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.
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.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
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.
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 classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2, where ∣χ1⟩=α∣0⟩+β∣1⟩, ∣χ2⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state σAB represent a classical state when n is odd while it represent a non-classical state when n is even. The purification of the state σAB is studied and found that the purification is possible if the spectral decomposition of the density matrices ρ1 and ρ2 represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity ⟨χ1∣χ2⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.
Classification and quantification of bipartite qubit, tripartite qubit, and bipartite qutrit entanglement are studied in detail based on the entanglement measure of multipartite pure states. The results show that different ways of pure state entanglement can be characterized by extremal values of the corresponding entanglement measure under the stochastic local operation and classical communication(SLOCC). Detailed analysis shows that there are three different kinds of entanglement under the SLOCC in both tripartite qubit and bipartite qutrit pure state systems. Comparison of the measure with the recently proposed entropy-product measure is made and briefly discussed.
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.