多体量子态的可分性和纠缠度量
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摘要
量子纠缠不但是量子力学区别于经典力学的重要特征之一，也是量子信息理论的重要组成部分。许多量子信息处理任务如量子隐形传态、量子稠密编码、量子密钥分发等，都需要量子纠缠，因此量子纠缠是一种重要的物理资源，而量化纠缠成为量子信息理论研究中的一个重要的课题。
近年来纠缠的量化虽然引起了广泛的关注，但是仅仅低维量子体系的纠缠量化得到了较好的解决，高维量子体系尤其是多体量子体系的纠缠量化仍旧是个悬而未解的问题。本论文主要研究了多体量子态的可分性以及纠缠度量。
论文共包括八章，其中我们的主要工作是第三章到第八章。第一章简单介绍了量子纠缠的研究的重要性，回顾了使量子纠缠得到广泛关注的EPR佯谬以及Schr(?)dinger猫态，介绍了几个典型的量子信息处理任务说明量子纠缠作为一个重要物理资源在量子信息理论中的应用，简单阐述了当前纠缠量化的研究现状，最后给出了本博士论文的主要研究内容和章节安排。
第二章给出了纠缠的基本概念，详细介绍了两体量子体系的可分性判据、纠缠度量的基本性质以及纠缠度量。
第三章详细介绍了两体量子态的concurrence矢量方法．以两体量子体系为例，将后面对多体量子体系所提出的concurrence下界的推导方法从数学上给以较为严格的阐述，为后面的应用奠定理论基础。尤其引入了Kronecker积近似技术，可以方便地得到concurrence的下界不同程度的近似。在最低阶近似下，可以给出concurrence解析的下界。
第四章利用concurrence作为纠缠度量来研究叠加态纠缠与被叠加态纠缠之间的关系。与Linden等人用约化密度矩阵的von Neumann熵作为纠缠度量相对比，利用concurrence作为纠缠度量可以给出叠加态纠缠的下界。
第五章对三体量子纯态引入了一种直观的描述一张量描述。通过这种直观的描述给出了三体二维量子态以及高维量子态的完全可分性判据。考虑到三体二维量子体系的真正三体纠缠度量，进一步利用这种直观的描述，给出了三体高维量子体系真正三体纠缠的存在性判据。
第六章简单介绍了几种多体纠缠度量。利用tilde内积研究三体2×2×n维量子体系的真正三体纠缠，给出了真正三体纠缠部分单调。通过将多体量子态进行不同的二分组，给出了多体自由纠缠度量，推广了多体量子体系的全局纠缠度量，证明了全局纠缠是一个纠缠单调。
第七章给出了三体2×2×n维量子纯态三体纠缠部分单调的一个简单的表达。利用这个简单表达，研究了具有三自旋相互作用的自旋1／2模型以及XXZ自旋链模型的基态纠缠与量子相变的关系。另外也提出了用一组EPR对来实现一个qudit的远态制备方案。此方案中如果把实空间中的qudit看作一个多体量子态，适当考虑这个量子态的可分性可以使得这个远态制备方案可能不受输入态空间维数的限制。
第八章根据不可区分光子的统计效应，利用线性光学器件来制备三个分别囚禁在不同光腔中的远距离原子的GHZ态、W态以及四原子空间中的一对qutrit的最大纠缠态。在Lamb-Dicke极限下，纠缠态的制备不需要光子的同时探测。
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.
近年来纠缠的量化虽然引起了广泛的关注，但是仅仅低维量子体系的纠缠量化得到了较好的解决，高维量子体系尤其是多体量子体系的纠缠量化仍旧是个悬而未解的问题。本论文主要研究了多体量子态的可分性以及纠缠度量。
论文共包括八章，其中我们的主要工作是第三章到第八章。第一章简单介绍了量子纠缠的研究的重要性，回顾了使量子纠缠得到广泛关注的EPR佯谬以及Schr(?)dinger猫态，介绍了几个典型的量子信息处理任务说明量子纠缠作为一个重要物理资源在量子信息理论中的应用，简单阐述了当前纠缠量化的研究现状，最后给出了本博士论文的主要研究内容和章节安排。
第二章给出了纠缠的基本概念，详细介绍了两体量子体系的可分性判据、纠缠度量的基本性质以及纠缠度量。
第三章详细介绍了两体量子态的concurrence矢量方法．以两体量子体系为例，将后面对多体量子体系所提出的concurrence下界的推导方法从数学上给以较为严格的阐述，为后面的应用奠定理论基础。尤其引入了Kronecker积近似技术，可以方便地得到concurrence的下界不同程度的近似。在最低阶近似下，可以给出concurrence解析的下界。
第四章利用concurrence作为纠缠度量来研究叠加态纠缠与被叠加态纠缠之间的关系。与Linden等人用约化密度矩阵的von Neumann熵作为纠缠度量相对比，利用concurrence作为纠缠度量可以给出叠加态纠缠的下界。
第五章对三体量子纯态引入了一种直观的描述一张量描述。通过这种直观的描述给出了三体二维量子态以及高维量子态的完全可分性判据。考虑到三体二维量子体系的真正三体纠缠度量，进一步利用这种直观的描述，给出了三体高维量子体系真正三体纠缠的存在性判据。
第六章简单介绍了几种多体纠缠度量。利用tilde内积研究三体2×2×n维量子体系的真正三体纠缠，给出了真正三体纠缠部分单调。通过将多体量子态进行不同的二分组，给出了多体自由纠缠度量，推广了多体量子体系的全局纠缠度量，证明了全局纠缠是一个纠缠单调。
第七章给出了三体2×2×n维量子纯态三体纠缠部分单调的一个简单的表达。利用这个简单表达，研究了具有三自旋相互作用的自旋1／2模型以及XXZ自旋链模型的基态纠缠与量子相变的关系。另外也提出了用一组EPR对来实现一个qudit的远态制备方案。此方案中如果把实空间中的qudit看作一个多体量子态，适当考虑这个量子态的可分性可以使得这个远态制备方案可能不受输入态空间维数的限制。
第八章根据不可区分光子的统计效应，利用线性光学器件来制备三个分别囚禁在不同光腔中的远距离原子的GHZ态、W态以及四原子空间中的一对qutrit的最大纠缠态。在Lamb-Dicke极限下，纠缠态的制备不需要光子的同时探测。
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.
引文
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