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第391行： + − 并非所有状态都是可分状态(因此也就是乘积状态)。修复一个基础 < math > scriptstyle { | i rangle _ a } </math > for 和一个基础 < math > scriptstyle { | j rangle _ b } </math > for。最普遍的状态是形式+ − + 第407行：
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第435行： − 如果组合系统处于这种状态，就不可能给任何一个系统或系统一个确定的纯状态。另一种说法是，尽管整个状态的冯纽曼熵为零(对于任何纯状态都是如此) ，但子系统的熵大于零。从这个意义上说，这两个系统是“纠缠”的。这对干涉测量法有具体的经验影响。上面的例子是四个贝尔态之一，它们是(最大)纠缠纯态(空间的纯态，但不能分离成每个和的纯态)。+
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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.<ref>{{Cite journal|url=http://www.nature.com/news/entangled-photons-make-a-picture-from-a-paradox-1.15781|title=Entangled photons make a picture from a paradox|journal=Nature|accessdate=13 October 2014|doi=10.1038/nature.2014.15781|year=2014|last1=Gibney|first1=Elizabeth|s2cid=124976589}}</ref>
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.<ref>{{Cite journal|url=http://www.nature.com/news/entangled-photons-make-a-picture-from-a-paradox-1.15781|title=Entangled photons make a picture from a paradox|journal=Nature|accessdate=13 October 2014|doi=10.1038/nature.2014.15781|year=2014|last1=Gibney|first1=Elizabeth|s2cid=124976589}}</ref>
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用光子“拍摄”物体，这些光子并没有与受试者发生相互作用，而是与确实与这些物体发生相互作用的光子纠缠在一起。来自维也纳大学的莱莫斯相信，这种新的量子成像技术可以在生物或医学成像等领域的低光成像领域得到应用。
Not all states are separable states (and thus product states). Fix a basis <math>\scriptstyle \{|i \rangle_A\}</math> for and a basis <math>\scriptstyle \{|j \rangle_B\}</math> for . The most general state in is of the form
Not all states are separable states (and thus product states). Fix a basis <math>\scriptstyle \{|i \rangle_A\}</math> for and a basis <math>\scriptstyle \{|j \rangle_B\}</math> for . The most general state in is of the form
并非所有的状态都是可分离的状态（因此产品状态也是如此）。修复的基础<math>\scriptstyle\{i\rangle\u a\}</math>，修复的基础<math>\scriptstyle\{j\rangle\u B\}</math>。最普遍的状态是
In 2015, Markus Greiner's group at Harvard performed a direct measurement of Renyi entanglement in a system of ultracold bosonic atoms.
In 2015, Markus Greiner's group at Harvard performed a direct measurement of Renyi entanglement in a system of ultracold bosonic atoms.
2015年，哈佛大学的马库斯 格瑞纳团队对超冷玻色子原子系统中的Renyi纠缠进行了直接测量。
<math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.
<math>|\psi\rangle_{AB} = \sum_{i,j} c_{ij} |i\rangle_A \otimes |j\rangle_B</math>.
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.<ref>{{Cite journal|last=Rozatkar|first=Gaurav|date=2018-08-16|title=Demonstration of quantum entanglement|url=https://osf.io/g8bpj/|journal=OSF|language=en}}</ref>
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.<ref>{{Cite journal|last=Rozatkar|first=Gaurav|date=2018-08-16|title=Demonstration of quantum entanglement|url=https://osf.io/g8bpj/|journal=OSF|language=en}}</ref>
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，包括量子纠缠。
This state is separable if there exist vectors <math>\scriptstyle [c^A_i], [c^B_j]</math> so that <math>\scriptstyle c_{ij}= c^A_ic^B_j,</math> yielding <math>\scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>\scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math>\scriptstyle [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math>\scriptstyle c^A_i,c^B_j</math> we have <math>\scriptstyle c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.
This state is separable if there exist vectors <math>\scriptstyle [c^A_i], [c^B_j]</math> so that <math>\scriptstyle c_{ij}= c^A_ic^B_j,</math> yielding <math>\scriptstyle |\psi\rangle_A = \sum_{i} c^A_{i} |i\rangle_A</math> and <math>\scriptstyle |\phi\rangle_B = \sum_{j} c^B_{j} |j\rangle_B.</math> It is inseparable if for any vectors <math>\scriptstyle [c^A_i],[c^B_j]</math> at least for one pair of coordinates <math>\scriptstyle c^A_i,c^B_j</math> we have <math>\scriptstyle c_{ij} \neq c^A_ic^B_j.</math> If a state is inseparable, it is called an 'entangled state'.
=== Mystery of time ===
=== Mystery of time 时间谜团===
For example, given two basis vectors <math>\scriptstyle \{|0\rangle_A, |1\rangle_A\}</math> of and two basis vectors <math>\scriptstyle \{|0\rangle_B, |1\rangle_B\}</math> of , the following is an entangled state:
For example, given two basis vectors <math>\scriptstyle \{|0\rangle_A, |1\rangle_A\}</math> of and two basis vectors <math>\scriptstyle \{|0\rangle_B, |1\rangle_B\}</math> of , the following is an entangled state:
If the composite system is in this state, it is impossible to attribute to either system or system 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 space, but which cannot be separated into pure states of each and ).
If the composite system is in this state, it is impossible to attribute to either system or system 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 space, but which cannot be separated into pure states of each and ).
如果复合系统处于这种状态，就不可能把某个系统或某个系统归于一个确定的纯状态。另一种说法是，虽然整个状态的冯诺依曼熵为零（就像任何纯状态一样），但子系统的熵大于零。从这个意义上说，系统是“纠缠”的。这对干涉测量法有具体的经验影响。上面的例子是四个贝尔态中的一个，它们是（最大）纠缠纯态（空间的纯态，但不能分为每个和的纯态）。
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{{by whom|date=August 2020}} 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.<ref name="medium.com"/>
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{{by whom|date=August 2020}} 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.<ref name="medium.com"/>