Some Reseach On Quantum Entanglement And Quantum Effect

Quantum entanglement is one of the most striking features in quantum mechanics. Quantum entanglement has become the most important resource in the rapid development of the field of quantum information, people make it to achieve the a lot of amazing applications, such as, quantum teleportation, quantum parallel computing, dense coding, quantum key distribution, entanglement swapping and remote state preparation and so on. We cannot do without the characteristics of quantum entanglem-ent to achieve these results. Although it is so widely used, it is a very difficult task that how to judge the entangled state. To solve the problem of the entanglement is equivalent to solve the divisibility problem. Therefore, characterizations of the separability of quantum states attract the extensive attention of numerous scholars.In this article, constructions of entanglement witnesses for some special entangled states are discussed; the relationships between entanglement witness and Bell inequality are disscussed. In addition, a kind of more exotic effect algebra is discussed and several distance characterizations of quantum effect algebras are also discussed. There are four chapters as follows.In Chapter1, we firstly recall the basic concepts of Hilbert space, tensor product and matrix singular value decompositions; secondly, the basic assumptions of quantum mechanics are listed; then the concepts of quantum states, density operator and reduced density operator are introduced; finally the concepts of quantum entangled states and quantum separated states are given.In Chapter2, the definition of entanglement witness is given. Then, with the help of the fact that entanglement witness makes a distinction entanglement and separability, we construct the entanglement witnesses for several special entangled states.In Chapter3, we firstly introduce the concepts of EPR paradox and Bell inequality. Then, we prove that an optimal entanglement witness can be expressed as partial transposition of the first system of the reduced density operator of arbitrary entangled state in2×2system. In addition, with the help of the Clauser-Horne-Shimony-Holtaa operator (called CHSH operator, for short), we discuss relationships between entanglement witness and Bell inequality in a special case.In Chapter4, we firstly introduce the concept of quantum effect algebra. Next, based on the trace distance of quantum states, quantum effect distance D(A,B) is defined. Then, we study some properties of D(A, B) analogy to study properties of the trace distance in quantum information theory. In addition, we define Dp(A, B) based on D(A,B)and obtain some properties. Finally, based on Dρ(A,B), we define distance dp(A,B), which is more convenient calculation, and prove related properties of dp(A,B).