论文翻译及原文

非惯性系中标量场和狄拉克场的非最大纠缠衰减潘启元荆继良中华人民共和国湖南 410081 长沙湖南师范大学教育部低维量子结构与调控重点实验室物理系物理研究所（2007 年7月18日接收；2008年2月18日发行）我们研究了两个相对加速的观测者所观察的自由标量场与狄拉克场两种形式之间的纠缠。

我们发现，受Unruh效应的影响，初始态参数α与其“归一化伙伴”两者的同一初始纠缠态将会沿着两条不同的轨道衰减，这里排除了体现闵科夫斯基和Rindler坐标系下自由场的不对等量化的最大纠缠态。

在无穷大加速度极限下，标量场中对于任意α的态没有可提纯的纠缠，但是对于狄拉克场总有依赖α的纠缠度。

有趣的是在加速度极限下互信息只有初始互信息的一半；这一结果与α和场的形式无关。

DOI: 10.1103/PhysRevA.77.024302 PACS number(s): 03.65.Ud, 04.62.+v, 04.70.D近年来量子信息理论取得了迅速的进展。

并且越来越多的工作已推广到相对论框架下去进行研究。

相对论背景下的纠缠已经得到了相当一部分人的注意，因为纠缠可作为主要的资源应用于很多量子信息操作中，如量子远程通信、量子计算等。

除了量子信息的潜在利益外，对量子纠缠的研究也能深化我们对黑洞热力学和黑洞信息疑难的理解。

因此，许多专家对不同场相对论框架下的量子纠缠进行了研究。

近期，Fuentes-Schuller and Alsing证实了纠缠取决于相对加速的观察者中的一个观察者在加速前的相对加速度的大小，而这两观察者共享一个最大的纠缠的玻色子或费米子对。

他们的结果也表明：不同类型的场对由Unruh效应产生的纠缠衰减在本质上有不同的影响。

在这简短的报告中，我们选取一个一般纠缠态作为初始纠缠态：α满足|α|∈(0,1)，α和互为“归一化伙伴”。

由于初始态参数α的存在，我们将试图观察对于两个相对加速的观察者这个不确定的初始纠缠态将会对纠缠的衰减产生怎样的影响。

我们需要注意，施瓦兹时空与无穷大加速度极限下的Rindler空间相类似。

因此，我们在这个极限下的研究结果可以用来讨论两个玻色子或是费米子模式间的纠缠，当一个观察者掉入黑洞，另外一个刚好以恒定的加速度逃逸时，这一纠缠可以被观测到。

Rindler坐标系适合于描述均匀加速运动的观察者。

在闵科夫斯基时空作均匀加速运动的观察者的世界线与原点左边（区域I）和右边（区域II）的双曲线相对应，该双曲线被限制在构成Rindler视界的类光渐近线内，因此两个Rindler区域相互之间理所当然的没有联系。

一个均匀加速的观察者被限制在Rindler区域I或区域II并且不能进入到另一个区域。

从惯性系来看方程（1）所描述的系统是双向的，但在非惯性系，区域II的一组额外的模变得相关。

因此我们将研究惯性系中的观察者Alice检测的模s态和以加速度a均匀加速的观察者Bob检测的模k态之间的纠缠混态。

玻色子纠缠。

对于自由标量场，闵科夫斯基真空态可用Rindler时空中的双模压缩态来描述，，k是波矢，r是加速度参量，和分别是Rindler区域I的粒子模和Rindler区域II的反粒子模。

运用方程(2)和第一激发态我们可以按照Alice使用闵科夫斯基模和Bob使用Rindler模来改写方程(1)，既然Bob与区域II因果不相连，我们写出这个区域的态从而得到。

部分转置标准为例[11]中纠缠的存在提供了充分的条件：部分转置图1.(彩色线)不同α的玻色子模的对数负度随加速度r的变化图。

矩阵至少存在一个负的本征值，密度矩阵才是纠缠的；但正转置矩阵的态仍然是纠缠的。

这是我们熟知的束缚纠缠态或是不可提纯的纠缠态。

通过交换Alice的量子比特，我们得到(n,n+1)范围内转置矩阵的本征值。

很明显在加速度有限的情况下()本征值总是负值。

因此，该混态对于Bob的任何有限加速度总是纠缠态。

在的极限下，负的本征值将会趋近于零。

进一步讨论，我们采用作为可提纯纠缠上界的对数负度。

此单调纠缠度定义为，是部分转置矩阵的矩阵迹范数。

所以我们得到对于零加速度()，。

在范围内，α越大，初始纠缠就越强；但在范围内，α越大，初始纠缠就越弱。

对于有限的加速度，α不同，随着r的增加而单调减少，这表明初始态纠缠度不受Unruh效应产生的热场的影响。

从图1我们惊奇地发现除了最大纠缠态外(|)，具有参数α和它的“归一化伙伴”的同一初始纠缠将沿着两条不同的的轨迹衰减。

这一现象是因α与关于r的双曲函数的关联而产生的，并表明对于闵科夫斯基坐标和Rindler坐标下的标量场的不对等量化。

在的极限下对于任一α对数负度都等于零，图2.(彩色线)不同α的玻色子模的互信息随加速度r的变化图。

这表明该态不存在可提纯纠缠。

互信息可以用来估计系统的任何两个子系统之间的关联程度，定义为：这里是密度矩阵的熵。

从方程(3)，我们可以得到该关联态的熵，对对Minkowski模求迹，我们得到Bob在区域I的密度矩阵；它的熵为用同样的方法，我们通过对Rindler I区进行求迹得到Alice的密度矩阵；它的熵为在图2中我们画出了对不同的α情况下的态的互信息随r的变化曲线。

对加速度为零的情况，互信息为在的范围，α越大，互信息就越强；但在范围内，α越大，互信息就越弱。

随着加速度的增大，互信息变得越弱。

有趣的是，除了最大纠缠态，对于α和的同一初始互信息将沿着两条不同的路线衰减。

然而，在有限的加速度极限下，互信息汇合到同一个点。

图3.(彩色线)不同α的费米子模的对数负度随加速度r的变化图(注意:).该点的互信息仅为的一半。

很显然如果越大，在此极限下，它将衰减到更高的值。

既然在有限加速度极限下可提纯纠缠度为零，那么可以毫不夸张的说在此极限下总关联是由经典关联加上束缚纠缠构成。

费米子纠缠。

采用Alsing使用的单模近似，费米子闵科夫斯基真空态可写作其唯一的激发态为这里，对于加速度参数r的范围是。

对闵科夫斯基粒子态和运用方程(8)和(9)并对区域II求迹，我们得到|mn⟩=.部分转置标准给两量子比特混态纠缠提供了一个充分必要条件：如果至少有一个部分转置矩阵的本征值为负值，那么密度矩阵是纠缠的。

交换Alice的量子比特，我们得到部分转置矩阵一个本征值,对于该值总是负的。

因此，对于Bob的任一一定的加速度该态总是纠缠态。

其对数负度为图4.(彩色线)不同α的费米子模的互信息随加速度r的变化图。

对为零的加速度(r=0)，对有限的加速度，正如图3所示，受到Unruh效应的影响，纠缠度会衰减。

我们发现在的范围内，α越大，初始纠缠度越大，但这并非总是退化到更高的纠缠度。

我们应该注意到对范围的初始态的最终纠缠度比最大纠缠态的纠缠度要大，该对数负度为对于最大的最终纠缠的对数负度为在的范围内，α越大，初始纠缠越弱并且最终纠缠度越小。

对于费米场情形，除了最大纠缠态，由于α与关于r的三角函数的耦合，在加速度极限下()对于α和的同一初始费米模的纠缠将沿着两条不同路线衰减并渐进的趋于两个不同的非零最小值。

在有限的加速度极限下，这意味着该态总是纠缠的。

这与玻色场形成了鲜明的对比，同时表明费米纠缠可作为量子源在加速度无穷大极限下进行量子信息任务操作。

我们可以用同样的方法得到费米子模的互信息的初始场态的互信息随加速度r的变化轨迹如图4所示。

加速度为零时，初始互信息等于，这和玻色模的一样。

随着加速度的增加互信息变得越小，同时我们又惊奇地发现除了最大纠缠态，对于α和的同一初始互信息将沿着两条不同的路线衰减。

在有限的加速度极限下互信息汇合到同一个点仅为的一半。

这一结果使我们联想到玻色子的情形，因此我们得出该结论独立于α和场的形式。

我们应该注意到如果我们把初始纠缠态设为我们将得到图1-4同一α的纠缠度衰减的结果。

总结，非惯性系中狄拉克场和标量场的纠缠度在Unruh效应的影响下随着Bob的加速度增加而衰减，但纠缠衰减会因初始态参数α的不同而不同。

我们又惊奇地发现除了最大纠缠态，对于α和的同一初始纠缠将沿着两条不同的路线衰减，这仅表明对于闵科夫斯基和Rindler坐标系下的自由标量场的不对等的量化。

在无限的加速度极限下，可应用到Alice掉进黑洞而Bob刚好逃离黑洞的情形，该态对于标量场的任意α都没有可提纯纠缠，但对于狄拉克场总有依赖α的纠缠度。

我们应该注意到，对于,在此极限下我们将得到费米子态的最大最终纠缠。

进一步的分析表明，随着加速度参量r的增大，互信息衰减到一个非零的最小值，该值依赖于这两个场的参量α。

然而，有趣的是无穷加速度极限下的互信息刚好等于初始互信息的一半，该结果独立于α和场的形式。

本工作得到以下单位的支持，国家自然科学基金编号10675045；全国优秀博士学位论文作者专项资金资助项目编号200317；湖南省自然科学基金编号07A0128；湖南省重点学科建设项目基金。

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Ohya, Information Dynamics and Open Systems—Classical and Quantum Approach (Kluwer Academic Publishers, Dordrecht, 1997).Degradation of nonmaximal entanglement of scalar andDirac fields in noninertial framesQiyuan Pan and Jiliang Jing*Institute of Physics and Department of Physics, and Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China(Received 18 July 2007; published 8 February 2008)The entanglement between two modes of free scalar and Dirac fields as seen by two relatively accelerated observers has been investigated. It is found that the same initial entanglement for an initial state parameter αand its “normalized partner”will be degraded by the Unruh effect along two different trajectoriesexcept for the maximally entangled state, which just shows the inequivalence of the quantization for a free field in Minkowski and Rindler coordinates. In the infinite-acceleration limit, the state does not have distillable entanglement for any αfor the scalar field, but always remains entangled to a degree that is dependent on αfor the Dirac field. It is also interesting to note that in this limit the mutual information equals just half of the initial mutual information; this result is independent of αand the type of field.DOI: 10.1103/PhysRevA.77.024302 PACS number(s): 03.65.Ud, 04.62.+v, 04.70.DQuantum-information theory has made rapid progress in recent years, and more and more efforts have been expended on the study of quantum information in the relativistic framework. In particular, entanglement in a relativistic setting has received considerable attention because it is considered to be a major resource for quantum-information tasks such as quantum teleportation, quantum computation, and so on. In addition to the potential interest for quantum information, the study of entanglement can also help us get a deeper understanding of black-hole thermo dynamics and the black-hole information paradox. Thus, many authors have investigated entanglement in relativistic frames (inertial or not) for various fields.More recently, Fuentes-Schuller et al.and Alsing et al. explicitly demonstrated that entanglement is a quantity depending on the relative acceleration of one of the observers who, before being accelerated, shared a maximally entangled bosonic or fermionic pair. Their results also showed that different types of field will have qualitatively different effects on the degradation of entanglement produced by the Unruh effect.In this Brief Report, we choose a generic state as the initial entangled state:where αis some real number that satisfies , and αand are the so-called ‘‘normalized partners.’’ We will try to see what effects this uncertain initial entangled state has on the degradation of entanglement for two relatively accelerated observers due to the presence of the initial state parameter α. Notice that the Schwarzschild space-time very close to the event horizon resembles the Rindler space in the infinite-acceleration limit. Hence, as in [6,7] our results in this limit can be applied to discuss the entanglement between two free bosonic or fermionic modes seen by observers when one observer falls into a black hole and the other barely escapes through eternal uniform acceleration.Rindler coordinates are appropriate for describing the viewpoint of an observer moving with uniform acceleration. The world lines of uniformly accelerated observers in the Minkowski coordinates correspond to hyperbolas to the left (region I) and right (region II) of the origin which are bounded by lightlike asymptotes constituting the Rindler horizon [6,7],so the two Rindler regions are causally disconnected from each other [10]. An observer undergoing uniform acceleration remains constrained to either Rindler region I or II and has no access to the other sector. The system in Eq.(1) is bipartite from an inertial perspective, but in a noninertial frame an extra set of modes in region II becomes relevant. Thus, we will study the mixed-state entanglement of the state as seen by an inertial observer Alice detecting the mode s and a uniformly accelerated observer Bob with proper acceleration a in region I detecting the second mode k.Bosonic entanglement. For a free scalar field, the Minkowski vacuum state can be expressed as a two-mode squeezed state in the Rindler frame [8,10],where , k is the wave vector, r is the acceleration parameter, and and indicate the Rindler-region-I particle mode and Rindler-region-II antiparticle mode, respectively. Using Eq.(2) and the first excited state [6,10]we can rewrite Eq.(1) in terms of Minkowski modes for Alice and Rindler modes for Bob. Since Bob is causally disconnected from region II,we will trace over the states in this region and obtainFIG. 1. (Color online) Logarithmic negativity of the bosonic modes versus r for different α.where . The partial transpose criterion provides a sufficient condition for the existence of entanglement in this case [11]: if at least one eigenvalue of the partial transpose is negative, the density matrix is entangled; but a state with positive partial transpose can still be entangled. It is the well-known bound or nondistillable entanglement [12].By interchanging Alice’s qubits, we get the eigenvalues of the partial transpose in the block:where . Obviously the eigenvalue is always negative for finite acceleration (). Hence, this mixed state is always entangled for any finite acceleration of Bob. In the limit , the negative eigenvalue will go to zero. To discuss this further, we will use the logarithmic negativity which serves as an upper bound on the entanglement of distillation [12]. This entanglement monotone is defined as , where is the trace norm of the partial transpose . We therefore findFor vanishing acceleration (), . In the range the larger α, the stronger the initialentanglement; but in the range the larger α, the weaker the initial entanglement. For finite acceleration, the monotonic decrease of with increasing r for different αmeans that the entanglement of the initial state is lost to the thermal fields generated by the Unruh effect. From Fig. 1 it is surprisingly found that the same initial entanglement for αand its normalized partner will be degraded along two different trajectories except for theFIG. 2. (Color online) Mutual information of the bosonic modes versus r for different α.maximally entangled state, i.e., |This phenomenon, due to the coupling of αand the hyperbolic functions related to r, just shows the inequivalence of the quantization for a scalar field in the Minkowski and Rindler coordinates. The logarithmic negativity is exactly zero for any αin the limit , which indicates that the state does not have distillableentanglement.The mutual information, which can be used to estimate the total amount of correlations between any two subsystems of the overall system, is defined as [13]where is the entropy of the density matrix . From Eq.(3), we can obtain the entropy of this joint state,Tracing over Alice’s states for , we get Bob’s density matrix in region I; its entropy isIn the same way, we have Alice’s density matrix by tracing over Bob’s states; its entropy is given byWe draw the behaviors of the mutual information versus r for different αin Fig. 2. For vanishing acceleration, the initial mutual information is . In the range the larger α, the stronger ; but in the range the larger α, the weaker . As the acceleration increases, the mutual information becomes smaller. It is interesting to note that, except for the maximally entangled state, the same initial mu-tual information for αand will be degraded along two different trajectories. However, in the infinite-acceleration limit, the mutual information converges to the same value again, i.e.,FIG. 3. (Color online) Logarithmic negativity of the fermionic modes versus r for different α(notice that ))., which equals just half of. Obviously if is higher, it is degraded to a higher degree in this limit. Since the distillable entanglement in the infinite-acceleration limit is zero, it is safe to say that the total correlations consist of classical correlations plus bound entanglement in this limit.Fermionic entanglement. With the single-mode approximation used by Alsing et al., the fermionic Minkowski vacuum can be written as [7]and the only excited state is given bywhere and the acceleration parameter r is in the range in this case. Using Eqs.(8) and (9) for the Minkowski particle states and and tracing over the modes in region II, we getwith |mn⟩=. The partial transpose criterion provides a necessary and sufficient condition for entanglement in a mixed state of two qubits [11]: if at least one eigenvalue of the partial transpose isnegative, the density matrix is entangled. Interchanging Ali ce’s qubits, we obtain an eigenvalue of the partial transpose ,which is always negative for . Thus, the state is always entangled for any uniform acceleration of Bob. The logarithmic negativity is expressed asFor vanishing acceleration (), . For finite acceleration, the entanglement is degraded by the Unruh effect just as shown in Fig. 3. We find that in the range , for larger , the initial entanglement is higher, but it is not always degraded to a higher degree. It should be noted that for the final entanglement of the initial state is higher than that of the maximally entangled state, i.e., , and forFIG. 4.(Color online)Mutual information of the fermionic modes versus r for different .the maximal final entanglement is . In the range , the larger , the weaker the initial entanglement and the lower the final entanglement. Unlike the behaviors of the bosonic case, except for the maximally entangled state, the same initial entanglement of the fermionic modes for and will be degraded along two different trajectories and asymptotically reach two different nonvanishing minimum values in the infinite-acceleration limit() due to the coupling of α and the trigonometric functions related to r. In the infinite-acceleration limit, which means that the state is always entangled. This is in strong contrast to the bosonic case and shows that the fermionic system can be used as a resource forperforming certain quantum-information-processing tasks.As for the bosonic case, we go through the same process again and get the mutual information for these fermionic modes,whose trajectories versus r for different αare shown by Fig. 4. For vanishing acceleration, the initial mutual information is, whose behaviors are thefor the bosonic modes. The mutual information same as those of Ibibecomes smaller as the acceleration increases, and again we surprisingly find that the same initial mutual information for and will be degraded along two different trajectories except for the maximally entangled state. In the infinite-acceleration limit the mutual information converges to , which is just half of . This behavior is reminiscent of that seen for the bosonic case, so we conclude thatwhich is independent of α and the type of field.It should be noted that, if we set the initial entangled state aswe will have the same behavior of the entanglement degradation for the same α that is shown in Figs. 1–4.Summarizing, the entanglement of the scalar and Dirac fields in noninertial frames is degraded by the Unruh effect as Bob’s rate of acceleration increases, but their behaviors of the degradation of entanglement are different for the same initial state parameter α. It is surprisingly found that the same initial entanglement for α andwill be degraded along two different trajectories except for the maximally entangled state, which just shows the inequivalence of the quantization for a free field in Minkowski and Rindler coordinates. In the infinite-acceleration limit, which can be applied to the case of Alice falling into a black hole while Bob barely escapes, the state does not have distillable entanglement for any α for the scalar field but always remains entangled to a degree that is dependent on α for the Dirac field. It should be noted that, for , we will have the maximal final entanglement for the fermionic state in this limit. Further analysis shows that, with increasing acceleration parameter r, the mutual information is degraded to a nonvanishing minimum value which is dependent on α forthese two fields. However, it is interesting to note that the mutual information in the infinite-acceleration limit equals just half of the initial mutual information, independently of α and the type of field.This work was supported by the National Natural Science Foundation of China under Grant No. 10675045; the FANEDD under Grant No. 200317; the Hunan Provincial Natural Science Foundation of China under Grant No. 07A0128; and the construct program of key disciplines in Hunan Province.References[1] A. Peres and D. R. Terno, Rev. Mod. Phys. 76, 93 (2004); D.Boschi, S. Branca, F. De Martini, L. Hardy, and S. Popescu, Phys. Rev. Lett. 80, 1121 (1998); J. W. Pan, C. Simon, C.Brukner, and A. Zeilinger, Nature (London) 410, 1067 (2001); S. J. van Enk and T. Rudolph, Quantum Inf. Comput. 3, 423 (2003).[2] D. Bouwmeester, A. Ekert, and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag, Berlin, 2000).[3] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev.D 34, 373 (1986); C. 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