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.
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.
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.
We present here an overview of our work concerning entanglement properties of composite quantum systems. The characterization of entanglement, i.e. the possibility to assert if a given quantum state is entangled with others and how much entangled it is, remains one of the most fundamental open questions in quantum information theory. We discuss our recent results related to the problem of separability and distillability for distinguishable particles, employing the tool of witness operators. Finally, we also state our results concerning quantum correlations for indistinguishable particles.
The study of entanglement in systems composed of identical particles raises interesting challenges with far-reaching implications in both, our fundamental understanding of the physics of composite quantum systems, and our capability of exploiting quantum indistinguishability as a resource in quantum information theory. Impressive theoretical and experimental advances have been made in the last decades that bring us closer to a deeper comprehension and to a better control of entanglement. Yet, when it involves composites of indistinguishable quantum systems, the very meaning of entanglement, and hence its characterization, still finds controversy and lacks a widely accepted definition. The aim of the present paper is to introduce, within an accessible and self-contained exposition, the basic ideas behind one of the approaches advanced towards the construction of a coherent definition of entanglement in systems of indistinguishable particles, with focus on fermionic systems. We also inquire whether the corresponding tools developed for studying entanglement in identical-fermion systems can be exploited when analysing correlations in distinguishable-party systems, in which the complete information of the individual parts is not available. Further, we open the discussion on the broader problem of constructing a suitable framework that accommodates entanglement in the presence of generalized statistics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.
We discuss particle entanglement in systems of indistinguishable bosons and fermions, in finite Hilbert spaces, with focus on operational measures of the entanglement of particles. We show how to use von Neumann entropy, negativity, and entanglement witnesses in these cases, proving interesting relations. We obtain analytic expressions to quantify the entanglement of particles in homogeneous D-dimensional Hamiltonians with certain symmetries.
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.
We present a unifying approach to the quantification of entanglement based on entanglement witnesses, which includes several already established entanglement measures such as the negativity, the concurrence, and the robustness of entanglement. We then introduce an infinite family of new entanglement quantifiers, having as its limits the best separable approximation measure and the generalized robustness. Gaussian states, states with symmetry, states constrained to super-selection rules, and states composed of indistinguishable particles are studied under the view of the witnessed entanglement. We derive new bounds to the fidelity of teleportation dmin, for the distillable entanglement ED and for the entanglement of formation. A particular measure, the PPT-generalized robustness, stands out due to its easy calculability and provides sharper bounds to dmin and ED than the negativity in most of the states. We illustrate our approach studying thermodynamical properties of entanglement in the Heisenberg XXX and dimerized models.