Entanglement Criteria For States In Infinite-Dimensional Multipartite Composite Ouantum Svstems

In quantum mechanics, a quantum system is associated with a separable complex Hilbert space H, and, a quantum state on H is a density operator which is positive and has trace1. Denote by S(H) the set of all states on H. Quantum information theory need to deal with multipartite composite quantum system. The underlying space H of a multipartite composite quantum system is a tensor product of underlying spaces Hi of its subsystems, that is H=H1(?)H2(?)…(?)Hn. In the case n=2the system is called a bipartite system. For the case that at least one of infinite dimension, if bipartite quantum state p∈S(H1(?)H2) can be expressed as the limit for convex combination of product states, called the state p is separable states, otherwise, p is said to be inseparable or entangled. For multipartite quantum state, we can define full separability, k-separability etc.It is very important but also difficult to determine whether or not a state in a com-posite system is separable. For the entanglement of bipartite quantum systems, many criteria have been proposed, however we know few entanglement criteria for multipar-tite case. The aim of the paper is to develop the formalism which could give general tools for testing the presence (or absence) of multipartite entanglement in given state of the system. In this paper, We discuss how to detecting multipartite quantum state full sep-arablity and partite separablity for infinite-dimensional multipartite compose system. For full separablity, we investigate the generalized partial transposition criterion of sep-arability, LPP elementary operator criterion, elementary witness criterion, etc. In addition, For partial separability, in the case of synchronous division we investigate partial separable entanglement witness criterion. This is a way of constructing entan-glement witness from LPP elementary operators. There are five chapters in this thesis, and the main results are as follows:1. A necessary condition are established for bi-and multipartite quantum state fully separably in infinite-dimension bi-and multipartite composite system, respectively, that is the generalized partial transposition criterion (or GPT criterion for short). We prove:let p be a state acting on infinite-dimensional complex Hilbert space HA(?)…(?) HN(HA(?)HB), if p is fully separable(separable), then‖pTy)‖Tr≤1, where Ty represent the generalized row or column transposition on chosen subsystems; in particular, if p is a pure state, then equal-sign hold. The criterion generalize well-known bi-or multipartite PPT criterion and bi-or multipartite realignment criterion for infinite-dimensional case. The last, a characterization of entanglement is given for infinite-dimensional multipartite quantum system, that is"Negativity of entanglement"; and some example are also presented to demonstrate how to apply multipartitealignment criterion to detect entanglement of tripartite states.2. LPP elementary operators criterion are established, then a necessary and suffi-cient necessary criterion of full separability for multipartite states are given. LPP ele-mentary operators criterion assert:a (n+1)-partite state p E S(H1(?)H2(?)…(?)Hn(?)n(?)K) is fully separable if and only if (Λ(?)Id)ρ>0for all (finite rank) LPP elementary op-erators A from B(H1(?)H2(?)…(?)Hn) into B(K). Furthermore, a characterization of LPP elementary operator is given. Some method of constructing LPP elementary oper-ators and LPP map by positive elementary operators and by an unextendible product basis(or UPB for short) are given, respectively. The last, an example is also presented to demonstrate how to apply the criterion to detect entanglement of tripartite states, as same as multipartite realignment criterion. 3. For multipartite composite quantum system, a multipartite state strong2-separable, strong k-separable and fully separable criterion are given, respectively. That is a sufficient and necessary condition criterion. These criterion can detect entanglem-rnt for given state by constructing an entanglement witness from finite rank positite elementary operator, finite rank LPP elementary operator and some finite rank entan-glement state, respectively. Another, we given two way to construction entanglement witness, one is by LPP elementary operator, another is by geometric methods. An example is also presented to demonstrate how to construction entanglement witnesses by geometric methods to detect entanglement of a new tripartite states.