point of view. The generalizations of entanglement Venn diagrams and the union I to N-qubit pure states are also discussed. Finally, based on the ranks of reduced density matrices, we discuss the separability of multiparticle arbitrary -dimensional pure and mixed states, respectively.
A geometric understanding of entanglement is proposed based on local measurements. Taking recourse to the general structure of density matrices in the framework of Euclidean geometry, we illustrate our approach for the two qubit states, which is subsequently generalized for qudits and arbitrary number of entangled qubits. It is demonstrated that separable states satisfy certain geometric constraints that entangled states do not. This approach leads to the known geometric measure of entanglement for pure states. The crucial role of local measurement in quantification of entanglement is explicated.
Abstract This chapter gives a detailed overview of entanglement for pure and mixed states. We begin with the Schmidt-decomposition theorem, which allows us to associate the subsystem mixedness of pure state with entanglement. We discuss the purification of mixed quantum states, the convexity of the set of separable states, and entanglement as a form of correlation. As an example we examine the Bloch decomposition for two qubits. For the detection of entanglement the Peres-Horodecki criterion is introduced, before we analyse the relation between entanglement and non-locality via the CHSH-Operator Criterion and the family of Werner states. We further consider Tsirelson’s bound and hidden non-locality. We then discuss important mathematical tools for entanglement detection, including the positive-map theorem, the Størmer-Woronowicz theorem, and the reduction criterion. In this context we examine the family of isotropic states. Finally, the Weyl states provide an example for which we study the geometry of two-qubit entanglement