Analysis On Quantum Entanglement

Quantum information science has emerged as one of the most exciting scientific developments of the past twenty years. The promise of new technologies like secure quantum cryptography, quantum communication and quantum computers, capable of handling otherwise untractable problems, has excited not only researchers from many different fields like physicists, mathematicians and computer scientists, but also a large public audience. In the field of quantum information and computation science entanglement has been recognized as a key resource and ingredient. As a result, it is quite natural and important to discover the mathematical structures underlying its theoretical description. Such a description aims to provide answers to three questions about entanglement, namely its characterization, manipulation and quantification. This thesis is a contribution to the analysis on these subjects of quantum entanglement.The structure of this thesis is as follows:Chapter 1 is the exordium. I simply introduce the general knowledge about quantum information and computation and some quantum information processing tasks based on quantum entanglement.In Chapter 2 I review basic aspects of entanglement including its characterization, detection, manipulation and quantifying. I also review some new progresses about quantum entanglement.In Chapter 3 an attempt is made to propose a new multipartite entanglement measure. Based on multipartite versions of the quantum mutual information I construct new multipartite entanglement measures. One is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state's extension and takes the infimum over such extensions. I prove additivity of such measures for two versions of the multipartite mutual information. The second one is based on taking classical extensions. I generalize the latter scheme to construct measures of entanglement based on the mixed convex roof of a quantity, which in contrast to the standard convex roof method involves optimization over all decompositions of a den- sity matrix rather than just the decompositions into pure states. Furthermore, I present a computable lower bound on the multipartite squashed entanglement. I also derive some inequalities relating the squashed entanglement to the other entanglement measure.In Chapter 4, attention turns to the question of the relation of the entanglement of a given multipartite superposition state in terms of the entanglement of its subsystems. I derive the lower and upper bounds on the entanglement of a given multipartite superposition state in terms of the entanglement of the states being superposed. The first entanglement measure we use is the geometric measure and the second is the squashed entanglement. These bounds allow us to estimate the amount of the multipartite entanglement of superpositions.In Chapter 5 I investigate a inherently quantum mechanical phenomena, namely complementarity and entanglement, from an information-theoretic perspective. Using a generalization of the invariant information introduced by Brukner and Zeilinger to high-dimensional systems, I introduce a complementarity relation between the local and nonlocal information for d×d systems under the isolated environment, where d is prime or the power of prime. I also analyze the dynamics of the local information in the decoherence process.In Chapter 6 I consider the problem of general entanglement-assisted transformation for bipartite pure quantum states. I introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. I also design an efficient algorithm to detect whether a k×k general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.Summary and some open problems are given in Chapter 7.