量子纠缠及其在强关联系统中的应用
摘要
量子信息论是研究量子规则下的信息论。纠缠是量子信息论中的核心问题,没有纠缠,许多量子任务不能实现,在这个意义上,纠缠是一种重要的资源。在最近的几年里,量子纠缠问题得到了充分的重视。本文主要就纠缠及其在强关联体系的应用做了一些研究。
第一章对量子信息论进行了简要的回顾。分别讨论了纯态和混合态的概念和一些性质,特别是讨论了纠缠的判别和度量的问题,分别论述了形成纠缠度、提纯纠缠度和相对熵纠缠度。另外,我们还讨论了几种比较实用和有效的纠缠度量方法,主要有von Neumann熵、线性熵、负度(Negativity)等。文中还讨论了热力学熵,它在研究有限温度下的两体系统的纠缠度时非常有用。随后讨论了量子信道问题并简要叙述了量子信息理论的三个实际应用：密集编码(dense coding)、离态传输(teleportation)、量子密码(quantum cryptography)。
第二章讨论了强关联体系的纠缠。以XXZ模型为例讨论了一维自旋链的纠缠量度,随后着重讨论了在量子相变点附近纠缠度的性质,利用共形场论讨论了在临界点附近纠缠熵与中心荷及链长度之间的关系。并且还讨论了在重整化群变换下纠缠熵减少的问题。此外还讨论了Entanglement witness的一些性质。
第三章在李代数的基础上对二能级体系下的一种纠缠度concurrence向高维情况进行了推广,我们称之为concurrence vector:我们可用此矢量的模长来量度一个高维二体纯态的纠缠度。我们还讨论了它和von Neumann熵和线性熵之间的联系。另一方面,我们也尝试着向混合态的情况进行了一些推广。对于没有束缚纠缠的混合态,我们给了如下一个纠缠度量公式：其中λ1αβ=max{λ1αβ尹,i=1,...,N}.λ1αβ是矩阵(?)αβ(?)αβ*的本征值的正平方根,(?)我们同样可用矢量的模长来刻划混合态的纠缠度。文中我们主要就对SU(3)态和SO(3)态的concurrence vector进行了比较细的讨论。在最后我们引入了纠缠边缘的概念,举了几个具体的例子。
第四章讨论了一个自旋为1和自旋为(?)的自旋对在有简并情况下的量子纠缠的性质。我们分别考虑了有外加磁场和没有外加磁场两种情况。在没有外加磁场时,系统是高度简并的,我们引入了平均纠缠的概念来量度此时的系统基态的纠缠性质。
第五章讨论了玻色-爱因斯坦凝聚体的纠缠,主要考虑氢原子和锂冷原子气中的外层电子自旋和原子核自旋之间的纠缠问题,此时系统的哈密顿量可写为H=J(Ix·Sx+Iy·Sy+Iz·Sz)+CSz+DIz.我们分别对其在零温和有限温度下情况进行了讨论,给出了纠缠度和温度、外场之间的关系式。通过调节外界的磁场和温度,可以对体系的纠缠态进行操作。
第六章中我们研究了两体qutrit系统的纠缠性质。首先构造了一个SU(3)关联体系,并考虑了其和外场之间的相互作用：我们讨论了系统纠缠的性质,特别是纠缠度随一些参数的变化而改变的情况,并且讨论了当纠缠度取最大值时的物理意义,还研究了关联函数在临界点附近的行为。在临界点,关联函数也同样取得最大值。另一方面,我们还讨论了其他的qutrit系统,如三能级Lipkin-Meshkov-Glick模型和SO(3)关联系统,分别进行了讨论和比较。
Quantum entanglement plays a very important role in the quantum informa-tion theory. Many quantum processing cannot occur without entanglement. In a sense, quantum entanglement is a kind of resource. In this dissertation, the properties of entanglement and its application in condensed matter physics are studied.
In chapter 1, a brief review of the quantum information theory is given. We discuss the concepts and some properties of pure state and mixed state respec-tively. In particular, we discuss how to detect and measure the entanglement. We give a brief introduction to the entanglement of formation, entanglement of distillation and the relative entropy of entanglement. Then we discuss some kinds feasible measures, including von Neumann entropy, linear entropy, Neg-ativity and so on. The quantum channel is then studied. In the last section of the chapter, we discuss the some applications for quantum entanglement, including dense coding, teleportation and cryptography.
In chapter 2, we discuss the entanglement in strongly correlated systems. The pairwise entanglement of XXZ model is studied. Much attention is devoted to study the properties in the vicinity of quantum phase transition points. From the viewpoint of conformal field theory, we discuss the relationship between entanglement entropy and central charges, the length of rings. We then discuss the information loss along the renormalization group trajectory.
In chapter 3, basing on the Lie algebra, we extend the notion of concurrence introduced by Hill and Wootters to high dimension Hilbert space. We call this generalized quantity "concurrence vector". For pure state, we can use the norm of concurrence vector to measure the entanglement of the state. We also try to extend the concurrence vector to study mixed state. In the chapter, we calculate the entanglement of 3×3 bipartite system as an example. We introduce the concept of "concurrence edge" and discuss the existence condition.
In chapter 4, we study a composite bipartite system which contains spin-1 and spin-1/2. We discuss the ground-state entanglement when the system is degenerate. We introduced one notion "average entanglement" to measure the degenerate ground state.
In chapter 5, we study the entanglement in Bose-Einstein Condensate, espe-cially the Hydrogen atom and Lithium cold atom gases. We calculate the en-tanglement of the system at zero temperature and finite temperature. At very low temperature, one can manipulate the entanglement of the atom through adjusting the external fields.
In chapter 6, we construct a system which carries out SU(3) representation. The system is interacted with external fields. We give the entanglement as some parameters are varied. We discuss the physical significance when the entanglement approaches its most maximally entangled. We also discuss the three-level Lipkin-Meshkov-Glick Model and spin-1 correlated spin.
第一章对量子信息论进行了简要的回顾。分别讨论了纯态和混合态的概念和一些性质,特别是讨论了纠缠的判别和度量的问题,分别论述了形成纠缠度、提纯纠缠度和相对熵纠缠度。另外,我们还讨论了几种比较实用和有效的纠缠度量方法,主要有von Neumann熵、线性熵、负度(Negativity)等。文中还讨论了热力学熵,它在研究有限温度下的两体系统的纠缠度时非常有用。随后讨论了量子信道问题并简要叙述了量子信息理论的三个实际应用：密集编码(dense coding)、离态传输(teleportation)、量子密码(quantum cryptography)。
第二章讨论了强关联体系的纠缠。以XXZ模型为例讨论了一维自旋链的纠缠量度,随后着重讨论了在量子相变点附近纠缠度的性质,利用共形场论讨论了在临界点附近纠缠熵与中心荷及链长度之间的关系。并且还讨论了在重整化群变换下纠缠熵减少的问题。此外还讨论了Entanglement witness的一些性质。
第三章在李代数的基础上对二能级体系下的一种纠缠度concurrence向高维情况进行了推广,我们称之为concurrence vector:我们可用此矢量的模长来量度一个高维二体纯态的纠缠度。我们还讨论了它和von Neumann熵和线性熵之间的联系。另一方面,我们也尝试着向混合态的情况进行了一些推广。对于没有束缚纠缠的混合态,我们给了如下一个纠缠度量公式：其中λ1αβ=max{λ1αβ尹,i=1,...,N}.λ1αβ是矩阵(?)αβ(?)αβ*的本征值的正平方根,(?)我们同样可用矢量的模长来刻划混合态的纠缠度。文中我们主要就对SU(3)态和SO(3)态的concurrence vector进行了比较细的讨论。在最后我们引入了纠缠边缘的概念,举了几个具体的例子。
第四章讨论了一个自旋为1和自旋为(?)的自旋对在有简并情况下的量子纠缠的性质。我们分别考虑了有外加磁场和没有外加磁场两种情况。在没有外加磁场时,系统是高度简并的,我们引入了平均纠缠的概念来量度此时的系统基态的纠缠性质。
第五章讨论了玻色-爱因斯坦凝聚体的纠缠,主要考虑氢原子和锂冷原子气中的外层电子自旋和原子核自旋之间的纠缠问题,此时系统的哈密顿量可写为H=J(Ix·Sx+Iy·Sy+Iz·Sz)+CSz+DIz.我们分别对其在零温和有限温度下情况进行了讨论,给出了纠缠度和温度、外场之间的关系式。通过调节外界的磁场和温度,可以对体系的纠缠态进行操作。
第六章中我们研究了两体qutrit系统的纠缠性质。首先构造了一个SU(3)关联体系,并考虑了其和外场之间的相互作用：我们讨论了系统纠缠的性质,特别是纠缠度随一些参数的变化而改变的情况,并且讨论了当纠缠度取最大值时的物理意义,还研究了关联函数在临界点附近的行为。在临界点,关联函数也同样取得最大值。另一方面,我们还讨论了其他的qutrit系统,如三能级Lipkin-Meshkov-Glick模型和SO(3)关联系统,分别进行了讨论和比较。
Quantum entanglement plays a very important role in the quantum informa-tion theory. Many quantum processing cannot occur without entanglement. In a sense, quantum entanglement is a kind of resource. In this dissertation, the properties of entanglement and its application in condensed matter physics are studied.
In chapter 1, a brief review of the quantum information theory is given. We discuss the concepts and some properties of pure state and mixed state respec-tively. In particular, we discuss how to detect and measure the entanglement. We give a brief introduction to the entanglement of formation, entanglement of distillation and the relative entropy of entanglement. Then we discuss some kinds feasible measures, including von Neumann entropy, linear entropy, Neg-ativity and so on. The quantum channel is then studied. In the last section of the chapter, we discuss the some applications for quantum entanglement, including dense coding, teleportation and cryptography.
In chapter 2, we discuss the entanglement in strongly correlated systems. The pairwise entanglement of XXZ model is studied. Much attention is devoted to study the properties in the vicinity of quantum phase transition points. From the viewpoint of conformal field theory, we discuss the relationship between entanglement entropy and central charges, the length of rings. We then discuss the information loss along the renormalization group trajectory.
In chapter 3, basing on the Lie algebra, we extend the notion of concurrence introduced by Hill and Wootters to high dimension Hilbert space. We call this generalized quantity "concurrence vector". For pure state, we can use the norm of concurrence vector to measure the entanglement of the state. We also try to extend the concurrence vector to study mixed state. In the chapter, we calculate the entanglement of 3×3 bipartite system as an example. We introduce the concept of "concurrence edge" and discuss the existence condition.
In chapter 4, we study a composite bipartite system which contains spin-1 and spin-1/2. We discuss the ground-state entanglement when the system is degenerate. We introduced one notion "average entanglement" to measure the degenerate ground state.
In chapter 5, we study the entanglement in Bose-Einstein Condensate, espe-cially the Hydrogen atom and Lithium cold atom gases. We calculate the en-tanglement of the system at zero temperature and finite temperature. At very low temperature, one can manipulate the entanglement of the atom through adjusting the external fields.
In chapter 6, we construct a system which carries out SU(3) representation. The system is interacted with external fields. We give the entanglement as some parameters are varied. We discuss the physical significance when the entanglement approaches its most maximally entangled. We also discuss the three-level Lipkin-Meshkov-Glick Model and spin-1 correlated spin.
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