This thesis poses a selection of recent research of the author in a common context. It starts with a selected review on research concerning the role entanglement might play at quantum phase transitions and introduces measures for entanglement used for this analysis. A selection of results from this research is given and proposed as evidence for the relevance of multipartite entanglement in this context. A constructive method for an SLOCC classification and quantification of multipartite qubit entanglement is outlined and results for convex roof extensions of the resulting measures are briefly discussed on a specific example. At the end, a transformation of antilinear expectation values into linear expectation values is presented which admits an expression of the aforementioned measures of genuine multipartite entanglement in terms of spin correlation function, hence making them experimentally accessible.
This thesis poses a selection of recent research of the author in a common context. It starts with a selected review on research concerning the role entanglement might play at quantum phase transitions and introduces measures for entanglement used for this analysis. A selection of results from this research is given and proposed as evidence for the relevance of multipartite entanglement in this context. A constructive method for an SLOCC classification and quantification of multipartite qubit entanglement is outlined and results for convex roof extensions of the resulting measures are briefly discussed on a specific example. At the end, a transformation of antilinear expectation values into linear expectation values is presented which admits an expression of the aforementioned measures of genuine multipartite entanglement in terms of spin correlation function, hence making them experimentally accessible.
The aspects of many particle systems, as far as their entanglement is concerned, is highlighted. To this end we briefly review the bipartite measures of entanglement and the entanglement of pairs both for systems of distinguishable and indistinguishable particles. The analysis of these quantities in macroscopic systems shows that close to quantum phase transitions, the entanglement of many particles typically dominates that of pairs. This leads to an analysis of a method to construct many-body entanglement measures. SL-invariant measures are a generalization to quantities as the concurrence, and can be obtained with a formalism containing two (actually three) orthogonal antilinear operators. The main drawback of this antilinear framework, namely to measure these quantities in the experiment, is resolved by a formula linking the antilinear formalism to an equivalent linear framework.
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.
Abstract We briefly review recent work on entanglement with particular emphasis on its behaviour in interacting spin systems. As already pointed out by several authors a general understanding of the properties of entanglement in condensed matter systems may lead to a deeper comprehension of both dynamics and thermodynamics of many‐body systems. After a brief resume of known means to identify and quantify entanglement, we take interacting spins on a network as an example. We briefly discuss critical properties of entanglement, if the system undergoes a quantum phase transition, and some aspects of its dynamical behaviour.
Quantum entanglement, a fundamental aspect of quantum mechanics, has captured significant attention in the era of quantum information science. In multipartite quantum systems, entanglement plays a crucial role in facilitating various quantum information processing tasks, such as quantum teleportation and dense coding. In this article, we review the theory of multipartite entanglement measures, with a particular focus on the genuine as well as the operational meaning of multipartite entanglement measures. By providing a thorough and valuable insight on this field, we hope that this review would inspire and guide researchers in their endeavors to further develop novel approaches for characterizing multipartite entanglement.
Chapter 14 transitions the focus to multipartite entanglement, a realm that broadens the discussion from bipartite systems to those involving multiple parties. This complex form of entanglement plays a crucial role in quantum computing, cryptography, and communication networks. The chapter introduces the foundational concepts of multipartite entanglement, including its characterization and the challenges associated with its classification. Significant attention is given to the classification of multipartite entangled states through SL-invariant polynomials, which provide tools for understanding the structure and properties of these states. Stochastic Local Operations and Classical Communication (SLOCC) are introduced as a means to classify entanglement. Furthermore, the chapter explores the entanglement of assistance and the monogamy of entanglement, two concepts that illustrate the limitations and potential for distributing entanglement among multiple parties. Through detailed explanations and examples in three and four qubits, this chapter offers insights into the intricate world of multipartite entanglement, revealing both its potential and challenges.
We review the theory of continuous-variable entanglement with special emphasis on foundational aspects, conceptual structures and mathematical methods. Much attention is devoted to the discussion of separability criteria and entanglement properties of Gaussian states, for their great practical relevance in applications to quantum optics and quantum information, as well as for the very clean framework that they allow for the study of the structure of nonlocal correlations. We give a self-contained introduction to phase-space and symplectic methods in the study of Gaussian states of infinite-dimensional bosonic systems. We review the most important results on the separability and distillability of Gaussian states and discuss the main properties of bipartite entanglement. These include the extremal entanglement, minimal and maximal, of two-mode mixed Gaussian states, the ordering of two-mode Gaussian states according to different measures of entanglement, the unitary (reversible) localization and the scaling of bipartite entanglement in multimode Gaussian states. We then discuss recent advances in the understanding of entanglement sharing in multimode Gaussian states, including the proof of the monogamy inequality of distributed entanglement for all Gaussian states. Multipartite entanglement of Gaussian states is reviewed by discussing its qualification by different classes of separability, and the main consequences of the monogamy inequality, such as the quantification of genuine tripartite entanglement in three-mode Gaussian states, the promiscuous nature of entanglement sharing in symmetric Gaussian states and the possible coexistence of unlimited bipartite and multipartite entanglement. We finally review recent advances and discuss possible perspectives on the qualification and quantification of entanglement in non-Gaussian states, a field of research that is to a large extent yet to be explored.