Multimode Gaussian State Entanglement

In recent years, Quantum Information Theory has been well developed and it will be widely applied in science and technology. The science of quantum information includes Quantum Theory , Information Science and Computer Science, while the quantum mechanics is applied into the electron and information technology.The essence of Quantum Information is entanglement. In this paper, we study the entanglement of multi-mode Gaussian state of continuous variable system. In recent papers, the determination of entanglement of a general two mode Gaussian state can just only depend on purity measurement, some studies have already extended to the case of quasi-symmetric multi-mode Gaussian state, which can reduce the problem to the equivalent two mode Gaussian state. In this paper, we make good use of these important results on the entanglement of quasi-symmetric multi-mode Gaussian state, which is quantified by Entanglement of Formation. Our purpose is to find the analytic description of the relation between the parameters that respectively belong to the quasi-symmetric multi-mode Gaussian state and the two mode Gaussian state.In the analysis, we find it is not unique to reduce a quasi-symmetric multi mode Gaussian state to a two mode Gaussian state. We also find that we neglect the eigenvalue of the least two mode Gaussian state, which means a two mode Gaussian state may correspond to many different multi-mode Gaussian states. In order to find how multi-mode Gaussian states reduce to two-mode Gaussian state, we take eight-mode Gaussian state for example and study the analytic description of relation between the parameters of the two kinds of states. By fixing some parameters of the multi-mode Gaussian state, we get the relation curve of the two kinds of states. However, the problem is how we choose the parameters and how they effect the relation, which needs further study.Furthermore, we study the entanglement of bipartite multi-mode Gaussian State and get some useful results. Because of time and competence, we can not analyse detailedly. However, with more and more people pay attention to Quantum Information, these problems will be solved one day.