Separability And Entanglement Measure Of Multipartite Quantum States

Quantum entanglement is not only one of the major characteristics that distinguishquantum from classical mechanics, but also an essential ingredient of quantum informationtheory. A lot of quantum information processing such as quantum teleportation, quantumdense coding, quantum key distribution and so on, can be achieved conditioned on quantumentanglement. Hence, quantum entanglement is an important physical resource and itsquantification becomes a vital subject of quantum information theory.In recent years, quantification of entanglement has attracted great attention. However,only the entanglement of low-dimensional quantum systems can be well quantified. Thequantification of entanglement of high-dimensional quantum systems, especially that ofmultipartite quantum systems, remains an open question.This dissertation is mainly focused on the separability and entanglement measure ofmultipartite quantum states. The dissertation includes eight chapters and our maincontributions are given in Chaps. 3 through 8. In Chap. 1, the importance of the investigationof quantum entanglement is first introduced. The EPR paradox and Schr(?)dinger cat throughwhich quantum entanglement has attracted much attention, are then reviewed. Several typicalquantum information protocols are also introduced in order to demonstrate that quantumentanglement is an important physical resource. The general situation of quantification ofentanglement is briefly described and the major research subjects and the organization of thedissertation are given at the end of this chapter.In Chap. 2, the concept of entanglement is introduced. The separability criteria ofbipartite quantum systems, the fundamental properties of entanglement measure andentanglement measure of bipartite quantum systems are described in detail.In Chap. 3, the concurrence vector for bipartite states is discussed in detail. Withbipartite systems as examples, the methods for deriving the lower bound of concurrence formultipartite quantum systems are rigorously given. These provide the theoretical backgroundfor the latter use. In particular, with Kronecker product approximation technique, the lowerbound of concurrence can be obtained up to various orders. To the lowest-order, an analyticlower bound of concurrence can be derived.In Chap. 4, concurrence is employed as an entanglement measure to study therelationship between the entanglement of the superposition state and that of the states being superposed. Compared with von Neumann entropy of reduced density matrix as anentanglement measure as employed by Linden et al., a lower bound of entanglement of thesuperposition state can be given with concurrence.In Chap. 5, an intuitionistic description—tensor description, is given to tripartitequantum pure states. On the basis of this description, full separability criteria are obtained fortripartite two-and higher-dimensional quantum systems. With the genuine tripartiteentanglement of tripartite quantum systems of qubits taken into consideration, the existencecriterion of genuine tripartite entanglement is also given for tripartite high-dimensionalquantum systems based on the intuitionistic description.In Chap. 6, several multipartite entanglement measures are introduced. By utilizing thetilde inner product, the genuine tripartite entanglement in 2×2×n-dimensional systems isstudied and the genuine tripartite entanglement semi-monotone is presented. Based on thedifferent bipartite grouping of a multipartite quantum state, free entanglement measure isobtained. Furthermore, the global entanglement is generalized and proved to be anentanglement monotone.In Chap. 7, a simple expression of the genuine tripartite entanglement semi-monotone for2×2×n-dimensional pure states is given. By making use of this simple expression, theconnection between the quantum phase transition and the entanglement of ground states of thespin-1/2XY model with three spin interactions and the XXZ model are considered. In addition, aprotocol is proposed in which a known qudit is remotely prepared onto a group of qubits byemploying a group of EPR pairs as quantum channels. In this protocol, if a qudit is consideredas a multipartite quantum state, it is possible that the protocol is not restricted by thedimension of input space with the separability of the multipartite state taken into account.In Chap. 8, based on the statistics of indistinguishable photons, schemes areproposed to use linear optical setups to prepare GHZ state and W state of three distant atomstrapped in different optical cavities. A scheme is also proposed to prepare a maximallyentangled state of two distant qutrits in the space spanned by four atoms. In the Lamb-Dickelimit, all the proposed schemes do not require the simultaneous detection of photons.