固态自旋系统中的量子纠缠和量子计算
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摘要
本文主要研究了一维固态自旋系统中的量子纠缠特性,通过不同的判据得到了系统存在热纠缠和基态纠缠的条件。从量子纠缠的本质出发,研究了多体量子系统中基态的临界行为。通过将这些典型的固态自旋体系应用到量子信息处理中,提出了可行的量子计算方案,并且分析了实际环境对量子计算的影响。
本文以量子纠缠的定义为基础,利用可测量的物理量构造出了有效纠缠判据。通过这些判据,我们研究了一维固态自旋链的纠缠特性,如:任意自旋的海森堡链、混合自旋链和实心玻色子Hubbard链。其中,我们得到了热纠缠存在的临界温度。当体系的物理温度低于此值时,相应的热平衡态就是纠缠的,但随着温度的升高,纠缠就逐渐消失。我们把这些判据与理论判别方法进行了比较,发现这些基于可测物理量的判据给出了纠缠存在的一个充分条件。数值计算的结果表明,在具有大量粒子数的自旋系统中,当温度较低的时候,实验上也能检测到纠缠态。对于多体量子系统,本文重点研究了多体和两体的基态纠缠,找到了全局纠缠与量子相变的联系,并且发现了在实际的固态自旋系统中,两体纠缠要比多体纠缠更容易存在。为了考虑实际环境对量子纠缠的影响,本文通过主方程分析了在真空热库中少量原子的量子态的含时演化过程。当其中只有一个原子处于激发态时,那些原先处于基态的原子对就会产生量子纠缠。纠缠大小随着时间的变化类似于带阻尼的拉比振荡,振幅则受到了原子自发衰减强度的调制。通过精确求解体系的量子态,我们给出了产生纠缠的物理机制。结果表明纠缠存在的主要因素取决于不同原子间激发态和基态的跃迁几率。
本文还讨论了基于固态自旋的量子计算方案。为了使系统模型更接近实际情况,我们重点考虑了非均匀和非对称的交换作用。其中,我们通过解析求解单量子位态的含时演化,得到了在海森堡XXZ模型中实现量子交换门的物理条件,分析了系统内的量子涨落对量子计算的影响,给出了可操作的实验方案。我们通过半导体量子点中电子自旋耦合的非对称性相互作用,利用单量子位旋转操作,构造出了两量子位控制非门,从而实现了精度较高的量子计算。在固态自旋系统中,人们要想在实验中实现量子信息处理,就需要克服操作距离短的困难。我们提出了以耦合自旋链作为量子计算总线的理论方案,当总线系统一直处于非简并的基态时,我们利用电极控制一些量子位与其产生弱耦合,这样我们就可以在这些量子位上间接得到有效的长程相互作用。在这个基础上,我们就构造出了一组通用量子门,从而实现长程量子计算的目的。
In this thesis, the entanglement properties of one-dimensional solid-state quantum spin systems are investigated. By means of some different entanglement witnesses, the conditions of the existence of thermal and ground-state entanglement are provided. From the origin of quantum entanglement, the quantum criticality of the ground state in many-body quantum systems is studied. Some typical solid-state quantum spin systems have been applied to the quantum information processing. Some feasible schemes of quantum computation are proposed. The effects of the realistic environment are taken into account.
In the thesis, based on the definition of quantum entanglement, some efficient witnesses can be obtained by some measurable physical qualities. Through these entanglement witnesses, the entanglement properties of one-dimensional solid-state quantum spin chains are investigated. The spin chains include a spin-s Heisenberg chain, mixed-spin chain, and hardcore Boson-Hubbard chain. The critical temperatures for the existence of the thermal entanglement are given. When the physical temperatures of the systems are lower than the critical temperature, the corresponding thermal equilibrium states are entangled. With the increase of the temperatures, the thermal entanglement vanishes gradually. Compared with other theoretical measures, these entanglement witnesses based on measurable qualities can provide one necessary condition for the existence of the entanglement. The numerical calculation shows that the entanglement in rather low temperatures in the solid state systems with a great number of spins can be detected. For the many-body quantum systems, the multipartite and bipartite ground state entanglement is investigated. The relation of global entanglement and quantum phase transition is obtained. In the real solid state spin system, the multipartite entanglement vanishes more rapidly than bipartite entanglement. In regard to the impacts of the environment on the entanglement, the time evolution of a small number of atoms in the vacuum reservoir is analyzed by means of the master equation. If there is only one excited atom in the ensemble, a pair of atoms originally at the ground states can be entangled. The dynamical behavior of the entanglement is similar to the damped Rabi oscillation whose amplitude is tuned by the spontaneous decay. The exact solution of the quantum states can help for the understanding of the physical essence of the entanglement existence. It is seen that the entanglement is dependent mainly on the transition probability between the excited state of one atom and ground state of another one.
In the thesis, some methods of solid spin-based quantum computation are proposed. The anisotropic and asymmetric exchange interactions are considered in the model which is close to the practical conditions. The conditions of the exact swap gate in the anisotropic Heisenberg XXZ model is presented by solving exactly the time evolution of single-qubit states. The effects of the quantum fluctuations in the systems are analyzed. The feasible experimental scheme is provided. Through the asymmetric interaction existing in the model of two coupled electron spins in semiconductor quantum dots and by means of single-qubit rotations, the two-qubit Controlled-NOT gate is constructed. The quantum computation with high accuracy can be implemented. In the experiments of solid-state quantum information processing using quantum spin system, the short distance of operations is regarded as one big obstacle. A scheme based on the quantum spin bus with two coupled chainsis proposed. When the bus is always frozen at the non-degenerate ground state, the effective long-range interaction can be obtained by the electrical control of the weak coupling between some qubits and the bus. Thus, a set of universal quantum logic gates can be set up to realize quantum computation in long distance.
本文以量子纠缠的定义为基础,利用可测量的物理量构造出了有效纠缠判据。通过这些判据,我们研究了一维固态自旋链的纠缠特性,如:任意自旋的海森堡链、混合自旋链和实心玻色子Hubbard链。其中,我们得到了热纠缠存在的临界温度。当体系的物理温度低于此值时,相应的热平衡态就是纠缠的,但随着温度的升高,纠缠就逐渐消失。我们把这些判据与理论判别方法进行了比较,发现这些基于可测物理量的判据给出了纠缠存在的一个充分条件。数值计算的结果表明,在具有大量粒子数的自旋系统中,当温度较低的时候,实验上也能检测到纠缠态。对于多体量子系统,本文重点研究了多体和两体的基态纠缠,找到了全局纠缠与量子相变的联系,并且发现了在实际的固态自旋系统中,两体纠缠要比多体纠缠更容易存在。为了考虑实际环境对量子纠缠的影响,本文通过主方程分析了在真空热库中少量原子的量子态的含时演化过程。当其中只有一个原子处于激发态时,那些原先处于基态的原子对就会产生量子纠缠。纠缠大小随着时间的变化类似于带阻尼的拉比振荡,振幅则受到了原子自发衰减强度的调制。通过精确求解体系的量子态,我们给出了产生纠缠的物理机制。结果表明纠缠存在的主要因素取决于不同原子间激发态和基态的跃迁几率。
本文还讨论了基于固态自旋的量子计算方案。为了使系统模型更接近实际情况,我们重点考虑了非均匀和非对称的交换作用。其中,我们通过解析求解单量子位态的含时演化,得到了在海森堡XXZ模型中实现量子交换门的物理条件,分析了系统内的量子涨落对量子计算的影响,给出了可操作的实验方案。我们通过半导体量子点中电子自旋耦合的非对称性相互作用,利用单量子位旋转操作,构造出了两量子位控制非门,从而实现了精度较高的量子计算。在固态自旋系统中,人们要想在实验中实现量子信息处理,就需要克服操作距离短的困难。我们提出了以耦合自旋链作为量子计算总线的理论方案,当总线系统一直处于非简并的基态时,我们利用电极控制一些量子位与其产生弱耦合,这样我们就可以在这些量子位上间接得到有效的长程相互作用。在这个基础上,我们就构造出了一组通用量子门,从而实现长程量子计算的目的。
In this thesis, the entanglement properties of one-dimensional solid-state quantum spin systems are investigated. By means of some different entanglement witnesses, the conditions of the existence of thermal and ground-state entanglement are provided. From the origin of quantum entanglement, the quantum criticality of the ground state in many-body quantum systems is studied. Some typical solid-state quantum spin systems have been applied to the quantum information processing. Some feasible schemes of quantum computation are proposed. The effects of the realistic environment are taken into account.
In the thesis, based on the definition of quantum entanglement, some efficient witnesses can be obtained by some measurable physical qualities. Through these entanglement witnesses, the entanglement properties of one-dimensional solid-state quantum spin chains are investigated. The spin chains include a spin-s Heisenberg chain, mixed-spin chain, and hardcore Boson-Hubbard chain. The critical temperatures for the existence of the thermal entanglement are given. When the physical temperatures of the systems are lower than the critical temperature, the corresponding thermal equilibrium states are entangled. With the increase of the temperatures, the thermal entanglement vanishes gradually. Compared with other theoretical measures, these entanglement witnesses based on measurable qualities can provide one necessary condition for the existence of the entanglement. The numerical calculation shows that the entanglement in rather low temperatures in the solid state systems with a great number of spins can be detected. For the many-body quantum systems, the multipartite and bipartite ground state entanglement is investigated. The relation of global entanglement and quantum phase transition is obtained. In the real solid state spin system, the multipartite entanglement vanishes more rapidly than bipartite entanglement. In regard to the impacts of the environment on the entanglement, the time evolution of a small number of atoms in the vacuum reservoir is analyzed by means of the master equation. If there is only one excited atom in the ensemble, a pair of atoms originally at the ground states can be entangled. The dynamical behavior of the entanglement is similar to the damped Rabi oscillation whose amplitude is tuned by the spontaneous decay. The exact solution of the quantum states can help for the understanding of the physical essence of the entanglement existence. It is seen that the entanglement is dependent mainly on the transition probability between the excited state of one atom and ground state of another one.
In the thesis, some methods of solid spin-based quantum computation are proposed. The anisotropic and asymmetric exchange interactions are considered in the model which is close to the practical conditions. The conditions of the exact swap gate in the anisotropic Heisenberg XXZ model is presented by solving exactly the time evolution of single-qubit states. The effects of the quantum fluctuations in the systems are analyzed. The feasible experimental scheme is provided. Through the asymmetric interaction existing in the model of two coupled electron spins in semiconductor quantum dots and by means of single-qubit rotations, the two-qubit Controlled-NOT gate is constructed. The quantum computation with high accuracy can be implemented. In the experiments of solid-state quantum information processing using quantum spin system, the short distance of operations is regarded as one big obstacle. A scheme based on the quantum spin bus with two coupled chainsis proposed. When the bus is always frozen at the non-degenerate ground state, the effective long-range interaction can be obtained by the electrical control of the weak coupling between some qubits and the bus. Thus, a set of universal quantum logic gates can be set up to realize quantum computation in long distance.
引文
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