CHARACTERIZATION OF ENTANGLEMENT IN HIGHER DIMENSIONAL BIPARTITE AS WELL AS MULTIPARTITE QUANTUM SYSTEM AND ITS APPLICATION

Abstract

In recent years considerable progress has been made towards developing a general theory of quantum entanglement. In particular, the criteria to decide whether a given quantum state is entangled are of high theoretical and practical interest. This problem is additionally com plicated by the existence of bound entanglement, which are weak entangled states and hard to detect. In this thesis, we have worked on the characterization of bipartite and tripartite en tanglement. We have established a few separability criteria that successfully detect negative partial transpose entangled states (NPTES) as well as positive partial transpose entangled states (PPTES). In this thesis, we propose some theoretical ideas to realize these entangle ment detection criteria experimentally. We have taken an analytical approach to construct a witness operator. To achieve this, we first constructed a linear map using the combination of partial transposition and realignmen t operation. Then we find some conditions on the parameters of the map for which the map represents a positive but not completely positive map. We then construct an entanglement wit ness operator, which is based on the Choi matrix. Finally, we prove its efficiency by detecting several bipartite bound entangled states that were previously undetected by some well-known separability criteria and find that our witness operator detects bound entangled states that are undetected by these criteria. It is known that the witness operator is useful in the detection as well as the quantification of entangled states. This motivated us for the construction of a family of witness operators that can detect many mixed entangled states. We have shown the significance of our constructed witness operator by detecting many bound entangled states and then, we use the expectation value of the witness operator to estimate the lower bound of the concurrence of those bound entangled states. Detection of entanglement through partial knowledge of the quantum state is a challenge to implement efficiently. Here we propose a separability criterion for detecting bipartite entangle ment in arbitrary dimensional quantum states using partial moments of the realigned density matrix. Our approach enables the detection of both distillable and bound entangled states through a common framework. We illustrate the efficiency of our method through examples of states belonging to both the above categories, which are not detectable using other schemes xiii relying on partial state information. Realignment criteria is a powerful tool for the detection of entangled states in bipartite and multipartite quantum systems. Since the matrix corresponding to the realignment map is indef inite, the experimental implementation of the map is an obscure task. We have approximated the realignment map to a positive map using the method of structural physical approximation (SPA) and then we have shown that the SPA of the realignment map (SPA-R) is completely positive. The positivity of the constructed map is characterized using moments that can be physically measured. Next, we develop a separability criterion based on our SPA-R map in the form of an inequality and have shown that the developed criterion not only detects NPTES but also PPTES. We have addressed the problem of experimental realization of the realignment operation. We first show that the realignment matrix can be expressed in terms of the partial transposition operation along with a permutation matrix which can be implemented via the SWAP operator acting on the density matrix. We have shown that the first moment of the realignment matrix can be expressed as the expectation value of a SWAP operator which indicates the possibility of its measurement. We defined a new matrix realignment operation for three-qubit states and have devised an entanglement criterion that can detect three-qubit genuine entangled states. Chapter 1 is introductory in nature. Chapters 2 – 6 are based on the research work published in the form of research papers in reputed refereed journals. Finally, we conclude with future scope and references. Each chapter begins with a brief outline of the work carried out in that chapter.