Separability Criteria And Entanglement Measures Of Quantum States

With the development of technology, physics especially quantum physics is closely linked with computer science, information science and so on, then it gives rise to quantum information science. Quantum entanglement plays a key role in the vast majority of applications in quantum information science as a unique resource. In recent decades quantum entanglement is attracting growing attention, people paid a huge effort to better understand the phenomenon of entanglement and try to make it play an important role in real world. For example, entanglement enables communicate safety to a new level.Quantum computing renews the concept of computation, as the basis for quantum computer will greatly change computer science. There is no doubt that this remarkable study,quantum entanglement, will greatly promote the progress of technology and society.On the background of quantum information theory and quantum mechanics, this dissertation mainly focuses on the detection and quantification of entanglement. Given a quantum state, how to determine it is entangled or separable? This requires the separability criterion. At present, there are many separability criterion. But until now, there is not a perfect separability criterion, each of them has particular applicable occasions and limitations. Furthermore, if there exists some known entangled states, can we quantify the degree of entanglement in order to compare which contains more entanglement?Because the more entangled states, the more reliable in application. This requires the concept of entanglement measure to quantify states. Existing entanglement measures are defined based on different perspectives. Nevertheless, most of them are difficult to be calculated. In summary, this dissertation aims to introduce new separability criterion and entanglement measure, which are easy to understand and convenient to calculate. The main innovations are as follows:1.It proposes a separability criterion for arbitrary multipartite pure state which is based on the rank of(reduced) density matrix. It is proved that the rank of(reduced) density matrix of a multipartite state is closely related to entanglement.In fact it can be used to characterize entanglement. By investigating the ranks of 29)-1- 1 independent reduced density matrices, it can tell a 9)-qudit state |?is a partial separable, fully separable or a genuine 9)-qudit entangled state. This separability criterion is a necessary and sufficient condition for entanglement.2.It presents a separability criterion according to rank of coefficient matrix of a multipartite state. This separability criterion is also proved to be a necessary and sufficient condition for entanglement. And it is equivalent with the method based on the rank of reduced density matrix. Both of them are simple to understand and calculate. Finally it is able to help us find the completely separable form of a multipartite pure state3.This paper introduces a new entanglement measure on the basis of coefficient matrix. Consider the rows and columns of coefficient matrices as vectors, it only needs to compute the vector lengths and the angles between vectors of every bipartition coefficient matrix. We found that our measure coincides with concurrence in the case of two-qubit system. This is helpful to illustrate the effectiveness of using vector lengths and the angles between vectors to quantify entanglement.In addition, we prove that our entanglement measure satisfies the three necessary conditions which are required for any entanglement measure. Therefore it is simple and feasible, with a remarkable geometric meaning.4.We introduce a new good entanglement measure which is presented based on reduced density matrices of multi-qudit pure states. It is also proved to be a feasible measure which is able to distinguish the relatively high entanglement and the maximal entanglement. It is good at characterizing multipartite entanglement.All of above methods are applied for some explicit examples. Finally our methods show advantage compared with some other methods in certain cases.