A great effort has been devoted to understand Quantum InformationTheory. This interest comes from the fact that this field has given us advanced methods to process and store information. Among these proposals we can cite the teleportation, the quantum cryptography, and the quantum computation. Interestingly, entanglement is the main resource that puts the quantum approach in a higher level than the classical counterpart. On the other hand, the theory of entanglement allows also a better understanding of physical phenomena already treated by the fields of condensed matter, statistical physics, and quantum optics among others. In this dissertation we aim to review the main topics concerning the Theory of Entanglement and propose new results on this theme as well. Firstly, we will discuss how the notion of entanglement raises naturally, how to determine whether a state is entangled, and how to quantify it. Moving on to the three qubit context,we will show a tomographic process able to determine the quantum stateof a system using only two detectors. It will be shown the connection between multipartite entanglement and the exceptions to our process. In the multipartite scenario we will obtain novel results in the formalism of the theory.Specifically, we study the question: are all maximally entangled statespure? While for bipartite systems almost all entanglement quantifiers agree in a positive answer, in a more general context this is not the case. Indeed, we will show a multipartite entanglement quantifier for which all maximally entangled states are pure. Finally, we address some questions on identical particle entanglement, i.e.: the entanglement coming just from the indistinguishabilityof quantum particles.
A great effort has been devoted to understand Quantum InformationTheory. This interest comes from the fact that this field has given us advanced methods to process and store information. Among these proposals we can cite the teleportation, the quantum cryptography, and the quantum computation. Interestingly, entanglement is the main resource that puts the quantum approach in a higher level than the classical counterpart. On the other hand, the theory of entanglement allows also a better understanding of physical phenomena already treated by the fields of condensed matter, statistical physics, and quantum optics among others. In this dissertation we aim to review the main topics concerning the Theory of Entanglement and propose new results on this theme as well. Firstly, we will discuss how the notion of entanglement raises naturally, how to determine whether a state is entangled, and how to quantify it. Moving on to the three qubit context,we will show a tomographic process able to determine the quantum stateof a system using only two detectors. It will be shown the connection between multipartite entanglement and the exceptions to our process. In the multipartite scenario we will obtain novel results in the formalism of the theory.Specifically, we study the question: are all maximally entangled statespure? While for bipartite systems almost all entanglement quantifiers agree in a positive answer, in a more general context this is not the case. Indeed, we will show a multipartite entanglement quantifier for which all maximally entangled states are pure. Finally, we address some questions on identical particle entanglement, i.e.: the entanglement coming just from the indistinguishabilityof quantum particles.
Quantum information science has grown from being a very small subfield in the 70s until being one of the most dynamic fields in physics, both in fundamentals and applications. In the theoretical section, perhaps the feature that has attracted most interest is the notion of entanglement, the ghostly relation between particles that dazzled Einstein and has provided fabulous challenges to build a coherent interpretation of quantum mechanics. While not completely solved, we have today learned enough to feel less uneasy with this fundamental problem, and the focus has shifted towards its potential powerful applications. Entanglement is now being studied from different perspectives as a resource for performing information processing tasks.\n\n\t\t\t\t With bipartite entanglement being largely understood nowadays, many questions remain unanswered in the multipartite case. The first part of this thesis deals with multipartite entanglement in different contexts. In the first chapters it is studied within the whole corresponding Hilbert space, and we investigate several entanglement measures searching for states that maximize them, including violations of Bell inequalities. Later, focus is shifted towards hamiltonians that have entangled ground states, and we investigate entanglement as a way to establish a distance between theories and we study frustration and methods to efficiently solve hamiltonians that exhibit it.\n\n\t\t\t\t In the practical section, the most promised upcoming technological advance is the advent of quantum computers. In the 90s some quantum algorithms improving the performance of all known classical algorithms for certain problems started to appear, while in the 2000s the first universal computers of few atoms began to be built, allowing implementation of those algorithms in small scales. The D-Wave machine already performs quantum annealing in thousands of qubits, although some controversy over the true quantumness of its internal workings surrounds it. Many countries in the planet are devoting large amounts of money to this field, with the recent European flagship and the involvement of the largest US technological companies giving reasons for optimism.\n\n\t\t\t\t The second part of this thesis deals with some aspects of quantum computation, starting with the creation of the field of cloud quantum computation with the appearance of the first computer available to the general public through internet, which we have used and analysed extensively. Also small incursions in quantum adiabatic computation and quantum thermodynamics are present in this second part.
In this work we studied the issue of quantification of entanglement in quantum systems performing a detailed analysis since its fundamental principles, i.e, the postulates of the theory of entanglement measures, and thus justifying the expected properties of "good quantifiers" of entanglement. We presented some examples of entanglement measures for distinguishable particles, and we approach finally the case where the particles are indistinguishable from each other. We saw that, in this case, the quantification of entanglement is a subject of great controversies with different proposals to their approach, each with its own singular interpretation. After discussing these approaches, analysing their main characteristics and foundations, we present our proposed method to quantify them based on the use of entanglement witnesses.
Since the 1980s, there has been a physical problem relating to the detection of multipartite entangled states that helps us to investigate fundamental point of view on interpretations of quantum mechanics. On the other hand, information theories based on the mathematical structure of quantum mechanics, for example, quantum cryptography and quantum computation, have been investigated by researchers whose number has been increasing day by day in the world. This field became to be called quantum information theories. A practical experimental realization of quantum information processing needs that we detect experimentally multipartite entangled states. The number of researchers for multipartite entangled states is increasing rapidly. Especially, Greenberger-Horne-Zeilinger (GHZ) states are useful for quantum information processing information processing using quantum mechanics, which improves the conventional information processing ability which had been based on classical theories. However, practical experimental realization of the theories and controlling quantum states have inevitable difficulties due to the interaction with environments. The requirement in maintaining multipartite entangled states in relation to the time that takes to make a practical experiment is also a problem. Many experimentalists are tackling this problem, and recently, several groups have reported that multipartite entangled states have been physically detected in their laboratories. The main difficulty is how to judge whether or not the observed quantum state is indeed a multipartite entangled state. For this purpose, it is necessary to construct theoretical models to analyze such experimental data. In this thesis, we apply a mathematical structure as to density operators described in a high-dimensional Hilbert space, and we employ mathematical theories to prove if the corresponding density operator represents a multipartite entangled state or not. The main aim for constructing such models is to analyze the raw experimental data. Moreover, this model helps experimentalists in the sense that it allows them to see how to confirm that observed states are indeed the desired state. With mathematical theories developed before this thesis, only limited applications for this purpose were allowed. Some theories cover two-dimensional Hilbert space only. It has not been even discussed how to analyze the fidelity to GHZ states of the observed state. This has led to some doubt of that experimental data might not always confirm three-partite entangled states. However, by using the method of this thesis, statistical check based on such experimental data of the existence of three particle-entangled states is possible under appropriate assumptions. This example symbolizes that our analytical knowledge has improved very much in the last few years. This thesis gives us an advanced mathematical model for analyzing experimental data.
The research field of quantum entanglement theory is comparatively new. While a basic understanding of the most simple systems in question (i.e. bipartite systems) has been established over the past few decades, multipartite entanglement still holds many unsolved questions and intriguing riddles. In particular, it is completely unclear how several concepts from the bipartite case can be generalised in a meaningful way to multipartite scenarios. In this work, the main issues of multipartite entanglement detection, characterisation and classification are discussed. The differences and similarities between the bipartite and the multipartite situation are reviewed, various possible generalisations are presented and results are obtained in several areas. The focus of this work particularly lies on a formalism - the so called HMGH framework, which has been developed and expanded for the past several years - which allows for construction of very specific separability criteria, capable of discriminating between different kinds of multipartite entanglement. By means of these criteria, the questions of partial separability, genuine multipartite entanglement and - ultimately - multipartite entanglement classification (which appear to contain the most striking differences to bipartite entanglement) are adressed and discussed. In order to illustrate the theoretical conclusions in these respects, several examples are given from different (and differently closely related) fields, showing the capabilities, strengths and weaknesses of the HMGH framework as well as giving insights into the current status of research in multipartite entanglement theory as a whole.
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.
The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.