一维自旋系统中的量子纠缠
摘要
量子信息与量子计算是量子理论与现代信息科学、计算科学相结合所产生的一门新兴交叉学科,现已成为物理学研究的热点。量子信息与计算科学在基础科学研究和应用方面都有非常重要的意义,它一方面可以增进人们对量子力学的认识;另一方面,基于量子力学的密码技术、通讯技术、计算与存储技术将对国防、商业等领域产生重大的影响。比如,量子不可克隆定理使得量子信息不能像经典信息那样可以被任意复制,这使得人们能够建立起绝对安全的量子密码系统。又如,利用量子态的相干叠加性,人们提出了量子并行算法,用以解决诸如大数因式分解等经典计算无法解决的问题,等等。
一直以来,科学家都在不断寻找并尝试各种能够实现量子信息处理过程的量子系统,如腔QED、离子阱、量子点、核磁共振等等,取得了很大进展。而实际应用的大规模的量子信息处理任务,需要在一种可扩展和易于集成的系统中实现,量子自旋系统就是其中一种非常有潜力的备选系统。目前,研究人员已经提出了很多基于自旋系统的量子信息处理方案,如量子计算、量子克隆、量子存储以及量子态传输等。而自旋系统中量子纠缠及其演化的研究则为上述应用提供理论支持,这也是我们的工作重点。
首先,我们研究了一维自旋系统在热态下的量子纠缠。我们知道,由于技术条件和环境的限制,一个实际的物理系统不可能总是处于其基态,而往往是处在某个温度的热平衡态下,所以研究热态下的纠缠对量子信息在实际条件下的应用非常重要。我们以XXZ模型为例,研究了一维自旋链中的两体热纠缠,并且给出了其纠缠临界温度随自旋链的格点数、格点自旋、各向异性参数以及外磁场强度而变化的规律,我们发现,(1)热纠缠的临界温度随着格点数的增加而不断减小,最后会趋于某个固定值:(2)随着格点自旋的增加,临界温度呈近似线性的增长,这表明,在温度较高的情况下,采用具有高格点自旋的系统是比较好的选择;(3)各向异性参数也会对热纠缠造成显著的影响,临界温度会随着各向异性参数的增加而减小,最后在接近于△=1的某点降为零:(4)外磁场同样会影响热纠缠的数值,但是不会影响临界温度的大小。
为了进一步讨论温度对量子信息处理过程的影响,我们研究了热态自旋链中的量子态传输,(?)条处于热态下的自旋链作为量子信道将一个已知量子态从链的一端传到另一端。计算结果显示,传输中所能得到的最大保真度并不是随着温度增加而单调减小的,而是取决于最大允许的传输时间。我们也比较了热信道和基态信道在传输相同量子态时的保真度,发现基态信道并不总是优于热态信道。这些结论表明,热噪声虽然会对量子态传输造成一定的影响,但是并不是灾难性的,态传输并不是必须要在非常苛刻的低温条件下进行。
最后,我们研究了自旋信道中量子纠缠的演化,发现在其演化过程中存在纠缠的突然死亡,并且得到了不同自旋链中纠缠死亡的参数区域。我们还讨论了存在外界环境时的纠缠演化,并且和没有环境的情形进行了比较,发现环境的出现将加快退相干的速度,进而使得纠缠死亡的区域面积变大。这些结果将帮助研究人员在量子信息处理过程中避开有害的纠缠死亡,或者对其加以利用。进一步,可以用外场来调控纠缠的演化,改变纠缠死亡的参数区域,这将是我们下一阶段的研究目标。
Quantum information and quantum computation, an interdisciplinary field which involves quantum mechanics, computer science and the theory of information, has attracted more and more interests. It is significant in both foundational science and application. It does not only enhance the understanding of the quantum theory, but also impact the field of national defense and commerce. For example, absolutely secure quantum cryptogram system have been established base on the quantum no-cloning theorem which said that the quantum information can't be copied as classical information. And, parallel quantum algorithm base on the quantum coherence, have been used to solve the hard problems for the classical computer, like the factorization of large numbers.
Many quantum system have been proposed in the quantum information processing (QIP), like cavity QED, ion trap, quantum dot, and NMR. Applied quantum information processing should been realized in systems that easy to be extended and integrated. Spin system is such a solid system, it has been proposed to been used in quantum information storing and quantum memories, quantum computation, quantum clone, and quantum state transfer and so on. Quantum entanglement is important resource in quantum information science, so the study on the entanglement in the spin system will be helpful to the practical application. It is also the main point of this thesis.
Firstly, we discuss the thermal entanglement in spin chain. A physically realizable quantum mechanisms are not always at zero temperature but often in a thermal equilibrium, so the study on the entanglement in thermal state is very important for the quantum information science. We choose the XXZ model to investigate the pairwise thermal entanglement in spin chain. The effects of spin, number of sites, anisotropic parameter, and magnetic field on the threshold temperature are discussed respectively. We find that: (l)The threshold temperature decreases with the increasing of the number of sites and finally trends to be a constant. (2)With the increasing of spin, threshold temperature inereases almost linearly and becomes sufficiently high, imply that the larger spin systems like molecular magnets should be better in the QIP with high temperature. (3)The effect of anisotropic parameter is observable, with the increasing of anisotropic parameter, threshold temperature decreases and vanishes at a point near△= 1. (4) The external magnetic field can also influence the entanglement but not change the threshold temperature.
For the further study on the thermal effect in the QIP, we discuss the quantum state transfer through a thermal spin chain, the quantum state is transmitted from one end of the chain to the other end. We find that the maximal fidelity does not decrease monotonously with the increasing of the temperature, but depends on the maximal allowed transfer time. We also compare the results of thermal channel with the transfer through ground state channel, and find that the thermal channel is sometimes better than the ground state channel. This results mean that, the thermal noise influences but does not destroy the quantum state transfer vitally, the transfer does not need to be processed in extreme low temperature.
Finally, We study the dynamic evolution of the entanglement in the spin channel with an XY-type interaction initially in an entangled state. We find that the phenomenon of entanglement sudden death (ESD) appears in the evolution of entanglement for some initial states. We calculate the entanglement and obtain the parameter regions of disentanglement for the chains with several numbers of sites. The influence of environment is also discussed, we find that the present of environment will speedup the decoherence and enlarge the regions of disentanglement. These results are helpful to apply or avoid the ESD in the QIP. Further more, we can use an additional external field to modulate the evolution of the entanglement, change the regions of disentanglement as needed, that is our research topic next stage.
一直以来,科学家都在不断寻找并尝试各种能够实现量子信息处理过程的量子系统,如腔QED、离子阱、量子点、核磁共振等等,取得了很大进展。而实际应用的大规模的量子信息处理任务,需要在一种可扩展和易于集成的系统中实现,量子自旋系统就是其中一种非常有潜力的备选系统。目前,研究人员已经提出了很多基于自旋系统的量子信息处理方案,如量子计算、量子克隆、量子存储以及量子态传输等。而自旋系统中量子纠缠及其演化的研究则为上述应用提供理论支持,这也是我们的工作重点。
首先,我们研究了一维自旋系统在热态下的量子纠缠。我们知道,由于技术条件和环境的限制,一个实际的物理系统不可能总是处于其基态,而往往是处在某个温度的热平衡态下,所以研究热态下的纠缠对量子信息在实际条件下的应用非常重要。我们以XXZ模型为例,研究了一维自旋链中的两体热纠缠,并且给出了其纠缠临界温度随自旋链的格点数、格点自旋、各向异性参数以及外磁场强度而变化的规律,我们发现,(1)热纠缠的临界温度随着格点数的增加而不断减小,最后会趋于某个固定值:(2)随着格点自旋的增加,临界温度呈近似线性的增长,这表明,在温度较高的情况下,采用具有高格点自旋的系统是比较好的选择;(3)各向异性参数也会对热纠缠造成显著的影响,临界温度会随着各向异性参数的增加而减小,最后在接近于△=1的某点降为零:(4)外磁场同样会影响热纠缠的数值,但是不会影响临界温度的大小。
为了进一步讨论温度对量子信息处理过程的影响,我们研究了热态自旋链中的量子态传输,(?)条处于热态下的自旋链作为量子信道将一个已知量子态从链的一端传到另一端。计算结果显示,传输中所能得到的最大保真度并不是随着温度增加而单调减小的,而是取决于最大允许的传输时间。我们也比较了热信道和基态信道在传输相同量子态时的保真度,发现基态信道并不总是优于热态信道。这些结论表明,热噪声虽然会对量子态传输造成一定的影响,但是并不是灾难性的,态传输并不是必须要在非常苛刻的低温条件下进行。
最后,我们研究了自旋信道中量子纠缠的演化,发现在其演化过程中存在纠缠的突然死亡,并且得到了不同自旋链中纠缠死亡的参数区域。我们还讨论了存在外界环境时的纠缠演化,并且和没有环境的情形进行了比较,发现环境的出现将加快退相干的速度,进而使得纠缠死亡的区域面积变大。这些结果将帮助研究人员在量子信息处理过程中避开有害的纠缠死亡,或者对其加以利用。进一步,可以用外场来调控纠缠的演化,改变纠缠死亡的参数区域,这将是我们下一阶段的研究目标。
Quantum information and quantum computation, an interdisciplinary field which involves quantum mechanics, computer science and the theory of information, has attracted more and more interests. It is significant in both foundational science and application. It does not only enhance the understanding of the quantum theory, but also impact the field of national defense and commerce. For example, absolutely secure quantum cryptogram system have been established base on the quantum no-cloning theorem which said that the quantum information can't be copied as classical information. And, parallel quantum algorithm base on the quantum coherence, have been used to solve the hard problems for the classical computer, like the factorization of large numbers.
Many quantum system have been proposed in the quantum information processing (QIP), like cavity QED, ion trap, quantum dot, and NMR. Applied quantum information processing should been realized in systems that easy to be extended and integrated. Spin system is such a solid system, it has been proposed to been used in quantum information storing and quantum memories, quantum computation, quantum clone, and quantum state transfer and so on. Quantum entanglement is important resource in quantum information science, so the study on the entanglement in the spin system will be helpful to the practical application. It is also the main point of this thesis.
Firstly, we discuss the thermal entanglement in spin chain. A physically realizable quantum mechanisms are not always at zero temperature but often in a thermal equilibrium, so the study on the entanglement in thermal state is very important for the quantum information science. We choose the XXZ model to investigate the pairwise thermal entanglement in spin chain. The effects of spin, number of sites, anisotropic parameter, and magnetic field on the threshold temperature are discussed respectively. We find that: (l)The threshold temperature decreases with the increasing of the number of sites and finally trends to be a constant. (2)With the increasing of spin, threshold temperature inereases almost linearly and becomes sufficiently high, imply that the larger spin systems like molecular magnets should be better in the QIP with high temperature. (3)The effect of anisotropic parameter is observable, with the increasing of anisotropic parameter, threshold temperature decreases and vanishes at a point near△= 1. (4) The external magnetic field can also influence the entanglement but not change the threshold temperature.
For the further study on the thermal effect in the QIP, we discuss the quantum state transfer through a thermal spin chain, the quantum state is transmitted from one end of the chain to the other end. We find that the maximal fidelity does not decrease monotonously with the increasing of the temperature, but depends on the maximal allowed transfer time. We also compare the results of thermal channel with the transfer through ground state channel, and find that the thermal channel is sometimes better than the ground state channel. This results mean that, the thermal noise influences but does not destroy the quantum state transfer vitally, the transfer does not need to be processed in extreme low temperature.
Finally, We study the dynamic evolution of the entanglement in the spin channel with an XY-type interaction initially in an entangled state. We find that the phenomenon of entanglement sudden death (ESD) appears in the evolution of entanglement for some initial states. We calculate the entanglement and obtain the parameter regions of disentanglement for the chains with several numbers of sites. The influence of environment is also discussed, we find that the present of environment will speedup the decoherence and enlarge the regions of disentanglement. These results are helpful to apply or avoid the ESD in the QIP. Further more, we can use an additional external field to modulate the evolution of the entanglement, change the regions of disentanglement as needed, that is our research topic next stage.
引文
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