Quantum Separability And Quantum Entanglement

Entanglement is the characteristic trait of quantum mechanics. It is responsiblefor many quantum tasks such as quantum teleportation, dense coding, swapping anderror correction. Quantum entangled states are the most important resources in quan-tum information processing. Although the nonclassical nature of entanglement has beenrecognized for many years, and considerable eforts have been taken to understand andcharacterize its properties recently, the physical character and mathematical structureof entangled states still have not been well understood. For example, how can we knowa given quantum state is entangled or not and how to qualify the entanglement of anarbitrary quantum state? How can we accomplish quantum information tasks perfectlyand efciently? Many important problems in the theory of quantum entanglement remainopen.This thesis contains five chapters. We give an introduction to quantum computationand quantum information in the first chapter, which mainly includes the history and someconcepts related to quantum information. Then we introduce some basic definitions andconcepts in linear algebra, quantum mechanics and the theory of quantum entanglement.Next, we mainly concern quantum entanglement that are the most important resource inphysical processing, and its application. The fourth chapter is devoted to the study of sep-arability of quantum states. We introduce several important separability criteria derivedin recent years: the Schmidt decomposition, positive partial transposition, realignmentcriteria, reduced density matrix, matrix positive map and entanglement witness.Concurrence, a well defined quantum entanglement measure, is studied in chapterfive. We study the concurrence of arbitrary dimensional bipartite quantum systems. Anexplicit analytical lower bound of concurrence is obtained, which detects entanglementfor some quantum states better than some well-known separability criteria, and improvesthe lower bounds such as from the PPT (positive partial transposition), realignmentcriteria and Breuer’s entanglement witness.