一维自旋链系统中量子纠缠的理论研究
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摘要
近年来的量子力学新进展为信息科学的发展提供了新的原理和方法,诞生了量子力学与信息科学相结合的新兴交叉学科一量子信息学,并已成为人们研究的热点领域。同时量子纠缠态作为量子力学最显著的奇妙特性之一,被认为是实现量子信息的基本资源,已被广泛的应用于量子隐形传态、稠密编码,量子密钥分配以及量子计算当中。因此对量子纠缠的深入研究,无论是对量子信息的基本理论还是未来的潜在应用都是非常有意义的。另一方面,真实的物理系统不可避免的与环境发生相互作用,结果导致系统的退相干及纠缠的退化,从而给纠缠的物理实现带来困难,所以研究量子退相干对纠缠演化的影响也是非常必要的。在本文中,基于自旋链系统中量子态的纠缠特性,我们理论上研究了一维自旋链系统中的量子纠缠及其演化。本论文分为五章,其中第二章至第五章为本人的工作,内容具体安排如下：
第一章首先介绍了量子信息学的历史背景及基本理论,然后对量子纠缠的定义及应用、量子退相干和一维自旋链模型进行了简单介绍。
第二章我们首先研究了(1／2,1)混合自旋系统中的热纠缠。与自旋1／2系统相比较,我们发现(1／2,1)混合自旋系统在热纠缠的产生方面更具有优越性。接着我们讨论了不同类型的海森堡模型中的热纠缠以及以热纠缠态为量子信道的隐形传态,给出了不同海森堡模型本身的性质和外部条件对热纠缠以及隐形传态保真度的影响。我们发现通过控制不同海森堡模型的各向异性参数,外加磁场,Dzyaloshinsky-Moriya (DM)相互作用等,可以提高热纠缠的值和优化隐形传态的效果。
第三章我们主要研究了纠缠sudden death现象。对于Tavis - Cummings模型中的两原子间的纠缠演化,我们发现初态为混合态中的双激发态是产生纠缠sudden death现象的主要原因,且纠缠sudden death的出现与选择的初态的形式有关。除此之外,我们还研究了四比特海森堡XX自旋链中的成对纠缠演化,结果表明周期性边界条件和外加非均匀磁场能有效地抑制纠缠sudden death对纠缠演化的影响。
第四章中我们研究了内在退相干下,不同海森堡模型中的纠缠演化及隐形传态。我们发现内在退相干对纠缠演化及隐形传态保真度的影响取决于选择的初态。通过控制自旋链本身及外界条件,如外加磁场,包括均匀磁场和非均匀磁场,以及DM相互作用,在纠缠演化过程中,可以产生和增强不随时间变化的稳定纠缠态,从而在很大程度上抑制内在退相干对纠缠演化及隐形传态的破坏影响。
第五章我们首先研究了热力学极限下,与处于有限温度的自旋环境相耦合的三比特自旋系统的纠缠演化和退相干。结果表明,成对纠缠演化与描述系统和环境的外界参数有关。通过调节两比特间的耦合常数和外磁场,以及系统和环境的耦合常数,可以有效地控制纠缠sudden death对纠缠演化的影响。此外,我们还讨论了有限温度下,与含DM相互作用的XY自旋链耦合的中心二体自旋系统的纠缠演化。我们发现DM相互作用不会影响XY自旋链由横场导致的量子相变的临界行为,不会引入新的临界点。其次,在纠缠演化过程中,DM相互作用可以减弱和增强纠缠,这取决于其本身的取值和横场强度以及自旋链环境的温度。
论文的最后,给出了全文的总结和展望。
The recent development in quantum mechanics provides new methods and principles for information science. As one of the most important applications, quantum information is a new combination of quantum mechanics and information science and has developed into a popular research field. Meanwhile, quantum entanglement is one of the most striking features of quantum mechanics and is regarded as a fundamental resource for quantum information processing. It has been widely used in quantum teleportation, superdense coding, quantum cryptographic key distribution, quantum computation and so on. So a further study on entanglement is very important for basic quantum information theory and future potential applications. On the other hand, a real quantum system will unavoidably interacts with the surrounding environment and thus leads to decoherence and disentanglement. This is a fundamental obstacle to the realization of quantum entanglement. Therefore, taking into account the effect of decoherence on entanglement dynamics is indispensable. In this thesis, based on the character of quantum state of spin chain system, the quantum entanglement and its evolution in one-dimensional spin chain system are investigated theoretically. The thesis is dived into five chapters, with our work included in chapters 2 through 5.
In chapter 1, the background and basic theory of quantum information are introduced, as well as the quantum entangled state and its application are briefly described, and then the concept of quantum decoherence and one-dimensional spin chain are given at the end of this chapter.
In chapter 2, the thermal entanglement in the (1/2,1) mixed-spin Heisenberg model is investigated firstly. Through comparison with the spin-half system, the (1/2,1) mixed-spin is superior in generation of thermal entanglement. And then the thermal entanglement in different Heisenberg models and quantum teleportation in the thermally entangled channel are studied. Here the effects of the intrinsic properties and external conditions of the Heisenberg model on the thermal entanglement and the teleportation fidelity are concerned. It is shown that the thermal entanglement and the quality of teleportation can be enhanced by adjusting the values of the anisotropy parameter, external magnetic field and Dzyaloshinsky-Moriya (DM) interaction.
In chapter 3, the phenomenon of entanglement sudden death is studied. As to the entanglement evolution between two Tavis-Cummings atoms, the results show that the initial portion of the double excited state in the initial states is responsible for the sudden death of entanglement, and the degree of this effect also depends on the form of the initial state。In addition, the dynamic evolution of pairwise entanglement in a four-qubit Heisenberg XX spin chain is also studied. It is shown that the entanglement sudden death effect can be weakened when the periodic boundary condition and the magnetic impurity are introduced.
In chapter 4, the entanglement evolution and teleportation in the different Heisenberg spin chains with intrinsic decoherence taken into account are investigated. The results show that the effects of intrinsic decoherence on the entanglement evolution and teleportation fidelity rely strongly on the initial state. Controlling the uniform or inhomogeneous magnetic fields and DM interaction can not only generate stationary entanglement, but also enhance the value of stationary entanglement. So the destructive effect of intrinsic decoherence on the entanglement evolution and teleportation can be moderated in this way.
In chapter 5, the entanglement dynamics and decoherence of a three-qubit system under a quantum spin environment at a finite temperature in the thermodynamical limit are studied firstly. It is shown that the evolution of pairwise entanglement depends on the parameters related to the system and the spin environment. In addition, an undesirable entanglement sudden death occurs in the process of entanglement evolution, and this effect can be controlled by the coupling constant between two qubits, external magnetic field, and the interaction between the system and the environment. In addition, the entanglement dynamics between two central spins coupled to an XY spin chain at finite temperature with the DM interaction is also investigated. By studying the entanglement decay of the central spins, the results suggest that the DM interaction does not affect the behavior of quantum phase transition induced by the external magnetic field and does not induce new critical regions in the XY model. Moreover, the DM interaction can efficiently enhance or suppress the entanglement between the central spins which depends on the values of the DM interaction, the magnetic intensity and the temperature of the environmental spin chain.
Finally, the results are summarized and suggestions for future research work are given.
第一章首先介绍了量子信息学的历史背景及基本理论,然后对量子纠缠的定义及应用、量子退相干和一维自旋链模型进行了简单介绍。
第二章我们首先研究了(1／2,1)混合自旋系统中的热纠缠。与自旋1／2系统相比较,我们发现(1／2,1)混合自旋系统在热纠缠的产生方面更具有优越性。接着我们讨论了不同类型的海森堡模型中的热纠缠以及以热纠缠态为量子信道的隐形传态,给出了不同海森堡模型本身的性质和外部条件对热纠缠以及隐形传态保真度的影响。我们发现通过控制不同海森堡模型的各向异性参数,外加磁场,Dzyaloshinsky-Moriya (DM)相互作用等,可以提高热纠缠的值和优化隐形传态的效果。
第三章我们主要研究了纠缠sudden death现象。对于Tavis - Cummings模型中的两原子间的纠缠演化,我们发现初态为混合态中的双激发态是产生纠缠sudden death现象的主要原因,且纠缠sudden death的出现与选择的初态的形式有关。除此之外,我们还研究了四比特海森堡XX自旋链中的成对纠缠演化,结果表明周期性边界条件和外加非均匀磁场能有效地抑制纠缠sudden death对纠缠演化的影响。
第四章中我们研究了内在退相干下,不同海森堡模型中的纠缠演化及隐形传态。我们发现内在退相干对纠缠演化及隐形传态保真度的影响取决于选择的初态。通过控制自旋链本身及外界条件,如外加磁场,包括均匀磁场和非均匀磁场,以及DM相互作用,在纠缠演化过程中,可以产生和增强不随时间变化的稳定纠缠态,从而在很大程度上抑制内在退相干对纠缠演化及隐形传态的破坏影响。
第五章我们首先研究了热力学极限下,与处于有限温度的自旋环境相耦合的三比特自旋系统的纠缠演化和退相干。结果表明,成对纠缠演化与描述系统和环境的外界参数有关。通过调节两比特间的耦合常数和外磁场,以及系统和环境的耦合常数,可以有效地控制纠缠sudden death对纠缠演化的影响。此外,我们还讨论了有限温度下,与含DM相互作用的XY自旋链耦合的中心二体自旋系统的纠缠演化。我们发现DM相互作用不会影响XY自旋链由横场导致的量子相变的临界行为,不会引入新的临界点。其次,在纠缠演化过程中,DM相互作用可以减弱和增强纠缠,这取决于其本身的取值和横场强度以及自旋链环境的温度。
论文的最后,给出了全文的总结和展望。
The recent development in quantum mechanics provides new methods and principles for information science. As one of the most important applications, quantum information is a new combination of quantum mechanics and information science and has developed into a popular research field. Meanwhile, quantum entanglement is one of the most striking features of quantum mechanics and is regarded as a fundamental resource for quantum information processing. It has been widely used in quantum teleportation, superdense coding, quantum cryptographic key distribution, quantum computation and so on. So a further study on entanglement is very important for basic quantum information theory and future potential applications. On the other hand, a real quantum system will unavoidably interacts with the surrounding environment and thus leads to decoherence and disentanglement. This is a fundamental obstacle to the realization of quantum entanglement. Therefore, taking into account the effect of decoherence on entanglement dynamics is indispensable. In this thesis, based on the character of quantum state of spin chain system, the quantum entanglement and its evolution in one-dimensional spin chain system are investigated theoretically. The thesis is dived into five chapters, with our work included in chapters 2 through 5.
In chapter 1, the background and basic theory of quantum information are introduced, as well as the quantum entangled state and its application are briefly described, and then the concept of quantum decoherence and one-dimensional spin chain are given at the end of this chapter.
In chapter 2, the thermal entanglement in the (1/2,1) mixed-spin Heisenberg model is investigated firstly. Through comparison with the spin-half system, the (1/2,1) mixed-spin is superior in generation of thermal entanglement. And then the thermal entanglement in different Heisenberg models and quantum teleportation in the thermally entangled channel are studied. Here the effects of the intrinsic properties and external conditions of the Heisenberg model on the thermal entanglement and the teleportation fidelity are concerned. It is shown that the thermal entanglement and the quality of teleportation can be enhanced by adjusting the values of the anisotropy parameter, external magnetic field and Dzyaloshinsky-Moriya (DM) interaction.
In chapter 3, the phenomenon of entanglement sudden death is studied. As to the entanglement evolution between two Tavis-Cummings atoms, the results show that the initial portion of the double excited state in the initial states is responsible for the sudden death of entanglement, and the degree of this effect also depends on the form of the initial state。In addition, the dynamic evolution of pairwise entanglement in a four-qubit Heisenberg XX spin chain is also studied. It is shown that the entanglement sudden death effect can be weakened when the periodic boundary condition and the magnetic impurity are introduced.
In chapter 4, the entanglement evolution and teleportation in the different Heisenberg spin chains with intrinsic decoherence taken into account are investigated. The results show that the effects of intrinsic decoherence on the entanglement evolution and teleportation fidelity rely strongly on the initial state. Controlling the uniform or inhomogeneous magnetic fields and DM interaction can not only generate stationary entanglement, but also enhance the value of stationary entanglement. So the destructive effect of intrinsic decoherence on the entanglement evolution and teleportation can be moderated in this way.
In chapter 5, the entanglement dynamics and decoherence of a three-qubit system under a quantum spin environment at a finite temperature in the thermodynamical limit are studied firstly. It is shown that the evolution of pairwise entanglement depends on the parameters related to the system and the spin environment. In addition, an undesirable entanglement sudden death occurs in the process of entanglement evolution, and this effect can be controlled by the coupling constant between two qubits, external magnetic field, and the interaction between the system and the environment. In addition, the entanglement dynamics between two central spins coupled to an XY spin chain at finite temperature with the DM interaction is also investigated. By studying the entanglement decay of the central spins, the results suggest that the DM interaction does not affect the behavior of quantum phase transition induced by the external magnetic field and does not induce new critical regions in the XY model. Moreover, the DM interaction can efficiently enhance or suppress the entanglement between the central spins which depends on the values of the DM interaction, the magnetic intensity and the temperature of the environmental spin chain.
Finally, the results are summarized and suggestions for future research work are given.
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