Entanglement Criteria For States In Infinite-Dimensional Bipartite Composite Quantum Systems

This thesis deals with the entanglement problem in infinite-dimensional bipartite composite quantum systems. After recalling some concepts and facts concerning entangle-ment, we investigate some entanglement criteria, discuss the properties of entanglement witnesses in detail and provide two methods of constructing entanglement witnesses.1. Several equivalent conditions of a pure state to be separable are proposed. We show that a pure state is separable if and only if it is invariant under the partial Hermitian conjugate (PHC) operation. Moreover, a pure state is separable if and only if it is a PPT state. Also, some other equivalent conditions are given. Based on the PHC criterion, we present a new entanglement measure—PHC measure, which coincides with concur-rence—an important entanglement measure. Finally, we gernerlize the Bell inequalities into infinite-dimensional case. Consequently, we obtain another sufficient and necessary condition that a pure state to be separable.2. The realignment operation of states and the computable cross norm of states in infinite-dimensional bipartite systems are established and the RCCN criterion is obtained. We prove that, the trace norm of the realignment operator equals the computable cross norm of the state. Moreover, for the separable state, the trace norm of the realignment operator is not larger than 1. Particularly, a pure state is separable if and only if the trace norm of the realignment operator is 1. Furthermore, two inequalities hold for all separable states are proposed.3. We discuss the properties of the entanglement witnesses in detail and investigate two methods of constructing entanglement witnesses. Firstly, we regard the set of entan-glement witnesses as a poset with respect to the containing relation of the set of states which can be detected by the entanglement witness. Consequently, we discuss the condi-tion that two entanglement witnesses with the 'finer' relation and we obtain the condition that two or more entanglement witnesses can detect the same entangled states simultane-ously. As a application, we derive the condition that a given entanglement witness is or not optimal. A method of constructing entanglement witnesses via the Hilbert-Schmidt bases is presented. At last, we give another way of constructing entanglement witnesses through the separable state which is nearest to the given entangled state with repect to the Hilbert-Schmidt distance.4. The partial trace is characterized:Assume that dim HA(?) HB =+∞. A map L:S(HA(?)HB)→S(Ha) is the partial-trace operation if and only if Tr(PL(p)) Tr((P(?)1B)ρ) holds for allρ∈S(HA(?) HB) and all rank-1 projections P acting on HA. The'if' part is proposed by Blanchard et al [Phys. Lett. A 355,180-187]. However, the proof there is wrong and we give a correct proof and also prove that 'only if' part is also true.