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.
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.
It is crucial to know whether the quantum state generated in the experiment is entangled. In the literature, this topic was studied extensively and researchers proposed different approaches for the detection of a mixed bipartite entangled state in arbitrary dimension. Proceeding in this line of research, we propose three different criteria for the detection of mixed bipartite negative partial transpose (NPT) entangled state in arbitrary dimension. Our criteria are based on the method of structural physical approximation (SPA) of partial transposition (PT). We show that the proposed criteria for the detection of a NPT entangled state can be realized experimentally. Two of the proposed criteria are given in terms of the concurrence of the given state in arbitrary dimension so it is essential to find out the concurrence. Thus, we provide lower and upper bounds of concurrence of the quantum state under investigation in terms of average fidelity of two quantum states, and hence these bounds can be realized experimentally. Moreover, we show how to perform a SPA map on a qutrit-qubit system and then explicitly calculate the matrix elements of the density matrix describing the SPA-PT of the qutrit-qubit system. We then illustrate our criteria for the detection of entanglement by considering a class of qubit-qubit system and a class of qutrit-qubit system.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
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.
A complete chapter on quantum entanglement measures and detection of entanglement is given because there are a wide variety of entanglement measures for pure and mixed states which are used very often. The interesting feature of these routines is the fact that, these operations can be done between any random set of qubits not necessarily contiguous qubits. Key recipes to perform operations like partial tracing and partial transpose are disscussed in detail. After that, it was decided to devote several routines to calculate the entanglement measures like concurrence and its variants, von Neumann entropy, block entropy, Renyi entropy, Q measure, negativity and log negativity, entanglement spectrum and residual entanglement (three tangle). Last but not least, certain entanglement detection procedures like Positive Partial Transpose (the Peres-Horodecki criteria) and the reduction criteria are also explained via examples.
Entanglement detection is an important problem in quantum information theory because quantum entanglement is a key resource in quantum information processing. Realignment criterion is a powerful tool for the detection of entangled states in bipartite and multipartite quantum systems. It works well not only for negative-partial-transpose entangled states (NPTESs) but also for positive-partial-transpose entangled states (PPTESs). Since the matrix corresponding to the realignment map is indefinite, the experimental implementation of the map is an obscure task. In this work, first, we approximate the realignment map to a positive map using the method of structural physical approximation, and then we show that the structural physical approximation of the realignment map (SPA-R) is completely positive. Positivity of the constructed map is characterized using moments which can be physically measured. Next, we develop a separability criterion based on our SPA-R map in the form of an inequality and show that the developed criterion not only detect NPTESs but also PPTESs. Further, we show that for a special class of states called Schmidt-symmetric states, the SPA-R separability criteria reduce to the original form of the realignment criteria. We provide some examples to support the results obtained. Moreover, we analyze the error that may occur because of approximating the realignment map.
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.