量子纠缠

此词条暂由Henry翻译。

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Spontaneous parametric down-conversion process can split photons into type II photon pairs with mutually perpendicular polarization.

Spontaneous parametric down-conversion process can split photons into type II photon pairs with mutually perpendicular polarization.

[自发参量下转换过程可以将光子分裂成具有相互垂直极化的 II 型光子对。]

Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

量子纠缠是一种物理现象，描述的是当一对或一组粒子被产生、相互作用或共享空间邻近性时（包括当粒子被大距离分离时），该对或该组粒子中的每个粒子的量子态都无法独立于其他粒子的态。量子纠缠是经典物理学和量子物理学之间差别悬殊的核心问题：纠缠是量子力学的一个主要特征，而经典力学却没有这种特征。

Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, will be found to be counterclockwise. However, this behavior gives rise to seemingly paradoxical effects: any measurement of a property of a particle results in an irreversible wave function collapse of that particle and will change the original quantum state. In the case of entangled particles, such a measurement will affect the entangled system as a whole.

Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, will be found to be counterclockwise. However, this behavior gives rise to seemingly paradoxical effects: any measurement of a property of a particle results in an irreversible wave function collapse of that particle and will change the original quantum state. In the case of entangled particles, such a measurement will affect the entangled system as a whole.

在某些情况下，对纠缠粒子的位置、动量、自旋和偏振等物理性质的测量的结果可以是完全相关的。例如，如果一对纠缠粒子的产生使得它们的总自旋已知为零，并且我们发现一个粒子在第一个轴上具有顺时针自旋，那么在同一个轴上测量的另一个粒子的自旋将会是逆时针的。然而，这种行为产生了看似矛盾的效应：对粒子性质的任何测量都会导致该粒子的不可逆波函数崩溃，并将改变原来的量子态。在粒子纠缠的情况下，这样的测量将影响整个纠缠系统。

Such phenomena were the subject of a 1935 paper by Albert Einstein, Boris Podolsky, and Nathan Rosen, and several papers by Erwin Schrödinger shortly thereafter, describing what came to be known as the EPR paradox. Einstein and others considered such behavior to be impossible, as it violated the local realism view of causality (Einstein referring to it as "spooky action at a distance") and argued that the accepted formulation of quantum mechanics must therefore be incomplete.

Later, however, the counterintuitive predictions of quantum mechanics were verified experimentally in tests in which polarization or spin of entangled particles were measured at separate locations, statistically violating Bell's inequality. In earlier tests, it couldn't be absolutely ruled out that the test result at one point could have been subtly transmitted to the remote point, affecting the outcome at the second location. However, so-called "loophole-free" Bell tests have been performed in which the locations were separated such that communications at the speed of light would have taken longer--in one case 10,000 times longer—than the interval between the measurements.

The special property of entanglement can be better observed if we separate the said two particles. Let's put one of them in the White House in Washington and the other in Buckingham Palace (think about this as a thought experiment, not an actual one). Now, if we measure a particular characteristic of one of these particles (say, for example, spin), get a result, and then measure the other particle using the same criterion (spin along the same axis), we find that the result of the measurement of the second particle will match (in a complementary sense) the result of the measurement of the first particle, in that they will be opposite in their values.

如果将这两种粒子分开，可以更好地观察到纠缠的特性。让我们把其中一个放在华盛顿的白宫，另一个放在白金汉宫。现在，如果我们测量其中一个粒子的特性(比如自旋) ，得到一个结果，然后用同样的标准(沿着同样的轴自旋)测量另一个粒子，我们发现第二个粒子的测量结果将匹配(在补充意义上)第一个粒子的测量结果，因为它们的值将相反。

According to some interpretations of quantum mechanics, the effect of one measurement occurs instantly. Other interpretations which don't recognize wavefunction collapse dispute that there is any "effect" at all. However, all interpretations agree that entanglement produces correlation between the measurements and that the mutual information between the entangled particles can be exploited, but that any transmission of information at faster-than-light speeds is impossible.

根据“一些”量子力学的解释，一次测量的效果瞬间发生。其他不承认波函数崩溃的解释则认为存在任何“效应”。然而，所有的解释都同意，纠缠在测量值之间产生了“相关”，并且纠缠粒子之间的互信息可以被利用，但是任何以高于光速的信息“传输”都是不可能的。

The above result may or may not be perceived as surprising. A classical system would display the same property, and a hidden variable theory (see below) would certainly be required to do so, based on conservation of angular momentum in classical and quantum mechanics alike. The difference is that a classical system has definite values for all the observables all along, while the quantum system does not. In a sense to be discussed below, the quantum system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measurement of the first particle. This may certainly be perceived as surprising in the case of spatially separated entangled particles.

上述结果可能会或不会被认为是令人惊讶的。一个经典系统也会表现出同样的性质，而一个隐藏变量理论（见下文）肯定会被要求这样做，它建立在经典力学和量子力学的角动量守恒的基础上。不同的是，一个经典系统对所有的可观测值都有确定的值，而量子系统则没有。在下文将要讨论的意义上，这里所考虑的量子系统似乎在测量第一个粒子时获得了沿另一粒子的任何轴的自旋测量结果的概率分布。这个概率分布通常不同于不测量第一个粒子时的概率分布。对于空间分离的纠缠粒子来说，这无疑是令人惊讶的。

Quantum entanglement has been demonstrated experimentally with photons, neutrinos, electrons, molecules as large as buckyballs, and even small diamonds. The utilization of entanglement in communication, computation and quantum radar is a very active area of research and development.

The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel faster than light) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the Copenhagen interpretation, the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured.

矛盾之处在于，对任一粒子的测量显然会使整个纠缠系统的状态崩溃，而且会瞬间崩溃，在关于测量结果的任何信息可以被传送到另一个粒子之前（假设信息不能比光传播得快），因此确保纠缠对的另一部分的测量结果是“正确的”。在哥本哈根解释中，对其中一个粒子的自旋测量的结果是坍缩成一种状态，其中每个粒子沿测量轴都有一个确定的自旋（向上或向下）。结果是随机的，每种可能性的概率为50%。然而，如果两个自旋沿同一轴测量，就会发现它们是反相关的。这意味着，对一个粒子进行测量的随机结果似乎已经传递给了另一个粒子，因此当它也被测量时，它可以做出“正确的选择”。

History 历史

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Article headline regarding the Einstein–Podolsky–Rosen paradox (EPR paradox) paper, in the May 4, 1935 issue of The New York Times.

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements spacelike, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of special relativity, it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events and there are inertial frames in which is first and others in which is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect.

我们可以选择测量的距离和时间，以便使两次测量之间的间隔像空间一样，因此，连接事件的任何因果效应都必须比光传播得更快。根据狭义相对论的原理，任何信息都不可能在两个这样的测量事件之间传递。甚至不可能说哪个测量值是第一个。对于两个分离的类空事件，存在惯性系，有惯性系在其中是第一位的，也有其他惯性系在其中是第一位的。因此，这两种测量之间的相关性不能解释为一种测量决定另一种测量：不同的观察者会对因果关系的作用产生分歧。

The counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. 1935年阿尔伯特 爱因斯坦与鲍里斯 波多斯基和纳兰 罗森在一篇联合论文中首次讨论了关于强关联系统的量子力学的反直觉预测。 (In fact similar paradoxes can arise even without entanglement: the position of a single particle is spread out over space, and two widely separated detectors attempting to detect the particle in two different places must instantaneously attain appropriate correlation, so that they do not both detect the particle.)

（事实上，即使没有纠缠，也会出现类似的悖论：单个粒子的位置分布在空间上，两个试图在两个不同位置检测粒子的大范围分离的探测器必须立即获得适当的相关性，这样它们就不会同时检测到粒子。） In this study, the three formulated the Einstein–Podolsky–Rosen paradox (EPR paradox), a thought experiment that attempted to show that quantum mechanical theory was incomplete. They wrote: "We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete." 在这项研究中，三人提出了爱因斯坦-波多尔斯基-罗森悖论（EPR悖论），一个思维实验，试图证明量子力学理论是不完全性。他们写道：“因此，我们被迫得出结论，波函数给出的物理实在的量子力学描述并不完整。”

However, the three scientists did not coin the word entanglement, nor did they generalize the special properties of the state they considered. Following the EPR paper, Erwin Schrödinger wrote a letter to Einstein in German in which he used the word Verschränkung (translated by himself as entanglement) "to describe the correlations between two particles that interact and then separate, as in the EPR experiment." 然而，这三位科学家并没有创造“纠缠”这个词，也没有概括出他们所考虑的状态的特殊性质。在EPR论文发表之后，埃尔温·薛定谔用德语给爱因斯坦写了一封信，信中他用“Verschränkung”（他自己翻译为“纠缠”）一词来描述两个相互作用然后分离的粒子之间的关联，就像EPR实验中那样。” A possible resolution to the paradox is to assume that quantum theory is incomplete, and the result of measurements depends on predetermined "hidden variables". The state of the particles being measured contains some hidden variables, whose values effectively determine, right from the moment of separation, what the outcomes of the spin measurements are going to be. This would mean that each particle carries all the required information with it, and nothing needs to be transmitted from one particle to the other at the time of measurement. Einstein and others (see the previous section) originally believed this was the only way out of the paradox, and the accepted quantum mechanical description (with a random measurement outcome) must be incomplete.

解决这一悖论的一个可能办法是假设量子理论是不完整的，测量结果取决于预先确定的“隐藏变量”。被测粒子的状态包含一些隐藏的变量，这些变量的值从分离的那一刻起就有效地决定了自旋测量的结果。这就意味着每个粒子都携带着所需的全部信息，在测量时不需要从一个粒子传输到另一个粒子。爱因斯坦和其他人（见上一节）最初认为这是摆脱悖论的唯一途径，而公认的量子力学描述（带有随机测量结果）肯定是不完整的。

Schrödinger shortly thereafter published a seminal paper defining and discussing the notion of "entanglement." In the paper, he recognized the importance of the concept, and stated: "I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought." 此后不久，薛定谔发表了一篇开创性的论文，对“纠缠”的概念进行了定义和讨论。在论文中，他认识到了这个概念的重要性，并指出：“我不会将[纠缠]称为‘一’，而是称之为[量子力学]的‘特性’。”，它完全背离了经典的思路。”

Local hidden variable theories fail, however, when measurements of the spin of entangled particles along different axes are considered. If a large number of pairs of such measurements are made (on a large number of pairs of entangled particles), then statistically, if the local realist or hidden variables view were correct, the results would always satisfy Bell's inequality. A number of experiments have shown in practice that Bell's inequality is not satisfied. However, prior to 2015, all of these had loophole problems that were considered the most important by the community of physicists. When measurements of the entangled particles are made in moving relativistic reference frames, in which each measurement (in its own relativistic time frame) occurs before the other, the measurement results remain correlated.

然而，当考虑沿不同轴的纠缠粒子自旋的测量时，局部隐变量理论是失败的。如果进行了大量成对的此类测量（在大量成对的纠缠粒子上），那么在统计上，如果局部现实主义或隐藏变量的观点是正确的，结果将始终满足贝尔不等式。大量的实验表明，贝尔不等式在实践中是不成立的。然而，在2015年之前，被物理学家群体认为是最关键的是所有这些实践都有漏洞问题，。当在运动的相对论参考系中对纠缠粒子进行测量时，每个测量（在它自己的相对论时间范围内）都发生在另一个之前，测量结果将保持相关。

Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, because it seemed to violate the speed limit on the transmission of information implicit in the theory of relativity. Einstein later famously derided entanglement as "spukhafte Fernwirkung" or "spooky action at a distance."

The fundamental issue about measuring spin along different axes is that these measurements cannot have definite values at the same time―they are incompatible in the sense that these measurements' maximum simultaneous precision is constrained by the uncertainty principle. This is contrary to what is found in classical physics, where any number of properties can be measured simultaneously with arbitrary accuracy. It has been proven mathematically that compatible measurements cannot show Bell-inequality-violating correlations, and thus entanglement is a fundamentally non-classical phenomenon.

沿不同轴线测量自旋的基本问题是，这些测量不可能同时具有确定的值——它们是不相容的，因为这些测量的最大同时精度受到不确定性原理的限制。这与经典物理学中的发现相反，在经典物理学中，任何数量的性质都可以以任意精度同时测量。从数学上证明了相容测量不能显示违反贝尔不等式的关联，因此纠缠是一个基本的非经典现象。

The EPR paper generated significant interest among physicists, which inspired much discussion about the foundations of quantum mechanics (perhaps most famously Bohm's interpretation of quantum mechanics), but produced relatively little other published work. Despite the interest, the weak point in EPR's argument was not discovered until 1964, when John Stewart Bell proved that one of their key assumptions, the principle of locality, as applied to the kind of hidden variables interpretation hoped for by EPR, was mathematically inconsistent with the predictions of quantum theory. EPR的论文引起了物理学家的极大兴趣，激发了许多关于量子力学基础的讨论（也许最著名的是量子力学的 Bohm表达），但其他发表的著作相对较少。尽管有人对此感兴趣，但直到1964年，约翰·斯图尔特·贝尔证明了他们的一个关键假设，局域性原理，即应用于EPR希望解释的隐藏变量，在数学上与量子理论的预测不一致时，EPR论点中的漏洞才被发现。

纠缠是保持不确定性原理所必需的，如 EPR 悖论所示。例如，假设一个高能光子衰变成一个电子/正电子对，然后测量电子的位置和正电子的动量。如果我们在物理描述中不允许纠缠，那么每个粒子的位置和动量就可以通过参考动量守恒来推导，这就违反了测不准原理。或者，如果我们要求不确定性原理保持真实，而仍然不允许在物理上描述对的纠缠，不确定性原理将会违反动量守恒定律，因为在位置和动量上强相关性是不可能的(也就是说，人们不能有效地推断电子的位置和动量，因为它们不能与正电子的位置和动量高度相关)。-->

Specifically, Bell demonstrated an upper limit, seen in Bell's inequality, regarding the strength of correlations that can be produced in any theory obeying local realism, and showed that quantum theory predicts violations of this limit for certain entangled systems. His inequality is experimentally testable, and there have been numerous relevant experiments, starting with the pioneering work of Stuart Freedman and John Clauser in 1972 and Alain Aspect's experiments in 1982. An early experimental breakthrough was due to Carl Kocher, who already in 1967 presented an apparatus in which two photons successively emitted from a calcium atom were shown to be entangled – the first case of entangled visible light. The two photons passed diametrically positioned parallel polarizers with higher probability than classically predicted but with correlations in quantitative agreement with quantum mechanical calculations. He also showed that the correlation varied only upon (as cosine square of) the angle between the polarizer settings and decreased exponentially with time lag between emitted photons. Kocher’s apparatus, equipped with better polarizers, was used by Freedman and Clauser who could confirm the cosine square dependence and use it to demonstrate a violation of Bell’s inequality for a set of fixed angles. All these experiments have shown agreement with quantum mechanics rather than the principle of local realism.

For decades, each had left open at least one loophole by which it was possible to question the validity of the results. However, in 2015 an experiment was performed that simultaneously closed both the detection and locality loopholes, and was heralded as "loophole-free"; this experiment ruled out a large class of local realism theories with certainty. Alain Aspect notes that the "setting-independence loophole" – which he refers to as "far-fetched", yet, a "residual loophole" that "cannot be ignored" – has yet to be closed, and the free-will / superdeterminism loophole is unclosable; saying "no experiment, as ideal as it is, can be said to be totally loophole-free."

In experiments in 2012 and 2013, polarization correlation was created between photons that never coexisted in time. The authors claimed that this result was achieved by entanglement swapping between two pairs of entangled photons after measuring the polarization of one photon of the early pair, and that it proves that quantum non-locality applies not only to space but also to time.

在2012年和2013年的实验中，在时间上从未共存的光子之间产生了偏振关联。作者认为，这一结果是在测量了一对纠缠光子的偏振态后，通过两对纠缠光子之间的纠缠交换得到的，证明了量子非定域性不仅适用于空间，也适用于时间。

A minority opinion holds that although quantum mechanics is correct, there is no superluminal instantaneous action-at-a-distance between entangled particles once the particles are separated.

In three independent experiments in 2013 it was shown that classically communicated separable quantum states can be used to carry entangled states. The first loophole-free Bell test was held in TU Delft in 2015 confirming the violation of Bell inequality.

2013年的三个独立实验表明，经典通信的可分离量子态可以用来携带纠缠态。第一次无漏洞贝尔试验于2015年在图代尔夫特举行，证实了贝尔不等式的不成立。

Bell's work raised the possibility of using these super-strong correlations as a resource for communication. It led to the 1984 discovery of quantum key distribution protocols, most famously BB84 by Charles H. Bennett and Gilles Brassard and E91 by Artur Ekert. Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

In August 2014, Brazilian researcher Gabriela Barreto Lemos and team were able to "take pictures" of objects using photons that had not interacted with the subjects, but were entangled with photons that did interact with such objects. Lemos, from the University of Vienna, is confident that this new quantum imaging technique could find application where low light imaging is imperative, in fields like biological or medical imaging.

2014年8月，巴西研究人员加布里埃拉·巴雷托·莱莫斯和他的团队能够使用光子“拍摄”物体，这些光子并没有与实验对象发生相互作用，而是与这些物体发生了纠缠。来自维也纳大学的勒莫斯相信，这种新的量子成像技术可以在微光成像势在必行的领域找到应用，比如生物或医学成像。

Concept 概念

In 2015, Markus Greiner's group at Harvard performed a direct measurement of Renyi entanglement in a system of ultracold bosonic atoms.

2015年，哈佛大学的马克斯·格雷纳团队直接测量了超冷玻色子原子系统中的Renyi纠缠。

Meaning of entanglement纠缠的意义

From 2016 various companies like IBM, Microsoft etc. have successfully created quantum computers and allowed developers and tech enthusiasts to openly experiment with concepts of quantum mechanics including quantum entanglement.

从2016年起，IBM、微软等多家公司成功创建了量子计算机，并允许开发人员和技术爱好者公开实验量子力学的概念，这其中就包括量子纠缠。

An entangled system is defined to be one whose quantum state cannot be factored as a product of states of its local constituents; that is to say, they are not individual particles but are an inseparable whole. In entanglement, one constituent cannot be fully described without considering the other(s). The state of a composite system is always expressible as a sum, or superposition, of products of states of local constituents; it is entangled if this sum necessarily has more than one term. 纠缠系统被定义为其量子态不能被分解为其局部成分的态的乘积；也就是说，它们不是单个粒子，而是一个不可分割的整体。在纠缠中，一个组分不能在不考虑其他组分的情况下被完全描述。复合系统的状态总是可以表示为局部成分的状态积的和，或叠加，如果这个和一定有一个以上的项，那么它是纠缠的。

Quantum systems can become entangled through various types of interactions. For some ways in which entanglement may be achieved for experimental purposes, see the section below on methods. Entanglement is broken when the entangled particles decohere through interaction with the environment; for example, when a measurement is made. 量子系统可以通过各种类型的相互作用而纠缠在一起。为了实验目的而实现纠缠的一些方法，请参见下面关于方法的部分。当纠缠粒子通过与环境的相互作用退相干时，例如在进行测量时，纠缠将被打破。

There have been suggestions to look at the concept of time as an emergent phenomenon that is a side effect of quantum entanglement.

有人建议把时间的概念看作是量子纠缠的副作用的一种自然现象。

In other words, time is an entanglement phenomenon, which places all equal clock readings (of correctly prepared clocks, or of any objects usable as clocks) into the same history. This was first fully theorized by Don Page and William Wootters in 1983.

换句话说，时间是一种纠缠现象，它将所有相等的时钟读数（正确准备的时钟或任何可用作时钟的物体的读数）放入同一个历史中。1983年，唐·佩奇和威廉·伍特斯首次提出了这一理论

As an example of entanglement: a subatomic particle decays into an entangled pair of other particles. The decay events obey the various conservation laws, and as a result, the measurement outcomes of one daughter particle must be highly correlated with the measurement outcomes of the other daughter particle (so that the total momenta, angular momenta, energy, and so forth remains roughly the same before and after this process). For instance, a spin-zero particle could decay into a pair of spin-½ particles. Since the total spin before and after this decay must be zero (conservation of angular momentum), whenever the first particle is measured to be spin up on some axis, the other, when measured on the same axis, is always found to be spin down. (This is called the spin anti-correlated case; and if the prior probabilities for measuring each spin are equal, the pair is said to be in the singlet state.) 作为纠缠的一个例子：一个亚原子粒子衰变变成一对纠缠的其他粒子。衰变事件遵循各种守恒定律，因此，一个子粒子的测量结果必须与另一个子粒子的测量结果高度相关（因此总动量、角动量、能量等在此过程前后保持大致相同）。例如，自旋-零粒子可以衰变为一对自旋为½的粒子。由于衰变前后的总自旋必须为零（角动量守恒），每当第一个粒子在某个轴上被测量为自旋向上，另一个粒子在同一个轴上被测量时，总是被发现为自旋向下。（这称为自旋反相关情况；如果测量每个自旋的先验概率相等，则称成对处于单态。） The Wheeler–DeWitt equation that combines general relativity and quantum mechanics – by leaving out time altogether – was introduced in the 1960s and it was taken up again in 1983, when Page and Wootters made a solution based on quantum entanglement. Page and Wootters argued that entanglement can be used to measure time.

20世纪60年代，惠勒-德威特方程引入了广义相对论和量子力学的概念，并于1983年再次引入，当时佩奇和伍特基于量子纠缠方程提出了一个解决方案。佩奇和伍特斯认为纠缠态可以用来测量时间。

The special property of entanglement can be better observed if we separate the said two particles. Let's put one of them in the White House in Washington and the other in Buckingham Palace (think about this as a thought experiment, not an actual one). Now, if we measure a particular characteristic of one of these particles (say, for example, spin), get a result, and then measure the other particle using the same criterion (spin along the same axis), we find that the result of the measurement of the second particle will match (in a complementary sense) the result of the measurement of the first particle, in that they will be opposite in their values. 将这两个粒子分开，可以更好地观察到纠缠的特殊性质。让我们把其中一个放在华盛顿的白宫，另一个放在白金汉宫（把这当成一个思维实验，而不是实际的实验）。现在，如果我们测量其中一个粒子的特定特性（例如，自旋），得到一个结果，然后使用相同的标准测量另一个粒子（沿相同的轴自旋），我们发现第二个粒子的测量结果将与第一个粒子的测量结果相匹配（在互补意义上）粒子，因为它们的值是相反的 In 2013, at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Turin, Italy, researchers performed the first experimental test of Page and Wootters' ideas. Their result has been interpreted to confirm that time is an emergent phenomenon for internal observers but absent for external observers of the universe just as the Wheeler-DeWitt equation predicts. The approach to entanglement would be from the perspective of the causal arrow of time, with the assumption that the cause of the measurement of one particle determines the effect of the result of the other particle's measurement.

2013年，在意大利都灵的国家理查尔卡计量研究所(INRIM) ，研究人员对佩奇和伍特的想法进行了首次实验测试。他们的结果被解释为证实了对于内部观察者来说时间是一种涌现的现象，但正如惠勒-德威特方程所预测的那样，对于宇宙的外部观察者来说时间是不存在的。纠缠的方法是从因果时间箭头的角度出发，假设一个粒子被测量的原因决定了另一个粒子测量结果的效应。

The above result may or may not be perceived as surprising. A classical system would display the same property, and a hidden variable theory (see below) would certainly be required to do so, based on conservation of angular momentum in classical and quantum mechanics alike. The difference is that a classical system has definite values for all the observables all along, while the quantum system does not. In a sense to be discussed below, the quantum system considered here seems to acquire a probability distribution for the outcome of a measurement of the spin along any axis of the other particle upon measurement of the first particle. This probability distribution is in general different from what it would be without measurement of the first particle. This may certainly be perceived as surprising in the case of spatially separated entangled particles. 上述结果可能会或不会被认为是令人惊讶的。一个经典系统将显示出相同的性质，而隐藏变量理论（见下文）肯定需要这样做，基于经典和量子力学中的角动量守恒。不同的是，一个经典系统对所有的可观测值都有确定的值，而量子系统则没有。在下文将要讨论的意义上，这里所考虑的量子系统似乎在测量第一个粒子时获得了沿另一粒子的任何轴的自旋测量结果的概率分布。这个概率分布通常不同于不测量第一个粒子时的概率分布。对于空间分离的纠缠粒子来说，这无疑是令人惊讶的。

Paradox矛盾

Based on AdS/CFT correspondence, Mark Van Raamsdonk suggested that spacetime arises as an emergent phenomenon of the quantum degrees of freedom that are entangled and live in the boundary of the space-time. Induced gravity can emerge from the entanglement first law.

基于AdS/CFT对应关系， Mark Van Raamsdonk提出时空是量子自由度的一种涌现现象，量子自由度纠缠在时空的边界上。诱导引力可以从纠缠第一定律中产生。

The paradox is that a measurement made on either of the particles apparently collapses the state of the entire entangled system—and does so instantaneously, before any information about the measurement result could have been communicated to the other particle (assuming that information cannot travel faster than light) and hence assured the "proper" outcome of the measurement of the other part of the entangled pair. In the Copenhagen interpretation, the result of a spin measurement on one of the particles is a collapse into a state in which each particle has a definite spin (either up or down) along the axis of measurement. The outcome is taken to be random, with each possibility having a probability of 50%. However, if both spins are measured along the same axis, they are found to be anti-correlated. This means that the random outcome of the measurement made on one particle seems to have been transmitted to the other, so that it can make the "right choice" when it too is measured. 矛盾之处在于，对任一粒子的测量显然会使整个纠缠系统的状态崩溃，而且会瞬间崩溃，在关于测量结果的任何信息可以被传送到另一个粒子之前（假设信息不能传播比光更快），从而确保纠缠对的另一部分的测量的“正确”结果。在哥本哈根解释中，其中一个粒子的自旋测量结果是坍缩成一种状态，在这种状态下，每个粒子沿测量轴都有一个确定的自旋（向上或向下）。结果是随机的，每种可能性的概率为50%。然而，如果两个自旋沿同一轴测量，就会发现它们是反相关的。这意味着，对一个粒子进行测量的随机结果似乎已经传递给了另一个粒子，因此当它也被测量时，它可以做出“正确的选择”。

The distance and timing of the measurements can be chosen so as to make the interval between the two measurements spacelike, hence, any causal effect connecting the events would have to travel faster than light. According to the principles of special relativity, it is not possible for any information to travel between two such measuring events. It is not even possible to say which of the measurements came first. For two spacelike separated events x₁ and x₂ there are inertial frames in which x₁ is first and others in which x₂ is first. Therefore, the correlation between the two measurements cannot be explained as one measurement determining the other: different observers would disagree about the role of cause and effect. 可以选择测量的距离和时间，以便使两次测量之间的间隔类太空，因此，任何与事件相关的因果效应都必须比光传播得更快。根据狭义相对论的原理，任何信息不可能在两个这样的测量事件之间传递。甚至不可能说哪个测量值是第一个。对于两个类空分离事件x'₁}和{math 和{mvar | B}，分别具有希尔伯特空间s{mvar | H_A}和{mvar | H_B}。复合系统的Hilbert空间是张量积 Following the definition above, for a bipartite composite system, mixed states are just density matrices on . That is, it has the general form

根据上面的定义，对于二部复合系统，混合态仅仅是上面的密度矩阵。也就是说，它有一般的形式

[math]\displaystyle{ H_A \otimes H_B. }[/math]

[math]\displaystyle{ \rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right]

[数学] rho = sum { i } w _ i 左[ sum _ { j } bar { c }{ ij }(| alpha _ { ij } rangle otimes | beta _ { ij } rangle)右]左[ sum _ k c _ { ik }(langle alpha _ ik } | otimes langle beta _ { ik } | 右]



 }[/math]

数学

If the first system is in state [math]\displaystyle{ \scriptstyle| \psi \rangle_A }[/math] and the second in state [math]\displaystyle{ \scriptstyle| \phi \rangle_B }[/math], the state of the composite system is

如果第一个系统处于状态[math]\displaystyle{ \scriptstyle |\psi\rangle_A }[/math]，第二个系统处于状态[math]\displaystyle{ \scriptstyle |\phi\rangle_B }[/math]，则复合系统的状态为

where the w_i are positively valued probabilities, [math]\displaystyle{ \sum_j |c_{ij}|^2=1 }[/math], and the vectors are unit vectors. This is self-adjoint and positive and has trace 1.

其中w_i是正值概率，[math]\displaystyle{ \sum|u j | c|ij}|^2=1 }[/math]，向量是单位向量。这是自伴正的，有迹1。

[math]\displaystyle{ |\psi\rangle_A \otimes |\phi\rangle_B. }[/math]

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

从纯粹情形扩展可分性的定义，我们说混合状态是可分的，如果它可以写成

States of the composite system that can be represented in this form are called separable states, or product states.

可以用这种形式表示的复合系统的状态称为可分离状态s或产品状态。

[math]\displaystyle{ \rho = \sum_i w_i \rho_i^A \otimes \rho_i^B,  }[/math]

(数学) rho = sum i w i rho i ^ a times rho i ^ b，(数学)

Not all states are separable states (and thus product states). Fix a basis [math]\displaystyle{ \scriptstyle \{|i \rangle_A\} }[/math] for H_A and a basis [math]\displaystyle{ \scriptstyle \{|j \rangle_B\} }[/math] for H_B. The most general state in H_A ⊗ H_B is of the form

where the are positively valued probabilities and the [math]\displaystyle{ \rho_i^A }[/math]'s and [math]\displaystyle{ \rho_i^B }[/math]'s are themselves mixed states (density operators) on the subsystems and respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that [math]\displaystyle{ \rho_i^A }[/math] and [math]\displaystyle{ \rho_i^B }[/math] are themselves pure ensembles. A state is then said to be entangled if it is not separable.

其中为正值概率和[math]\displaystyle{ \rho i^A }[/math]和[math]\displaystyle{ \rho i^B }[/math]分别为子系统和上的混合态（密度算子）。换句话说，如果一个状态是不相关状态或乘积状态的概率分布，那么它是可分离的。通过将密度矩阵写成纯系综的和并展开，我们可以假定[math]\displaystyle{ \rho i^A }[/math]和[math]\displaystyle{ \rho i^B }[/math]本身就是纯系综。如果一个态是不可分离的，它就被称为纠缠态。

[math]\displaystyle{ |\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B }[/math].

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard. For the and cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.

一般来说，要判断一个混合态是否是纠缠态是很困难的。一般的二部格被证明是NP-困难的。对于和种情形，利用著名的正偏转子(PPT)条件给出了可分性的一个充要条件。

This state is separable if there exist vectors [math]\displaystyle{ \scriptstyle [c^A_i], [c^B_j] }[/math] so that [math]\displaystyle{ \scriptstyle c_{ij}= c^A_ic^B_j, }[/math] yielding [math]\displaystyle{ \scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A }[/math] and [math]\displaystyle{ \scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B. }[/math] It is inseparable if for any vectors [math]\displaystyle{ \scriptstyle [c^A_i],[c^B_j] }[/math] at least for one pair of coordinates [math]\displaystyle{ \scriptstyle c^A_i,c^B_j }[/math] we have [math]\displaystyle{ \scriptstyle c_{ij} \neq c^A_ic^B_j. }[/math] If a state is inseparable, it is called an 'entangled state'.

如果存在向量[math]\displaystyle{ \scriptstyle[c^A\u i]，[c^B\u j] }[/math]，则此状态是可分离的，因此[math]\displaystyle{ \scriptstyle c\u{ij}=c^A\u ic^B\u j， }[/math]产生[math]\displaystyle{ \scriptstyle |\psi\rangle | A=\sum{i}c^A{i}i\rangle | A }[/math]和[math]\displaystyle{ \scriptstyle |\phi\rangle | B=\sum{j}c^B{j\rangle B. }[/math]对于任何向量[math]\displaystyle{ \scriptstyle[c^A | i]，[c^B | j] }[/math]至少对于一对坐标[math]\displaystyle{ \scriptstyle c^A | i，如果一个态是不可分的，它就叫做“纠缠态” For example, given two basis vectors \lt math\gt \scriptstyle \{|0\rangle_A, |1\rangle_A\} }[/math] of H_A and two basis vectors [math]\displaystyle{ \scriptstyle \{|0\rangle_B, |1\rangle_B\} }[/math] of H_B, the following is an entangled state:

The idea of a reduced density matrix was introduced by Paul Dirac in 1930. Consider as above systems and each with a Hilbert space . Let the state of the composite system be

约化密度矩阵的概念是由保罗·狄拉克在1930年提出的。考虑以上系统，每个系统都有一个希尔伯特空间。设复合系统的状态为

[math]\displaystyle{ \tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right ). }[/math]

[math]\displaystyle{  |\Psi \rangle \in H_A \otimes H_B.  }[/math]

[数学] | Psi 在 h _ a 和 h _ b 之间。数学

If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state. Another way to say this is that while the von Neumann entropy of the whole state is zero (as it is for any pure state), the entropy of the subsystems is greater than zero. In this sense, the systems are "entangled". This has specific empirical ramifications for interferometry. The above example is one of four Bell states, which are (maximally) entangled pure states (pure states of the H_A ⊗ H_B space, but which cannot be separated into pure states of each H_A and H_B).

As indicated above, in general there is no way to associate a pure state to the component system . However, it still is possible to associate a density matrix. Let

如上所述，通常没有办法将纯状态关联到组件系统。然而，仍然有可能将密度矩阵联系起来。让

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If in the entangled state given above Alice makes a measurement in the [math]\displaystyle{ \scriptstyle \{|0\rangle, |1\rangle\} }[/math] eigenbasis of A, there are two possible outcomes, occurring with equal probability:

[math]\displaystyle{ \rho_T = |\Psi\rangle \; \langle\Psi| }[/math].

1. Alice measures 0, and the state of the system collapses to [math]\displaystyle{ \scriptstyle |0\rangle_A |1\rangle_B }[/math].

Alice测量0，系统的状态将塌陷为[math]\displaystyle{ \scriptstyle | 0\rangle|A | 1\rangle|B }[/math] which is the projection operator onto this state. The state of is the partial trace of over the basis of system :

也就是这个状态的投影操作符。状态是系统基础上的部分轨迹:

1. Alice measures 1, and the state of the system collapses to [math]\displaystyle{ \scriptstyle |1\rangle_A |0\rangle_B }[/math].

Alice测量1，系统的状态将塌陷为[math]\displaystyle{ \scriptstyle | 1\rangle | A | 0\rangle | B }[/math]。

[math]\displaystyle{ \rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T. }[/math]

(| Psi rangle langle Psi | right) | j rangle b = hbox { Tr } _ b; rho _ t. </math >

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1. If the latter occurs, (Alice measures 1) then Bob's measurement will return 0 with certainty. Thus, system B has been altered by Alice performing a local measurement on system A. This remains true even if the systems A and B are spatially separated. This is the foundation of the EPR paradox. 如果发生了前者，那么Bob在相同的基础上执行的任何后续测量都将始终返回1。如果出现后者，（Alice测量1），那么Bob的测量值肯定会返回0。因此，通过Alice对系统{mvar | a}执行本地测量，系统{mvar | B}已经改变。即使系统{mvar | A}}和{mvar | B}在空间上是分开的，这仍然是正确的。这是[EPR悖论]的基础。

is sometimes called the reduced density matrix of  on subsystem . Colloquially, we "trace out" system  to obtain the reduced density matrix on .

有时被称为子系统的约化密度矩阵。通俗地说，我们“追踪”系统，以获得约化密度矩阵。

The outcome of Alice's measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. Causality is thus preserved, in this particular scheme. For the general argument, see no-communication theorem. 爱丽丝的测量结果是随机的。Alice无法决定将复合系统折叠到哪个状态，因此无法通过操作她的系统将信息传输给Bob。因此，在这个特殊的方案中，因果关系得以保留。关于一般的论点，请参见无通信定理。

For example, the reduced density matrix of for the entangled state

例如，纠缠态的约化密度矩阵

Ensembles集成

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has less information about the system, then one calls it an 'ensemble' and describes it by a density matrix, which is a positive-semidefinite matrix, or a trace class when the state space is infinite-dimensional, and has trace 1. Again, by the spectral theorem, such a matrix takes the general form: 如上所述，量子系统的状态由希尔伯特空间中的单位向量给出。更一般地说，如果系统的信息较少，则称之为“系综”，并用密度矩阵来描述，它是半正定矩阵，或迹类，当状态空间是无限维的，且有迹1时。同样，根据谱定理，这样的矩阵具有一般形式：

[math]\displaystyle{ \tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right), }[/math]

左(| 0 rangle _ a otimes | 1 rangle _ b-| 1 rangle _ a otimes | 0 rangle _ b right) ，</math >

[math]\displaystyle{ \rho = \sum_i w_i |\alpha_i\rangle \langle\alpha_i|, }[/math]

discussed above is

以上所讨论的是

where the w_i are positive-valued probabilities (they sum up to 1), the vectors α_i are unit vectors, and in the infinite-dimensional case, we would take the closure of such states in the trace norm. We can interpret ρ as representing an ensemble where w_i is the proportion of the ensemble whose states are [math]\displaystyle{ |\alpha_i\rangle }[/math]. When a mixed state has rank 1, it therefore describes a 'pure ensemble'. When there is less than total information about the state of a quantum system we need density matrices to represent the state. 其中“w”_i是正值概率（它们的总和为1），向量{mvar |α_i}是单位向量，在无限维的情况下，我们将在迹范数中取这类状态的闭包。我们可以将{mvar |ρ}解释为表示一个系综，其中{mvar | w_i}是状态为[math]\displaystyle{ \alpha\u i\rangle }[/math]的系综的比例。当一个混合态有秩1时，它就描述了一个“纯系综”。当一个量子系统的状态信息不足时，我们需要密度矩阵来表示这个状态。

[math]\displaystyle{ \rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right ) }[/math]

左(| 0 rangle 0 | a + | 1 rangle 1 | a right) </math >

Experimentally, a mixed ensemble might be realized as follows. Consider a "black box" apparatus that spits electrons towards an observer. The electrons' Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state [math]\displaystyle{ |\mathbf{z}+\rangle }[/math] with spins aligned in the positive z direction, and the other with state [math]\displaystyle{ |\mathbf{y}-\rangle }[/math] with spins aligned in the negative y direction. Generally, this is a mixed ensemble, as there can be any number of populations, each corresponding to a different state. 实验上，混合系综可以实现如下。考虑一个向观察者吐出电子s的“黑匣子”装置。电子的希尔伯特空间是相同。这个装置可能产生所有处于相同状态的电子；在这种情况下，观察者接收到的电子就是一个纯系综。然而，这种装置可以产生不同状态的电子。例如，它可以产生两个电子群：一个电子群的态[math]\displaystyle{ |\mathbf{z}+\rangle }[/math]， spins在正z'方向对齐，另一个电子群的态[math]\displaystyle{ |\mathbf{y}-\rangle }[/math]，自旋在负{math |y'}方向对齐。一般来说，这是一个混合集合，因为可以有任意数量的总体，每个总体对应于不同的状态。 This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of for the pure product state [math]\displaystyle{ |\psi\rangle_A \otimes |\phi\rangle_B }[/math] discussed above is

这表明，正如预期的那样，一个纠缠纯系综的约化密度矩阵是一个混合系综。同样不足为奇的是，上面讨论的纯乘积态的密度矩阵

Following the definition above, for a bipartite composite system, mixed states are just density matrices on H_A ⊗ H_B. That is, it has the general form

[math]\displaystyle{ \rho_A = |\psi\rangle_A \langle\psi|_A }[/math].

我不知道，但是我知道，我知道。

[math]\displaystyle{ \rho =\sum_{i} w_i\left[\sum_{j} \bar{c}_{ij} (|\alpha_{ij}\rangle\otimes|\beta_{ij}\rangle)\right]\left[\sum_k c_{ik} (\langle\alpha_{ik}|\otimes\langle\beta_{ik}|)\right] In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure. 一般情况下，二体纯态 ρ 纠缠当且仅当其约化态是混合态而不是纯态。 }[/math]

where the w_i are positively valued probabilities, [math]\displaystyle{ \sum_j |c_{ij}|^2=1 }[/math], and the vectors are unit vectors. This is self-adjoint and positive and has trace 1. 其中“w”_i是正值概率，[math]\displaystyle{ \sum|u j | c|ij}|^2=1 }[/math]，向量是单位向量。这是自伴正的，有迹1。 Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional AKLT spin chain: the ground state can be divided into a block and an environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

在不同的基态自旋链中显式计算了约化密度矩阵。一维 AKLT 自旋链就是一个例子: 基态可以分为一个区块和一个环境。块的约化密度矩阵与另一个哈密顿量的简并基态成正比。

Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as

The reduced density matrix also was evaluated for XY spin chains, where it has full rank. It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence in this case.

并对 XY 自旋链的全秩约化密度矩阵进行了计算。证明了在热力学极限中，大块自旋的约化密度矩阵的谱在这种情况下是一个精确的几何序列。

[math]\displaystyle{ \rho = \sum_i w_i \rho_i^A \otimes \rho_i^B, }[/math]

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called LOCC (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states.

在量子信息理论中，纠缠态被认为是一种“资源”，也就是说，生产成本高，可以实现有价值的转换。这种观点最明显的背景是“遥远的实验室”，即标记为“A”和“B”的两个量子系统，每个量子系统上都可以执行任意的量子操作，但它们之间没有量子力学的相互作用。唯一允许的相互作用是经典信息的交换，它与最一般的局部量子操作相结合，产生了一类称为局部操作和经典通信的操作。这些操作不允许在系统A和B之间产生纠缠态。但是如果A和B具有纠缠态的供应，那么这些操作与LOCC操作一起可以实现更大类别的变换。例如，a的一个量子位和B的一个量子位之间的相互作用可以通过首先将a的量子位传送到B，然后让它与B的量子位相互作用（现在是LOCC操作，因为两个量子位都在B的实验室里），然后将量子位传送回a来实现。在这个过程中，两个量子位的两个最大纠缠态被耗尽。因此，纠缠态是一种资源，能够在只有LOCC可用的情况下实现量子相互作用（或量子通道），但它们在过程中被消耗。在其他应用中，纠缠可以被视为一种资源，例如，私人通信或区分量子态。

where the w_i are positively valued probabilities and the [math]\displaystyle{ \rho_i^A }[/math]'s and [math]\displaystyle{ \rho_i^B }[/math]'s are themselves mixed states (density operators) on the subsystems A and B respectively. In other words, a state is separable if it is a probability distribution over uncorrelated states, or product states. By writing the density matrices as sums of pure ensembles and expanding, we may assume without loss of generality that [math]\displaystyle{ \rho_i^A }[/math] and [math]\displaystyle{ \rho_i^B }[/math] are themselves pure ensembles. A state is then said to be entangled if it is not separable.

其中，{mvar | w_i}是正值概率，[math]\displaystyle{ \rho | i^A }[/math]和[math]\displaystyle{ \rho | i^B }[/math]分别是子系统{mvar | A}和{mvar | B}上的混合态（密度算子）。换句话说，如果一个状态是不相关状态或乘积状态的概率分布，那么它是可分离的。通过将密度矩阵写成纯系综的和并展开，我们可以假定[math]\displaystyle{ \rho i^A }[/math]和[math]\displaystyle{ \rho i^B }[/math]本身就是纯系综。如果一个态是不可分离的，它就被称为纠缠态

In general, finding out whether or not a mixed state is entangled is considered difficult. The general bipartite case has been shown to be NP-hard. For the 2 × 2 and 2 × 3 cases, a necessary and sufficient criterion for separability is given by the famous Positive Partial Transpose (PPT) condition.

Reduced density matrices约化密度矩阵

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement.

在这一节中，我们将讨论混合态的熵，以及如何将其视为量子纠缠的度量。

The idea of a reduced density matrix was introduced by Paul Dirac in 1930. Consider as above systems A and B each with a Hilbert space H_A, H_B. Let the state of the composite system be

[math]\displaystyle{ |\Psi \rangle \in H_A \otimes H_B. }[/math]

The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.

二分子2能级纯态的冯纽曼熵与本征值的图。当本征值为5时，冯纽曼熵处于最大值，相当于最大纠缠度。

In classical information theory , the Shannon entropy, is associated to a probability distribution,[math]\displaystyle{ p_1, \cdots, p_n }[/math], in the following way:

在经典的信息论中，香农熵，是与概率分布相关联的，如下:

As indicated above, in general there is no way to associate a pure state to the component system A. However, it still is possible to associate a density matrix. Let 如上所述，一般来说，无法将纯状态与组件系统{mvar | a}相关联。但是，仍然可以关联密度矩阵。让

[math]\displaystyle{ H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i. }[/math]

[ math ] h (p _ 1，cdots，p _ n) =-sum _ i p _ i log _ 2 p _ i. [ math ]

[math]\displaystyle{ \rho_T = |\Psi\rangle \; \langle\Psi| }[/math].

Since a mixed state is a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:

由于混合状态是一个概率分布超过一个总体，这自然导致了冯纽曼熵的定义:

which is the projection operator onto this state. The state of A is the partial trace of ρ_T over the basis of system B: 它是这个状态的投影操作符。{mvar | A}}的状态是{mvar |ρ_T}在系统{mvar | B}基础上的[[部分迹]：

[math]\displaystyle{ S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right). }[/math]

(rho) =-hbox { Tr } left (rho log _ 2{ rho } right)

[math]\displaystyle{ \rho_A \ \stackrel{\mathrm{def}}{=}\ \sum_j \langle j|_B \left( |\Psi\rangle \langle\Psi| \right) |j\rangle_B = \hbox{Tr}_B \; \rho_T. }[/math]

In general, one uses the Borel functional calculus to calculate a non-polynomial function such as . If the nonnegative operator acts on a finite-dimensional Hilbert space and has eigenvalues [math]\displaystyle{ \lambda_1, \cdots, \lambda_n }[/math], turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues [math]\displaystyle{ \log_2(\lambda_1), \cdots, \log_2(\lambda_n) }[/math]. The Shannon entropy is then:

一般来说，人们使用 Borel 函数演算来计算一个非多项式函数，如。如果非负算子作用于有限维希尔伯特空间，并且具有本征值 < math > lambda _ 1，那么 cdots，lambda _ n </math > ，结果只不过是具有相同本征向量的算子，但本征值 < math > log _ 2(lambda _ 1) ，点，log _ 2(lambda _ n) </math > 。那么香农熵就是:

ρ_A is sometimes called the reduced density matrix of ρ on subsystem A. Colloquially, we "trace out" system B to obtain the reduced density matrix on A.

{mvar |ρ_A}有时被称为子系统{mvar |ρ}上{mvar |ρ}的约化密度矩阵。通俗地说，我们“追踪”系统{mvar | B}，得到{mvar | A}上的约化密度矩阵。

[math]\displaystyle{ S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i }[/math].

(rho) =-hbox { Tr } left (rho log 2{ rho } right) =-sum _ i lambda _ i log _ 2 lambda _ i </math > .

For example, the reduced density matrix of A for the entangled state 例如，纠缠态{mvar | A}的约化密度矩阵

Since an event of probability 0 should not contribute to the entropy, and given that

因为一个概率为0的事件不应该对熵有贡献，并且假设

[math]\displaystyle{ \tfrac{1}{\sqrt{2}} \left ( |0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B \right), }[/math]

[math]\displaystyle{ \lim_{p \to 0} p \log p = 0, }[/math]

[ math > lim _ { p to 0} p log p = 0，</math >

discussed above is

the convention 0}} is adopted. This extends to the infinite-dimensional case as well: if has spectral resolution

约定0}被采用。这也延伸到无限维情况: 如果有光谱分辨率

[math]\displaystyle{ \rho_A = \tfrac{1}{2} \left ( |0\rangle_A \langle 0|_A + |1\rangle_A \langle 1|_A \right ) }[/math]

[math]\displaystyle{  \rho = \int \lambda d P_{\lambda}, }[/math]

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of A for the pure product state [math]\displaystyle{ |\psi\rangle_A \otimes |\phi\rangle_B }[/math] discussed above is 这表明，与预期一样，纠缠纯系综的约化密度矩阵是一个混合系综。同样不奇怪的是，上面讨论的纯积态{mvar | A}}的密度矩阵是

assume the same convention when calculating

在计算时采用相同的约定

[math]\displaystyle{ \rho_A = |\psi\rangle_A \langle\psi|_A }[/math].

[math]\displaystyle{  \rho \log_2 \rho = \int \lambda \log_2 \lambda d P_{\lambda}. }[/math]

[数学] rho log 2 rho = int lambda log 2 lambda d { lambda }

In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.

一般来说，二部纯态ρ是纠缠的当且仅当它的约化态是混合态而不是纯态。

As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is (which can be shown to be the maximum entropy for mixed states).

就像统计力学一样，系统的不确定性(微观状态的数量)越多，熵就越大。例如，任何纯态的熵都为零，这并不奇怪，因为处于纯态的系统没有不确定性。上面讨论的纠缠态的两个子系统中的任何一个的熵都是(混合态的最大熵)。

Two applications that use them 两种使用它们的应用

Reduced density matrices were explicitly calculated in different spin chains with unique ground state. An example is the one-dimensional AKLT spin chain: the ground state can be divided into a block and an environment. The reduced density matrix of the block is proportional to a projector to a degenerate ground state of another Hamiltonian.

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist. If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

熵提供了一个可以用来量化纠缠的工具，尽管还存在其他的纠缠度量方法。如果整个系统是纯系统，则可以用一个子系统的熵来衡量其与其他子系统的纠缠程度。

The reduced density matrix also was evaluated for XY spin chains, where it has full rank. It was proved that in the thermodynamic limit, the spectrum of the reduced density matrix of a large block of spins is an exact geometric sequence in this case.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

对于两体纯态，减少态的冯纽曼熵是唯一的纠缠度量，因为它是满足纠缠度量所要求的特定公理的态家族中唯一的函数。

Entanglement as a resource 作为资源的纠缠

In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows to implement valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labeled "A" and "B" on each of which arbitrary quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called LOCC (local operations and classical communication). These operations do not allow the production of entangled states between the systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations. For example, an interaction between a qubit of A and a qubit of B can be realized by first teleporting A's qubit to B, then letting it interact with B's qubit (which is now a LOCC operation, since both qubits are in B's lab) and then teleporting the qubit back to A. Two maximally entangled states of two qubits are used up in this process. Thus entangled states are a resource that enables the realization of quantum interactions (or of quantum channels) in a setting where only LOCC are available, but they are consumed in the process. There are other applications where entanglement can be seen as a resource, e.g., private communication or distinguishing quantum states. 在量子信息理论中，纠缠态被认为是一种“资源”，也就是说，生产成本高，可以实现有价值的转换。这种观点最明显的背景是“遥远的实验室”，即标记为“A”和“B”的两个量子系统，在每个量子系统上可以执行任意的量子操作s，但它们之间不以量子力学方式相互作用。唯一允许的相互作用是经典信息的交换，它与最一般的局部量子操作相结合，产生了一类称为LOCC的操作（局部操作和经典通信）。这些操作不允许在系统A和B之间产生纠缠态。但是如果A和B具有纠缠态的供应，那么这些操作与LOCC操作一起可以实现更大类别的变换。例如，a的一个量子位和B的一个量子位之间的相互作用可以通过首先将a的量子位传送到B，然后让它与B的量子位相互作用（现在是LOCC操作，因为两个量子位都在B的实验室里），然后将量子位传送回a来实现。在这个过程中，两个量子位的两个最大纠缠态被耗尽。因此，纠缠态是一种资源，能够在只有LOCC可用的情况下实现量子相互作用（或量子通道），但它们在过程中被消耗。在其他应用中，纠缠可以被视为一种资源，例如，私人通信或区分量子态。 It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state is said to be a maximally entangled state if the reduced state of is the diagonal matrix

一个经典的结果是，香农熵在均匀概率分布{1/n，... ，1/n }处达到最大值。因此，如果二分纯态的约化态是对角矩阵，则称二分纯态为最大纠缠态

Classification of entanglement 纠缠分类

[math]\displaystyle{ \begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}. }[/math]

< math > begin { bmatrix } frac {1}{ n } & & ddots & frac {1}{ n } end { bmatrix } . </math >

Not all quantum states are equally valuable as a resource. To quantify this value, different entanglement measures (see below) can be used, that assign a numerical value to each quantum state. However, it is often interesting to settle for a coarser way to compare quantum states. This gives rise to different classification schemes. Most entanglement classes are defined based on whether states can be converted to other states using LOCC or a subclass of these operations. The smaller the set of allowed operations, the finer the classification. Important examples are: 并不是所有的量子态都具有同等的资源价值。为了量化这个值，可以使用不同的纠缠度量（见下文），为每个量子态分配一个数值。然而，用一种更粗糙的方法来比较量子态是很有趣的。这就产生了不同的分类方案。大多数纠缠类的定义是基于是否可以使用LOCC或这些操作的子类将状态转换为其他状态。允许的操作集越小，分类就越精细。重要的例子有：

• If two states can be transformed into each other by a local unitary operation, they are said to be in the same LU class. This is the finest of the usually considered classes. Two states in the same LU class have the same value for entanglement measures and the same value as a resource in the distant-labs setting. There is an infinite number of different LU classes (even in the simplest case of two qubits in a pure state).

如果两个状态可以通过局部幺正运算相互转换，则称它们为同一“LU类”。这是通常认为最好的一类。同一LU类中的两个态具有相同的纠缠度量值，并且在远程实验室设置中具有相同的资源值。有无限多个不同的LU类（即使是在纯态中两个量子比特的最简单情况下）。 For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.

对于混合态，简化冯纽曼熵并不是唯一合理的纠缠度量。

• If two states can be transformed into each other by local operations including measurements with probability larger than 0, they are said to be in the same 'SLOCC class' ("stochastic LOCC"). Qualitatively, two states [math]\displaystyle{ \rho_1 }[/math] and [math]\displaystyle{ \rho_2 }[/math] in the same SLOCC class are equally powerful (since I can transform one into the other and then do whatever it allows me to do), but since the transformations [math]\displaystyle{ \rho_1\to\rho_2 }[/math] and [math]\displaystyle{ \rho_2\to\rho_1 }[/math] may succeed with different probability, they are no longer equally valuable. E.g., for two pure qubits there are only two SLOCC classes: the entangled states (which contains both the (maximally entangled) Bell states and weakly entangled states like [math]\displaystyle{ |00\rangle+0.01|11\rangle }[/math]) and the separable ones (i.e., product states like [math]\displaystyle{ |00\rangle }[/math]).

如果两个状态可以通过局部操作（包括概率大于0的测量）相互转换，则它们被称为同一个“SLOCC类”（“随机LOCC”）。从质量上讲，同一SLOCC类中的两个状态[math]\displaystyle{ \rho\u 1 }[/math]和[math]\displaystyle{ \rho\u 2 }[/math]是同等强大的（因为我可以将一个状态转换为另一个状态，然后执行它允许我执行的任何操作），但是由于转换[math]\displaystyle{ \rho\u 1\到\rho\u 2 }[/math]和[math]\displaystyle{ \rho\u 2\到\rho\u 1 }[/math]可能以不同的概率成功，它们不再是同样有价值。E、 例如，对于两个纯量子位，只有两个SLOCC类：纠缠态（包含（最大纠缠）贝尔态和弱纠缠态，如[math]\displaystyle{ | 00\rangle+0.01 | 11\rangle }[/math]）和可分离态（即乘积态，如[math]\displaystyle{ | 00\rangle }[/math]）

• Instead of considering transformations of single copies of a state (like [math]\displaystyle{ \rho_1\to\rho_2 }[/math]) one can define classes based on the possibility of multi-copy transformations. E.g., there are examples when [math]\displaystyle{ \rho_1\to\rho_2 }[/math] is impossible by LOCC, but [math]\displaystyle{ \rho_1\otimes\rho_1\to\rho_2 }[/math] is possible. A very important (and very coarse) classification is based on the property whether it is possible to transform an arbitrarily large number of copies of a state [math]\displaystyle{ \rho }[/math] into at least one pure entangled state. States that have this property are called distillable. These states are the most useful quantum states since, given enough of them, they can be transformed (with local operations) into any entangled state and hence allow for all possible uses. It came initially as a surprise that not all entangled states are distillable, those that are not are called 'bound entangled'.

我们可以根据多副本转换的可能性来定义类，而不是考虑状态的单个副本的转换（如从[math]\displaystyle{ \rho\u1\到\rho\u2 }[/math]）。E、 例如，有这样的例子：LOCC不可能实现[math]\displaystyle{ \rho\u 1\到\rho\u 2 }[/math]，但有时可以实现[math]\displaystyle{ \rho\u 1\到\rho\u 2 }[/math]。一个非常重要（而且非常粗糙）的分类是基于这样一个性质：是否有可能将一个态的任意多个拷贝[math]\displaystyle{ \rho }[/math]转换成至少一个纯纠缠态。具有这种性质的态称为可蒸馏。这些态是最有用的量子态，因为只要有足够的量子态，它们就可以（通过局部操作）转换成任何纠缠态，从而允许所有可能的用途。最初令人惊讶的是，并非所有的纠缠态都是可提取的，那些不可提取的被称为“束缚纠缠”。 As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the Boltzmann constant 1}}). For example, by properties of the Borel functional calculus, we see that for any unitary operator ,

顺便说一句，信息论的定义与统计力学意义上的熵密切相关(比较在当前语境下的两个定义，通常设置波兹曼常数1})。例如，通过 Borel 泛函微积分的性质，我们可以看到，对于任何幺正算符,

A different entanglement classification is based on what the quantum correlations present in a state allow A and B to do: one distinguishes three subsets of entangled states: (1) the non-local states, which produce correlations that cannot be explained by a local hidden variable model and thus violate a Bell inequality, (2) the steerable states that contain sufficient correlations for A to modify ("steer") by local measurements the conditional reduced state of B in such a way, that A can prove to B that the state they possess is indeed entangled, and finally (3) those entangled states that are neither non-local nor steerable. All three sets are non-empty.

[math]\displaystyle{ S(\rho) = S \left (U \rho U^* \right). }[/math]

s (rho) = s left (u rho u ^ * right) . </math >

Entropy熵

Indeed, without this property, the von Neumann entropy would not be well-defined.

事实上，如果没有这个属性，冯纽曼熵就不会有明确的定义。

In this section, the entropy of a mixed state is discussed as well as how it can be viewed as a measure of quantum entanglement. 在本节中，我们将讨论混合态的熵，以及如何将其视为量子纠缠的量度。

In particular, could be the time evolution operator of the system, i.e.,

特别是，可以是系统的时间演化算子，即,

Definition 定义

文件:Von Neumann entropy for bipartite system plot.svg

The plot of von Neumann entropy Vs Eigenvalue for a bipartite 2-level pure state. When the eigenvalue has value .5, von Neumann entropy is at a maximum, corresponding to maximum entanglement.

[math]\displaystyle{ U(t) = \exp \left(\frac{-i H t }{\hbar}\right), }[/math]

[ math ] u (t) = exp left (frac {-i h t }{ hbar } right) ，[ math ]

In classical information theory H, the Shannon entropy, is associated to a probability distribution,[math]\displaystyle{ p_1, \cdots, p_n }[/math], in the following way:

where is the Hamiltonian of the system. Here the entropy is unchanged.

这个系统的哈密顿量在哪里。这里熵不变。

[math]\displaystyle{ H(p_1, \cdots, p_n ) = - \sum_i p_i \log_2 p_i. }[/math]

The reversibility of a process is associated with the resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. Therefore, the march of the arrow of time towards thermodynamic equilibrium is simply the growing spread of quantum entanglement.

一个过程的可逆性与由此产生的熵变有关，也就是说，一个过程是可逆的，当且仅当它使系统的熵不变。因此，时间之箭向热力学平衡的前进只不过是量子纠缠的蔓延。

Since a mixed state ρ is a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:

This provides a connection between quantum information theory and thermodynamics.

这提供了量子信息理论和热力学之间的联系。

[math]\displaystyle{ S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right). }[/math]

Rényi entropy also can be used as a measure of entanglement.

熵也可以用来度量纠缠。

In general, one uses the Borel functional calculus to calculate a non-polynomial function such as log₂(ρ). If the nonnegative operator ρ acts on a finite-dimensional Hilbert space and has eigenvalues [math]\displaystyle{ \lambda_1, \cdots, \lambda_n }[/math], log₂(ρ) turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues [math]\displaystyle{ \log_2(\lambda_1), \cdots, \log_2(\lambda_n) }[/math]. The Shannon entropy is then:

Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature

量子纠缠度量了量子态(通常被视为双体)中纠缠的数量。如前所述，纠缠熵是纯态的标准量度(但不再是混合态的量度)。对于混合态，文献中有一些纠缠度量

[math]\displaystyle{ S(\rho) = - \hbox{Tr} \left( \rho \log_2 {\rho} \right) = - \sum_i \lambda_i \log_2 \lambda_i }[/math].

Since an event of probability 0 should not contribute to the entropy, and given that

The Reeh-Schlieder theorem of quantum field theory is sometimes seen as an analogue of quantum entanglement.

量子场论的 Reeh-Schlieder 定理有时被看作是量子纠缠的类比。

[math]\displaystyle{ \lim_{p \to 0} p \log p = 0, }[/math]

the convention 0 log(0) = 0 is adopted. This extends to the infinite-dimensional case as well: if ρ has spectral resolution

Entanglement has many applications in quantum information theory. With the aid of entanglement, otherwise impossible tasks may be achieved.

纠缠态在量子信息理论中有许多应用。在纠缠的帮助下，否则不可能完成的任务就可能实现。

[math]\displaystyle{ \rho = \int \lambda d P_{\lambda}, }[/math]

Among the best-known applications of entanglement are superdense coding and quantum teleportation.

其中最著名的应用是超稠密编码和量子遥传纠缠。

assume the same convention when calculating 计算时假设相同的约定 Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some).

大多数研究人员认为量子纠缠对于实现量子计算是必要的(尽管有些人对此有争议)。

[math]\displaystyle{ \rho \log_2 \rho = \int \lambda \log_2 \lambda d P_{\lambda}. }[/math]

Entanglement is used in some protocols of quantum cryptography. This is because the "shared noise" of entanglement makes for an excellent one-time pad. Moreover, since measurement of either member of an entangled pair destroys the entanglement they share, entanglement-based quantum cryptography allows the sender and receiver to more easily detect the presence of an interceptor.

纠缠被用于量子密码学的一些协议中。这是因为纠缠的“共享噪音”造就了绝佳的一次性衬垫。此外，由于测量纠缠对的任何一个成员都会破坏它们共享的纠缠，基于纠缠的量子密码学可以让发送方和接收方更容易地检测到拦截器的存在。

As in statistical mechanics, the more uncertainty (number of microstates) the system should possess, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is log(2) (which can be shown to be the maximum entropy for 2 × 2 mixed states).

在统计力学中，系统应具有的不确定性（微观状态数）越多，熵就越大。例如，任何纯态的熵都是零，这并不奇怪，因为纯态下的系统没有不确定性。上面讨论的纠缠态的两个子系统中的任何一个子系统的熵是{math | log（2）}（这可以显示为{math | 2×2}混合态的最大熵）

In interferometry, entanglement is necessary for surpassing the standard quantum limit and achieving the Heisenberg limit.

在干涉术中，纠缠态对于超越标准量子极限和达到海森堡极限是必要的。

As a measure of entanglement作为纠缠的测量

Entropy provides one tool that can be used to quantify entanglement, although other entanglement measures exist. If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems.

There are several canonical entangled states that appear often in theory and experiments.

在理论和实验中经常会出现几种典型的纠缠态。

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure. 对于二部纯态，约化态的von Neumann熵是唯一的纠缠度量，因为它是满足纠缠度量所要求的某些公理的态族上的唯一函数。 For two qubits, the Bell states are

对于两个量子比特，贝尔态是

It is a classical result that the Shannon entropy achieves its maximum at, and only at, the uniform probability distribution {1/n,...,1/n}. Therefore, a bipartite pure state ρ ∈ H_A ⊗ H_B is said to be a maximally entangled state if the reduced state模板:Clarify of ρ is the diagonal matrix

[math]\displaystyle{ |\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B) }[/math]

< math > | Phi ^ pm rangle = frac {1}{ sqrt {2}(| 0 rangle _ a o times | 0 rangle _ b | 1 rangle _ a o times | 1 rangle _ b) </math >

[math]\displaystyle{ |\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B) }[/math].

< math > | Psi ^ pm rangle = frac {1}{ sqrt {2}(| 0 rangle _ a o times | 1 rangle _ b pm | 1 rangle _ a o times | 0 rangle _ b) </math > .

[math]\displaystyle{ \begin{bmatrix} \frac{1}{n}& & \\ & \ddots & \\ & & \frac{1}{n}\end{bmatrix}. }[/math]

These four pure states are all maximally entangled (according to the entropy of entanglement) and form an orthonormal basis (linear algebra) of the Hilbert space of the two qubits. They play a fundamental role in Bell's theorem.

这四个纯态都是最大纠缠态(根据纠缠熵) ，并且形成了两个量子位的希尔伯特空间的标准正交基(线性代数)。它们在贝尔定理中起着基本的作用。

For mixed states, the reduced von Neumann entropy is not the only reasonable entanglement measure.

For M>2 qubits, the GHZ state is

对于 m > 2量子位，GHZ 态是

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions in the present context, it is customary to set the Boltzmann constant k = 1). For example, by properties of the Borel functional calculus, we see that for any unitary operator U,

[math]\displaystyle{ |\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}, }[/math]

< math > | mathrm { GHZ } rangle = frac { | 0 rangle ^ { otimes m } + | 1 rangle ^ { otimes m }{ sqrt {2} ，</math >

[math]\displaystyle{ S(\rho) = S \left (U \rho U^* \right). }[/math]

which reduces to the Bell state [math]\displaystyle{ |\Phi^+\rangle }[/math] for [math]\displaystyle{ M=2 }[/math]. The traditional GHZ state was defined for [math]\displaystyle{ M=3 }[/math]. GHZ states are occasionally extended to qudits, i.e., systems of d rather than 2 dimensions.

它缩小到贝尔状态。传统的 GHZ 状态定义为 < math > m = 3 </math > 。GHZ 状态偶尔会扩展到 qudit，即 d 而不是2维系统。

Indeed, without this property, the von Neumann entropy would not be well-defined.

Also for M>2 qubits, there are spin squeezed states. Spin squeezed states are a class of squeezed coherent states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled. Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.

对于 m > 2量子位，也存在自旋压缩态。自旋压缩态是一类对自旋测量不确定度满足一定限制的压缩相干态，它必然是纠缠态。自旋压缩态是利用量子纠缠增强精密测量的理想候选态。

In particular, U could be the time evolution operator of the system, i.e.,

For two bosonic modes, a NOON state is

对于两个玻色模态，NOON 状态是

[math]\displaystyle{ U(t) = \exp \left(\frac{-i H t }{\hbar}\right), }[/math]

[math]\displaystyle{ |\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \,  }[/math]

[数学] | psi _ text { NOON } rangle = frac { | n rangle _ a | 0 rangle _ b + | {0} rangle _ a | { n } rangle _ b }{ sqrt {2} ，，</math >

where H is the Hamiltonian of the system. Here the entropy is unchanged.

This is like the Bell state [math]\displaystyle{ |\Psi^+\rangle }[/math] except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".

这就像贝尔态 < math > | Psi ^ + rangle </math > 除了基函数0和1已经被“ n 个光子处于一种模式”和“ n 个光子处于另一种模式”所取代。

The reversibility of a process is associated with the resulting entropy change, i.e., a process is reversible if, and only if, it leaves the entropy of the system invariant. Therefore, the march of the arrow of time towards thermodynamic equilibrium is simply the growing spread of quantum entanglement.

This provides a connection between quantum information theory and thermodynamics.

Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam splitter. They are the sum of multiple of NOON states, and can used to achieve the Heisenberg limit.

最后，还存在玻色子模式的双 Fock 态，它可以通过将 Fock 态输入到两个导致分束器的臂来产生。它们是 NOON 态的倍数之和，可以用来实现海森堡极限。

Rényi entropy also can be used as a measure of entanglement.

For the appropriately chosen measure of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.

对于适当选择的纠缠度量，Bell、 GHZ 和 NOON 态是最大纠缠态，而自旋压缩态和双 Fock 态只是部分纠缠。部分纠缠态通常更容易在实验上准备。

Entanglement measures

Entanglement measures quantify the amount of entanglement in a (often viewed as a bipartite) quantum state. As aforementioned, entanglement entropy is the standard measure of entanglement for pure states (but no longer a measure of entanglement for mixed states). For mixed states, there are some entanglement measures in the literature and no single one is standard.

Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is spontaneous parametric down-conversion to generate a pair of photons entangled in polarisation. Other methods include the use of a fiber coupler to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a quantum dot, the use of the Hong–Ou–Mandel effect, etc., In the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades.

纠缠通常是由亚原子粒子间的直接相互作用产生的。这些相互作用可以有多种形式。最常用的方法之一是用自发参量下转换产生一对纠缠在偏振中的光子。其他方法包括使用光纤耦合器来限制和混合光子，量子点中双激子衰变级联发射的光子，Hong-Ou-Mandel 效应的使用等等。在贝尔定理最早的测试中，纠缠粒子是利用原子级联产生的。

• Entanglement cost

• Distillable entanglement

It is also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.

通过使用纠缠交换，也有可能在不直接相互作用的量子系统之间创造纠缠。如果它们的波函数在空间上仅仅重叠，至少是部分重叠，那么它们也可以相互纠缠全同粒子。

• Entanglement of formation

• Relative entropy of entanglement

• Squashed entanglement

• Logarithmic negativity

A density matrix ρ is called separable if it can be written as a convex sum of product states, namely

密度矩阵 ρ 称为可分的，如果它可以写成乘积态的凸和，即

Most (but not all) of these entanglement measures reduce for pure states to entanglement entropy, and are difficult (NP-hard) to compute.

[math]\displaystyle{ \displaystyle{\rho=\sum_j p_j \rho_j^{(A)}\otimes\rho_j^{(B)}} }[/math]

显示方式{ rho = sum _ j p _ j rho _ j ^ {(a)}次 rho _ j ^ {(b)}} </math >

Quantum field theory

The Reeh-Schlieder theorem of quantum field theory is sometimes seen as an analogue of quantum entanglement.

with [math]\displaystyle{ 1\ge p_j\ge 0 }[/math] probabilities. By definition, a state is entangled if it is not separable.

概率为1 ge p _ j ge 0 </math > 。根据定义，如果一个态不可分离，它就是纠缠态。

Applications

For 2-Qubit and Qubit-Qutrit systems (2 × 2 and 2 × 3 respectively) the simple Peres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized. Other separability criteria include (but not limited to) the range criterion, reduction criterion, and those based on uncertainty relations. See Ref. for a review of separability criteria in discrete variable systems.

对于2量子比特和2 × 2量子比特-量子特里特系统(分别为2 × 2和2 × 3) ，简单的 Peres-horowitz 准则为分离提供了一个必要和充分的判据，从而无意识地提供了检测纠缠的判据。然而，对于一般情形，该判据仅仅是可分性的必要条件，因为问题一经推广就变成了 np 难问题。其他可分性标准包括(但不限于)范围标准、归约标准和基于不确定关系的标准。参见参考文献。回顾了离散变量系统的可分性准则。

Entanglement has many applications in quantum information theory. With the aid of entanglement, otherwise impossible tasks may be achieved.

A numerical approach to the problem is suggested by Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement". Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built-in Peres-Horodecki criterion testing) is "StateSeparator" web-app.

Jon Magne Leinaas，Jan Myrheim 和 Eirik Ovrum 在他们的论文“纠缠的几何方面”中提出了一个数值方法来解决这个问题。莱纳斯等。提供一个数值方法，迭代精炼一个估计的可分离状态朝向要测试的目标状态，并检查目标状态是否确实能够到达。该算法的一个实现(包括内置的 peres-horowitz 标准测试)是[ StateSeparator http://phweb.technion.ac.il/~StateSeparator/] web-app。

Among the best-known applications of entanglement are superdense coding and quantum teleportation.

In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for [math]\displaystyle{ 1\oplus1 }[/math]-mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for [math]\displaystyle{ 1\oplus n }[/math]-mode Gaussian states, but no longer sufficient for [math]\displaystyle{ 2\oplus2 }[/math]-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators or by using entropic measures.

在连续变量系统中，Peres-Horodecki 准则也适用。具体地说，Simon 根据正则算符的二阶矩，制定了 Peres-Horodecki 准则的一个特定版本，并表明它对于 < math > 1 oplus1 </math >-mode Gaussian 状态是必要的和充分的。看似不同，但本质上等价的方法)。后来发现，Simon 的条件对于 < math > 1 oplus n </math >-mode Gaussian 状态也是必要和充分的，但是对于 < math > 2 oplus2 </math >-mode Gaussian 状态不再是充分的。Simon 条件可以通过考虑正则算子的高阶矩或者用熵测度来推广。

Most researchers believe that entanglement is necessary to realize quantum computing (although this is disputed by some).

In 2016 China launched the world’s first quantum communications satellite. The $100m Quantum Experiments at Space Scale (QUESS) mission was launched on Aug 16, 2016, from the Jiuquan Satellite Launch Center in northern China at 01:40 local time.

2016年，中国发射了世界上第一颗量子通信卫星。耗资1亿美元的空间量子实验任务于2016年8月16日当地时间01:40从中国北方的酒泉卫星发射中心空间站发射升空。

Entanglement is used in some protocols of quantum cryptography. This is because the "shared noise" of entanglement makes for an excellent one-time pad. Moreover, since measurement of either member of an entangled pair destroys the entanglement they share, entanglement-based quantum cryptography allows the sender and receiver to more easily detect the presence of an interceptor.

For the next two years, the craft – nicknamed "Micius" after the ancient Chinese philosopher – will demonstrate the feasibility of quantum

在接下来的两年里，这艘以中国古代哲学家墨子命名的飞船将展示量子化的可行性

communication between Earth and space, and test quantum entanglement over unprecedented distances.

地球和太空之间的通信，并在前所未有的距离上测试量子纠缠。

In interferometry, entanglement is necessary for surpassing the standard quantum limit and achieving the Heisenberg limit.

In the June 16, 2017, issue of Science, Yin et al. report setting a new quantum entanglement distance record of 1,203 km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching a CHSH valuation of 2.37 ± 0.09, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.

在2017年6月16日的《科学》杂志上。在严格的爱因斯坦定域条件下，从墨丘利卫星到 Lijian、云南和 Delingha、 Quinhai 的基地的 CHSH 估值为2.37 ± 0.09，证明了双光子对的存在和对 Bell 不等式的违反，从而提高了数量级通过光纤实验的传输效率。

Entangled states

There are several canonical entangled states that appear often in theory and experiments.

For two qubits, the Bell states are

The electron shells of multi-electron atoms always consist of entangled electrons. The correct ionization energy can be calculated only by consideration of electron entanglement.

多电子原子的电子壳层总是由纠缠电子组成。只有考虑到电子纠缠，才能计算出正确的电离能。

[math]\displaystyle{ |\Phi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B) }[/math]

[math]\displaystyle{ |\Psi^\pm\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B) }[/math].

It has been suggested that in the process of photosynthesis, entanglement is involved in the transfer of energy between light-harvesting complexes and photosynthetic reaction centers where light (energy) is harvested in the form of chemical energy. Without such a process, the efficient conversion of light into chemical energy cannot be explained. Using femtosecond spectroscopy, the coherence of entanglement in the Fenna-Matthews-Olson complex was measured over hundreds of femtoseconds (a relatively long time in this regard) providing support to this theory.

研究表明，在光合作用过程中，纠缠参与了捕光复合物与光合反应中心之间的能量传递，而光(能)是以化学能的形式获得的。没有这样一个过程，光转化为化学能的有效性就无从解释。利用飞秒光谱技术，我们测量了 Fenna-Matthews-Olson 复合体中纠缠态的相干性，时间长达数百飞秒，为这一理论提供了支持。

These four pure states are all maximally entangled (according to the entropy of entanglement) and form an orthonormal basis (linear algebra) of the Hilbert space of the two qubits. They play a fundamental role in Bell's theorem.

However, critical follow-up studies question the interpretation of these results and assign the reported signatures of electronic quantum coherence to nuclear dynamics in the chromophores.

然而，关键的后续研究对这些结果的解释提出了质疑，并将报告的电子量子相干特征赋予了发色团中的核动力学。

For M>2 qubits, the GHZ state is

In 2020 researchers reported the quantum entanglement between the motion of a millimetre-sized mechanical oscillator and a disparate distant spin system of a cloud of atoms.

2020年，研究人员报告了一个毫米大小的机械振荡器的运动和一个原子云的不同距离的自旋系统之间的量子纠缠。

[math]\displaystyle{ |\mathrm{GHZ}\rangle = \frac{|0\rangle^{\otimes M} + |1\rangle^{\otimes M}}{\sqrt{2}}, }[/math]

which reduces to the Bell state [math]\displaystyle{ |\Phi^+\rangle }[/math] for [math]\displaystyle{ M=2 }[/math]. The traditional GHZ state was defined for [math]\displaystyle{ M=3 }[/math]. GHZ states are occasionally extended to qudits, i.e., systems of d rather than 2 dimensions.

In October 2018, physicists reported producing quantum entanglement using living organisms, particularly between photosynthetic molecules within living bacteria and quantized light.

2018年10月，物理学家报告说，他们利用活体生物制造量子纠缠，特别是利用活体细菌中的光合分子和量子化的光。

Also for M>2 qubits, there are spin squeezed states. Spin squeezed states are a class of squeezed coherent states satisfying certain restrictions on the uncertainty of spin measurements, and are necessarily entangled. Spin squeezed states are good candidates for enhancing precision measurements using quantum entanglement.

Living organisms (green sulphur bacteria) have been studied as mediators to create quantum entanglement between otherwise non-interacting light modes, showing high entanglement between light and bacterial modes, and to some extent, even entanglement within the bacteria.

生物体(绿色硫细菌)已被研究作为介质，在非相互作用的光模式之间创造量子纠缠，表明光和细菌模式之间的高度纠缠，甚至在某种程度上纠缠在细菌内部。

For two bosonic modes, a NOON state is

[math]\displaystyle{ |\psi_\text{NOON} \rangle = \frac{|N \rangle_a |0\rangle_b + |{0}\rangle_a |{N}\rangle_b}{\sqrt{2}}, \, }[/math]

This is like the Bell state [math]\displaystyle{ |\Psi^+\rangle }[/math] except the basis kets 0 and 1 have been replaced with "the N photons are in one mode" and "the N photons are in the other mode".

Finally, there also exist twin Fock states for bosonic modes, which can be created by feeding a Fock state into two arms leading to a beam splitter. They are the sum of multiple of NOON states, and can used to achieve the Heisenberg limit.

For the appropriately chosen measure of entanglement, Bell, GHZ, and NOON states are maximally entangled while spin squeezed and twin Fock states are only partially entangled. The partially entangled states are generally easier to prepare experimentally.

Methods of creating entanglement

Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is spontaneous parametric down-conversion to generate a pair of photons entangled in polarisation. Other methods include the use of a fiber coupler to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a quantum dot, the use of the Hong–Ou–Mandel effect, etc., In the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades.

It is also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping. Two independently prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.

Testing a system for entanglement

A density matrix ρ is called separable if it can be written as a convex sum of product states, namely

[math]\displaystyle{ \displaystyle{\rho=\sum_j p_j \rho_j^{(A)}\otimes\rho_j^{(B)}} }[/math]

with [math]\displaystyle{ 1\ge p_j\ge 0 }[/math] probabilities. By definition, a state is entangled if it is not separable.

For 2-Qubit and Qubit-Qutrit systems (2 × 2 and 2 × 3 respectively) the simple Peres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus—inadvertently—for detecting entanglement. However, for the general case, the criterion is merely a necessary one for separability, as the problem becomes NP-hard when generalized. Other separability criteria include (but not limited to) the range criterion, reduction criterion, and those based on uncertainty relations. See Ref. for a review of separability criteria in discrete variable systems.

A numerical approach to the problem is suggested by Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement". Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built-in Peres-Horodecki criterion testing) is "StateSeparator" web-app.

In continuous variable systems, the Peres-Horodecki criterion also applies. Specifically, Simon formulated a particular version of the Peres-Horodecki criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for [math]\displaystyle{ 1\oplus1 }[/math]-mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for [math]\displaystyle{ 1\oplus n }[/math]-mode Gaussian states, but no longer sufficient for [math]\displaystyle{ 2\oplus2 }[/math]-mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators or by using entropic measures.

In 2016 China launched the world’s first quantum communications satellite. The $100m Quantum Experiments at Space Scale (QUESS) mission was launched on Aug 16, 2016, from the Jiuquan Satellite Launch Center in northern China at 01:40 local time.

For the next two years, the craft – nicknamed "Micius" after the ancient Chinese philosopher – will demonstrate the feasibility of quantum

communication between Earth and space, and test quantum entanglement over unprecedented distances.

In the June 16, 2017, issue of Science, Yin et al. report setting a new quantum entanglement distance record of 1,203 km, demonstrating the survival of a two-photon pair and a violation of a Bell inequality, reaching a CHSH valuation of 2.37 ± 0.09, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.

Naturally entangled systems

The electron shells of multi-electron atoms always consist of entangled electrons. The correct ionization energy can be calculated only by consideration of electron entanglement.

Photosynthesis

It has been suggested that in the process of photosynthesis, entanglement is involved in the transfer of energy between light-harvesting complexes and photosynthetic reaction centers where light (energy) is harvested in the form of chemical energy. Without such a process, the efficient conversion of light into chemical energy cannot be explained. Using femtosecond spectroscopy, the coherence of entanglement in the Fenna-Matthews-Olson complex was measured over hundreds of femtoseconds (a relatively long time in this regard) providing support to this theory.

However, critical follow-up studies question the interpretation of these results and assign the reported signatures of electronic quantum coherence to nuclear dynamics in the chromophores.

Entanglement of macroscopic objects

In 2020 researchers reported the quantum entanglement between the motion of a millimetre-sized mechanical oscillator and a disparate distant spin system of a cloud of atoms.

Entanglement of elements of living systems

In October 2018, physicists reported producing quantum entanglement using living organisms, particularly between photosynthetic molecules within living bacteria and quantized light.

Living organisms (green sulphur bacteria) have been studied as mediators to create quantum entanglement between otherwise non-interacting light modes, showing high entanglement between light and bacterial modes, and to some extent, even entanglement within the bacteria.

See also

• CNOT gate

• Bound entanglement

• Concurrence (quantum computing)

• Einstein's thought experiments

• Entanglement distillation

• Entanglement witness

• Faster-than-light communication

• Ghirardi–Rimini–Weber theory

• Multipartite entanglement

• Normally distributed and uncorrelated does not imply independent

• Observer effect (physics)

• Quantum coherence

• Quantum discord

• Quantum phase transition

• Quantum computing

• Quantum network

Category:Quantum information science

类别: 量子信息科学

• Quantum pseudo-telepathy

Category:Quantum mechanics

类别: 量子力学

• Quantum teleportation

Category:Unsolved problems in physics

类别: 物理学中未解决的问题

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