低维自旋系统的量子纠缠与磁学性质
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摘要
低维量子自旋系统作为一个联系着量子信息学和凝聚态物理两个领域的多体系统,其量子纠缠性质和磁学性质多年来一直受到广泛而深入的研究。
由爱因斯坦等提出的量子纠缠在量子计算与量子通讯的很多应用方面起着非常重要的作用,此外,凝聚态物理中的量子自旋链中的纠缠倍受关注。本文首先讨论了磁场方向和温度对两量子位Ising链纠缠与密集编码能力的影响,发现仅改变磁场方向时系统纠缠度与最优密集编码容量发生变化,且铁磁与反铁磁耦合呈现出不同的纠缠特性和密集编码能力。另外,还发现磁场方向只有在一定范围内变化时,才能进行有效的密集编码。同时我们进一步讨论了磁场方向对两量子位Ising链量子隐形传态的影响,发现通过调整磁场方向可使传输热纠缠和保真度达到最大。并且,为实现纠缠传输,信道纠缠不得小于某一临界值。对于铁磁Ising链,无论磁场取何方向,传输保真度总是小于2/3。有趣的是信道纠缠不能完全反映传输纠缠和保真度,较大的信道纠缠并不意味着能较好地传输纠缠。此外,在某一温度下,对磁场作用下的两量子位XX链,若磁场大小不变,则存在一个最优方向,使得系统的纠缠最大,我们称之为磁场方向诱导纠缠。
其次,量子相变是指量子多体系统因量子涨落而在不同相之间的转变,在凝聚态物理中起着重要作用。近来,量子纠缠与量子相变之间的关系引起了人们的广泛关注,对某些自旋系统,量子相变可由量子纠缠的临界特性来表征。然而,量子相变的发生一般不能由形成纠缠的奇导性来判断,形成纠缠的奇异性与量子临界点之间并没有一一对应的关系。本文以具有阻挫作用的1/2自旋反铁磁梯子为例,讨论了磁场对形成纠缠的影响,发现在量子相变点形成纠缠是连续的,因而从另一角度说明了量子临界点的存在与形成纠缠的奇异性之间不存在一一对应关系。此外,磁化平台与纠缠平台一一对应。
第三,具有多自旋相互作用的低维量子自旋链理论上引起了人们的兴趣,多自旋相互作用产生量子阻挫,从而引起量子相变。具有3-自旋相互作用的自旋-1/2XY链可严格地求解,本文我们进一步考虑4-自旋相互作用的影响,发现两格点纠缠熵可有效地探索量子相变。并且,系统除了出现自旋能隙和非对称亚铁磁序外,还出现了类“自旋波”相。在三相临界点,系统发生一级相变。
The low-dimensional quantum spin systems, which bridge the quantum information theory and condensed matter physics, have attracted considerable attention for its characteristics of quantum entanglement and magnetic properties.
Quantum entanglement, firstly proposed by Einstein et.al., plays an important role in many aspects of applications of quantum computation and quantum information. Furthermore, the quantum entanglement in spin chains have been extensively investigated in condensed matter physics. First, we analyze and discuss the effects of the direction of an external magnetic field and the temperature on entanglement and the dense coding capacity in a two-qubit Ising spin chain. It is found that the entanglement behavior and the maximal dense coding capacity of the system can be varied by controlling the direction of magnetic field. Likewise, it is also different for antiferromagnetic(AF) and ferromagnetic(FM) couplings. Thus, only in a special region for the direction of magnetic field, the optimal dense coding is feasible through the quantum channel.
Besides, the effects of direction of magnetic field on. quantum teleportation via a two-qubit Ising spin chain are also discussed. The teleported thermal concurrence and average fidelity can reach a maximum value by controlling the direction of magnetic field. A minimal entanglement of the thermal state is needed to carried out the entanglement teleportation. Nevertheless, the average fidelity is always less than classical limit 2/3, independent of the direction of magnetic field for two-qubit ferromagnetic Ising channel. It is interesting that the entanglement of the channel cannot completely reflect the teleported concurrence and average fidelity. Larger amount of entanglement does not mean better teleportation, for a fixed entanglement corresponding to infinite teleported concurrences or average fidelities.
In addition, at a certain temperature, the entanglement in a two-qubit XX chain can reach maximum by adjusting the magnetic field with fixed magnitude to an optimal direction, which can be regarded as magnetic direction induced entanglement.
Second, quantum phase transitions (QPTs) which are the evolvements between different ground states of quantum many-body systems, driven solely by quantum fluctuations, play an important role in modern condensed matter physics. Recently, a great deal of effort has been devoted to the understanding of the connection between quantum entanglement and QPTs. For a number of spin systems, it has been shown that QPTs are signaled by a critical behavior of concurrence. However, in general, QPTs can not be detected completely through the analysis of the singularities of concurrence, which are not one-to-one correspondence to the critical points. Herein, the effect of magnetic field on entanglement of a spin-1/2 antiferromagnetic spin ladder with frustration has been investigated. It is found that the concurrence is continuous at the critical point, where the QPT takes place, suggesting no one-to-one correspondence between QPT and the non-analyticity property of the concurrence from another aspect. Moreover, the concurrence shows plateaus in the same region where the magnetization plateaus occur.
The third, the multi-spin interactions in low-dimensional quantum spin chains have emerged in the theory study. Such interactions may cause quantum frustration, which eventually leads to the quantum phase transition (QPT) in the system. The spin-1/2 XY chains with three-spin interactions have been studied rigorously. Further considering the influence of four-spin interactions, we show that the two-site entanglement entropy can be exploited as a useful tool to detect the QPTs. In addition to the spin-gapped behavior and incommensurate ferrimagnetic ordering, the "spin waves" modulated by the four-spin interactions is uncovered. The system undergoes the second order QPTs except for the first order QPT occurring at the tricritical point with four-spin couplings.
由爱因斯坦等提出的量子纠缠在量子计算与量子通讯的很多应用方面起着非常重要的作用,此外,凝聚态物理中的量子自旋链中的纠缠倍受关注。本文首先讨论了磁场方向和温度对两量子位Ising链纠缠与密集编码能力的影响,发现仅改变磁场方向时系统纠缠度与最优密集编码容量发生变化,且铁磁与反铁磁耦合呈现出不同的纠缠特性和密集编码能力。另外,还发现磁场方向只有在一定范围内变化时,才能进行有效的密集编码。同时我们进一步讨论了磁场方向对两量子位Ising链量子隐形传态的影响,发现通过调整磁场方向可使传输热纠缠和保真度达到最大。并且,为实现纠缠传输,信道纠缠不得小于某一临界值。对于铁磁Ising链,无论磁场取何方向,传输保真度总是小于2/3。有趣的是信道纠缠不能完全反映传输纠缠和保真度,较大的信道纠缠并不意味着能较好地传输纠缠。此外,在某一温度下,对磁场作用下的两量子位XX链,若磁场大小不变,则存在一个最优方向,使得系统的纠缠最大,我们称之为磁场方向诱导纠缠。
其次,量子相变是指量子多体系统因量子涨落而在不同相之间的转变,在凝聚态物理中起着重要作用。近来,量子纠缠与量子相变之间的关系引起了人们的广泛关注,对某些自旋系统,量子相变可由量子纠缠的临界特性来表征。然而,量子相变的发生一般不能由形成纠缠的奇导性来判断,形成纠缠的奇异性与量子临界点之间并没有一一对应的关系。本文以具有阻挫作用的1/2自旋反铁磁梯子为例,讨论了磁场对形成纠缠的影响,发现在量子相变点形成纠缠是连续的,因而从另一角度说明了量子临界点的存在与形成纠缠的奇异性之间不存在一一对应关系。此外,磁化平台与纠缠平台一一对应。
第三,具有多自旋相互作用的低维量子自旋链理论上引起了人们的兴趣,多自旋相互作用产生量子阻挫,从而引起量子相变。具有3-自旋相互作用的自旋-1/2XY链可严格地求解,本文我们进一步考虑4-自旋相互作用的影响,发现两格点纠缠熵可有效地探索量子相变。并且,系统除了出现自旋能隙和非对称亚铁磁序外,还出现了类“自旋波”相。在三相临界点,系统发生一级相变。
The low-dimensional quantum spin systems, which bridge the quantum information theory and condensed matter physics, have attracted considerable attention for its characteristics of quantum entanglement and magnetic properties.
Quantum entanglement, firstly proposed by Einstein et.al., plays an important role in many aspects of applications of quantum computation and quantum information. Furthermore, the quantum entanglement in spin chains have been extensively investigated in condensed matter physics. First, we analyze and discuss the effects of the direction of an external magnetic field and the temperature on entanglement and the dense coding capacity in a two-qubit Ising spin chain. It is found that the entanglement behavior and the maximal dense coding capacity of the system can be varied by controlling the direction of magnetic field. Likewise, it is also different for antiferromagnetic(AF) and ferromagnetic(FM) couplings. Thus, only in a special region for the direction of magnetic field, the optimal dense coding is feasible through the quantum channel.
Besides, the effects of direction of magnetic field on. quantum teleportation via a two-qubit Ising spin chain are also discussed. The teleported thermal concurrence and average fidelity can reach a maximum value by controlling the direction of magnetic field. A minimal entanglement of the thermal state is needed to carried out the entanglement teleportation. Nevertheless, the average fidelity is always less than classical limit 2/3, independent of the direction of magnetic field for two-qubit ferromagnetic Ising channel. It is interesting that the entanglement of the channel cannot completely reflect the teleported concurrence and average fidelity. Larger amount of entanglement does not mean better teleportation, for a fixed entanglement corresponding to infinite teleported concurrences or average fidelities.
In addition, at a certain temperature, the entanglement in a two-qubit XX chain can reach maximum by adjusting the magnetic field with fixed magnitude to an optimal direction, which can be regarded as magnetic direction induced entanglement.
Second, quantum phase transitions (QPTs) which are the evolvements between different ground states of quantum many-body systems, driven solely by quantum fluctuations, play an important role in modern condensed matter physics. Recently, a great deal of effort has been devoted to the understanding of the connection between quantum entanglement and QPTs. For a number of spin systems, it has been shown that QPTs are signaled by a critical behavior of concurrence. However, in general, QPTs can not be detected completely through the analysis of the singularities of concurrence, which are not one-to-one correspondence to the critical points. Herein, the effect of magnetic field on entanglement of a spin-1/2 antiferromagnetic spin ladder with frustration has been investigated. It is found that the concurrence is continuous at the critical point, where the QPT takes place, suggesting no one-to-one correspondence between QPT and the non-analyticity property of the concurrence from another aspect. Moreover, the concurrence shows plateaus in the same region where the magnetization plateaus occur.
The third, the multi-spin interactions in low-dimensional quantum spin chains have emerged in the theory study. Such interactions may cause quantum frustration, which eventually leads to the quantum phase transition (QPT) in the system. The spin-1/2 XY chains with three-spin interactions have been studied rigorously. Further considering the influence of four-spin interactions, we show that the two-site entanglement entropy can be exploited as a useful tool to detect the QPTs. In addition to the spin-gapped behavior and incommensurate ferrimagnetic ordering, the "spin waves" modulated by the four-spin interactions is uncovered. The system undergoes the second order QPTs except for the first order QPT occurring at the tricritical point with four-spin couplings.
引文
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