# 爱因斯坦错了：测量光子会影响其位置

来源：百度文库 编辑：超级军网 时间：2022/05/24 01:48:28

量子力学是当今科学中接受实验分析最多的理论之一，也是科学家试图通过实验证明爱因斯坦错误的为数不多的领域之一。近日，来自澳大利亚格里菲斯大学和日本东京大学的研究团队宣称做到了这一点，他们的实验显示，测量的确会影响被测量粒子的位置。

关于这一奇特现象背后的“不确定性原理”，爱因斯坦早在20世纪20到30年代就表达了不支持的态度。他称这是“鬼魅般的远距作用”，即一个粒子可以同时在两个地方出现，直到有人对这个粒子进行测量时，它才具有确定的位置，但是在这一过程中，似乎并没有信号以超光速的速度传输给这个粒子。当这个粒子的位置确定时，物理学家称之为该粒子的波函数坍缩。

这一现象超出了同时代科学家的物理学经验，而且似乎与相对论相悖，后者认为光速是任何信息传播速度的上限。爱因斯坦提出，这个粒子并不是处于叠加态，也不是同时出现在两个地方，更可能的情况是它具有一个“真实”的位置，只是人们无法观测到。

这一现象可以通过一个思想实验来说明。假设有一道光束被分离成两半，一半发射到爱丽丝的实验室，另一半发射到鲍勃的实验室。然后，爱丽丝要指出是否探测到了光子，以及如果探测到的话光子处于什么状态——此时或许可以用波包(wave packet)来描述这个光子。在数学上，此时光子处于“叠加”(superposition)状态，即它可以同时出现在两个(或更多)地方。它的波函数似乎表明光子并没有确定的位置。

格里菲斯大学量子力学中心主任霍华德·怀斯曼(Howard Wiseman)说：“爱丽丝的测量导致叠加态坍缩。”这意味着在测量时光子存在于此处或别处，而不是在两处同时出现。如果爱丽丝探测到一个光子，意味着在鲍勃的实验室里，光子的量子态出现了坍缩，进入了零光子状态；但如果爱丽丝并没有探测到光子，那鲍勃这边的光子就坍缩到单光子状态。

在3月24日的《自然-通讯》(Nature Communications)杂志上，怀斯曼等对这一实验进行了描述。他说：“你会觉得这一现象合理吗？我觉得不会，因为爱因斯坦肯定不认为它合理。他认为这很疯狂。”从描述上看，爱丽丝的测量似乎对鲍勃的观测结果起着支配作用。

这个悖论在提出来几年之后得到了部分解答。有实验显示，尽管两个量子粒子之间的相互作用瞬时发生的，即二者信息传递速度比光速还快，但还是没有办法利用这一现象来进行信息传递。

霍华德·怀斯曼主持了新的实验，他和研究团队的成员们希望能更进一步，揭示出波函数坍缩的过程——即爱丽丝“选择”了测量并因此影响到了鲍勃的探测——是确实发生的。已经有其他实验揭示出两个粒子之间的纠缠状态，但在这项新研究中，光子是和它自身纠缠在一起。

在实验中，科学家发射的光子束经过一个分离器，一半的光传递出去到达一个实验室，另一半则经过反射到达另一个实验室(相当于前述思想实验中的爱丽丝和鲍勃的实验室)。光每一次的传播都是以单个光子的形式，因此这个光子分成了两个。在被测量之前，它以叠加态存在。

其中一个实验室利用激光作为参照，对光子的相位进行测量。如果把光想象成一道重复的正弦波，相位就是进行测量时的角度，从0到180度。当改变参照激光的角度时，就会得到对光子的不同测量结果：它或者处于某个特定的相位，或者根本就不会出现。

在另一个实验室中，科学家对光子的测量结果与前一个实验室出现了反相关性，即如果这边观测到了光子，那另一边就观测不到光子，反之亦然。后一个实验室的光子状态取决于前一个实验室的测量结果；但是，在经典物理学中这种现象是不应该出现的，两个粒子应该处于各自独立的状态。

http://tech.sina.com.cn/d/i/2015-04-07/doc-icczmvun8652741.shtml量子力学是当今科学中接受实验分析最多的理论之一，也是科学家试图通过实验证明爱因斯坦错误的为数不多的领域之一。近日，来自澳大利亚格里菲斯大学和日本东京大学的研究团队宣称做到了这一点，他们的实验显示，测量的确会影响被测量粒子的位置。

关于这一奇特现象背后的“不确定性原理”，爱因斯坦早在20世纪20到30年代就表达了不支持的态度。他称这是“鬼魅般的远距作用”，即一个粒子可以同时在两个地方出现，直到有人对这个粒子进行测量时，它才具有确定的位置，但是在这一过程中，似乎并没有信号以超光速的速度传输给这个粒子。当这个粒子的位置确定时，物理学家称之为该粒子的波函数坍缩。

这一现象超出了同时代科学家的物理学经验，而且似乎与相对论相悖，后者认为光速是任何信息传播速度的上限。爱因斯坦提出，这个粒子并不是处于叠加态，也不是同时出现在两个地方，更可能的情况是它具有一个“真实”的位置，只是人们无法观测到。

这一现象可以通过一个思想实验来说明。假设有一道光束被分离成两半，一半发射到爱丽丝的实验室，另一半发射到鲍勃的实验室。然后，爱丽丝要指出是否探测到了光子，以及如果探测到的话光子处于什么状态——此时或许可以用波包(wave packet)来描述这个光子。在数学上，此时光子处于“叠加”(superposition)状态，即它可以同时出现在两个(或更多)地方。它的波函数似乎表明光子并没有确定的位置。

格里菲斯大学量子力学中心主任霍华德·怀斯曼(Howard Wiseman)说：“爱丽丝的测量导致叠加态坍缩。”这意味着在测量时光子存在于此处或别处，而不是在两处同时出现。如果爱丽丝探测到一个光子，意味着在鲍勃的实验室里，光子的量子态出现了坍缩，进入了零光子状态；但如果爱丽丝并没有探测到光子，那鲍勃这边的光子就坍缩到单光子状态。

在3月24日的《自然-通讯》(Nature Communications)杂志上，怀斯曼等对这一实验进行了描述。他说：“你会觉得这一现象合理吗？我觉得不会，因为爱因斯坦肯定不认为它合理。他认为这很疯狂。”从描述上看，爱丽丝的测量似乎对鲍勃的观测结果起着支配作用。

这个悖论在提出来几年之后得到了部分解答。有实验显示，尽管两个量子粒子之间的相互作用瞬时发生的，即二者信息传递速度比光速还快，但还是没有办法利用这一现象来进行信息传递。

霍华德·怀斯曼主持了新的实验，他和研究团队的成员们希望能更进一步，揭示出波函数坍缩的过程——即爱丽丝“选择”了测量并因此影响到了鲍勃的探测——是确实发生的。已经有其他实验揭示出两个粒子之间的纠缠状态，但在这项新研究中，光子是和它自身纠缠在一起。

在实验中，科学家发射的光子束经过一个分离器，一半的光传递出去到达一个实验室，另一半则经过反射到达另一个实验室(相当于前述思想实验中的爱丽丝和鲍勃的实验室)。光每一次的传播都是以单个光子的形式，因此这个光子分成了两个。在被测量之前，它以叠加态存在。

其中一个实验室利用激光作为参照，对光子的相位进行测量。如果把光想象成一道重复的正弦波，相位就是进行测量时的角度，从0到180度。当改变参照激光的角度时，就会得到对光子的不同测量结果：它或者处于某个特定的相位，或者根本就不会出现。

在另一个实验室中，科学家对光子的测量结果与前一个实验室出现了反相关性，即如果这边观测到了光子，那另一边就观测不到光子，反之亦然。后一个实验室的光子状态取决于前一个实验室的测量结果；但是，在经典物理学中这种现象是不应该出现的，两个粒子应该处于各自独立的状态。

http://tech.sina.com.cn/d/i/2015-04-07/doc-icczmvun8652741.shtml

关于这一奇特现象背后的“不确定性原理”，爱因斯坦早在20世纪20到30年代就表达了不支持的态度。他称这是“鬼魅般的远距作用”，即一个粒子可以同时在两个地方出现，直到有人对这个粒子进行测量时，它才具有确定的位置，但是在这一过程中，似乎并没有信号以超光速的速度传输给这个粒子。当这个粒子的位置确定时，物理学家称之为该粒子的波函数坍缩。

这一现象超出了同时代科学家的物理学经验，而且似乎与相对论相悖，后者认为光速是任何信息传播速度的上限。爱因斯坦提出，这个粒子并不是处于叠加态，也不是同时出现在两个地方，更可能的情况是它具有一个“真实”的位置，只是人们无法观测到。

这一现象可以通过一个思想实验来说明。假设有一道光束被分离成两半，一半发射到爱丽丝的实验室，另一半发射到鲍勃的实验室。然后，爱丽丝要指出是否探测到了光子，以及如果探测到的话光子处于什么状态——此时或许可以用波包(wave packet)来描述这个光子。在数学上，此时光子处于“叠加”(superposition)状态，即它可以同时出现在两个(或更多)地方。它的波函数似乎表明光子并没有确定的位置。

格里菲斯大学量子力学中心主任霍华德·怀斯曼(Howard Wiseman)说：“爱丽丝的测量导致叠加态坍缩。”这意味着在测量时光子存在于此处或别处，而不是在两处同时出现。如果爱丽丝探测到一个光子，意味着在鲍勃的实验室里，光子的量子态出现了坍缩，进入了零光子状态；但如果爱丽丝并没有探测到光子，那鲍勃这边的光子就坍缩到单光子状态。

在3月24日的《自然-通讯》(Nature Communications)杂志上，怀斯曼等对这一实验进行了描述。他说：“你会觉得这一现象合理吗？我觉得不会，因为爱因斯坦肯定不认为它合理。他认为这很疯狂。”从描述上看，爱丽丝的测量似乎对鲍勃的观测结果起着支配作用。

这个悖论在提出来几年之后得到了部分解答。有实验显示，尽管两个量子粒子之间的相互作用瞬时发生的，即二者信息传递速度比光速还快，但还是没有办法利用这一现象来进行信息传递。

霍华德·怀斯曼主持了新的实验，他和研究团队的成员们希望能更进一步，揭示出波函数坍缩的过程——即爱丽丝“选择”了测量并因此影响到了鲍勃的探测——是确实发生的。已经有其他实验揭示出两个粒子之间的纠缠状态，但在这项新研究中，光子是和它自身纠缠在一起。

在实验中，科学家发射的光子束经过一个分离器，一半的光传递出去到达一个实验室，另一半则经过反射到达另一个实验室(相当于前述思想实验中的爱丽丝和鲍勃的实验室)。光每一次的传播都是以单个光子的形式，因此这个光子分成了两个。在被测量之前，它以叠加态存在。

其中一个实验室利用激光作为参照，对光子的相位进行测量。如果把光想象成一道重复的正弦波，相位就是进行测量时的角度，从0到180度。当改变参照激光的角度时，就会得到对光子的不同测量结果：它或者处于某个特定的相位，或者根本就不会出现。

在另一个实验室中，科学家对光子的测量结果与前一个实验室出现了反相关性，即如果这边观测到了光子，那另一边就观测不到光子，反之亦然。后一个实验室的光子状态取决于前一个实验室的测量结果；但是，在经典物理学中这种现象是不应该出现的，两个粒子应该处于各自独立的状态。

http://tech.sina.com.cn/d/i/2015-04-07/doc-icczmvun8652741.shtml量子力学是当今科学中接受实验分析最多的理论之一，也是科学家试图通过实验证明爱因斯坦错误的为数不多的领域之一。近日，来自澳大利亚格里菲斯大学和日本东京大学的研究团队宣称做到了这一点，他们的实验显示，测量的确会影响被测量粒子的位置。

关于这一奇特现象背后的“不确定性原理”，爱因斯坦早在20世纪20到30年代就表达了不支持的态度。他称这是“鬼魅般的远距作用”，即一个粒子可以同时在两个地方出现，直到有人对这个粒子进行测量时，它才具有确定的位置，但是在这一过程中，似乎并没有信号以超光速的速度传输给这个粒子。当这个粒子的位置确定时，物理学家称之为该粒子的波函数坍缩。

这一现象超出了同时代科学家的物理学经验，而且似乎与相对论相悖，后者认为光速是任何信息传播速度的上限。爱因斯坦提出，这个粒子并不是处于叠加态，也不是同时出现在两个地方，更可能的情况是它具有一个“真实”的位置，只是人们无法观测到。

这一现象可以通过一个思想实验来说明。假设有一道光束被分离成两半，一半发射到爱丽丝的实验室，另一半发射到鲍勃的实验室。然后，爱丽丝要指出是否探测到了光子，以及如果探测到的话光子处于什么状态——此时或许可以用波包(wave packet)来描述这个光子。在数学上，此时光子处于“叠加”(superposition)状态，即它可以同时出现在两个(或更多)地方。它的波函数似乎表明光子并没有确定的位置。

格里菲斯大学量子力学中心主任霍华德·怀斯曼(Howard Wiseman)说：“爱丽丝的测量导致叠加态坍缩。”这意味着在测量时光子存在于此处或别处，而不是在两处同时出现。如果爱丽丝探测到一个光子，意味着在鲍勃的实验室里，光子的量子态出现了坍缩，进入了零光子状态；但如果爱丽丝并没有探测到光子，那鲍勃这边的光子就坍缩到单光子状态。

在3月24日的《自然-通讯》(Nature Communications)杂志上，怀斯曼等对这一实验进行了描述。他说：“你会觉得这一现象合理吗？我觉得不会，因为爱因斯坦肯定不认为它合理。他认为这很疯狂。”从描述上看，爱丽丝的测量似乎对鲍勃的观测结果起着支配作用。

这个悖论在提出来几年之后得到了部分解答。有实验显示，尽管两个量子粒子之间的相互作用瞬时发生的，即二者信息传递速度比光速还快，但还是没有办法利用这一现象来进行信息传递。

霍华德·怀斯曼主持了新的实验，他和研究团队的成员们希望能更进一步，揭示出波函数坍缩的过程——即爱丽丝“选择”了测量并因此影响到了鲍勃的探测——是确实发生的。已经有其他实验揭示出两个粒子之间的纠缠状态，但在这项新研究中，光子是和它自身纠缠在一起。

在实验中，科学家发射的光子束经过一个分离器，一半的光传递出去到达一个实验室，另一半则经过反射到达另一个实验室(相当于前述思想实验中的爱丽丝和鲍勃的实验室)。光每一次的传播都是以单个光子的形式，因此这个光子分成了两个。在被测量之前，它以叠加态存在。

其中一个实验室利用激光作为参照，对光子的相位进行测量。如果把光想象成一道重复的正弦波，相位就是进行测量时的角度，从0到180度。当改变参照激光的角度时，就会得到对光子的不同测量结果：它或者处于某个特定的相位，或者根本就不会出现。

在另一个实验室中，科学家对光子的测量结果与前一个实验室出现了反相关性，即如果这边观测到了光子，那另一边就观测不到光子，反之亦然。后一个实验室的光子状态取决于前一个实验室的测量结果；但是，在经典物理学中这种现象是不应该出现的，两个粒子应该处于各自独立的状态。

http://tech.sina.com.cn/d/i/2015-04-07/doc-icczmvun8652741.shtml

看来 爱因斯坦的理论 也不过就是 低光速条件下适用

显然，写这篇报道的人啥都没搞懂

不理解单个光子如何分成2个

这个举例我在什么书上看到过，爱丽丝和鲍勃，这两个名字在一起好熟。似乎还提到了量子咖啡屋，还有测量赛车的长度以及旋转木马等。反正这不是什么新玩意。

我总觉得量子力学方面，10年前的书和现在的虽有区别，但不是很大，这点和计算机比起来简直就是在停顿。

哎！被三体的智子困住了吗？

我总觉得量子力学方面，10年前的书和现在的虽有区别，但不是很大，这点和计算机比起来简直就是在停顿。

哎！被三体的智子困住了吗？

显然，写这篇报道的人啥都没搞懂

有没有办法搞到原版论文看看？

有没有办法搞到原版论文看看？

coolfile 发表于 2015-4-11 00:53

有没有办法搞到原版论文看看？

http://www.nature.com/ncomms/201 ... ull/ncomms7665.html

Experimental proof of nonlocal wavefunction collapse for a single particle using homodyne measurements

Maria Fuwa,

Shuntaro Takeda,

Marcin Zwierz,

Howard M. Wiseman

& Akira Furusawa

Nature Communications 6, Article number: 6665 doi:10.1038/ncomms7665

有没有办法搞到原版论文看看？

http://www.nature.com/ncomms/201 ... ull/ncomms7665.html

Experimental proof of nonlocal wavefunction collapse for a single particle using homodyne measurements

Maria Fuwa,

Shuntaro Takeda,

Marcin Zwierz,

Howard M. Wiseman

& Akira Furusawa

Nature Communications 6, Article number: 6665 doi:10.1038/ncomms7665

http://www.nature.com/ncomms/2015/150324/ncomms7665/full/ncomms7665.html

Experimental proof of ...

22镑呀，买不起。

Experimental proof of ...

22镑呀，买不起。

coolfile 发表于 2015-4-11 16:26

22镑呀，买不起。

Abstract

A single quantum particle can be described by a wavefunction that spreads over arbitrarily large distances; however, it is never detected in two (or more) places. This strange phenomenon is explained in the quantum theory by what Einstein repudiated as ‘spooky action at a distance’: the instantaneous nonlocal collapse of the wavefunction to wherever the particle is detected. Here we demonstrate this single-particle spooky action, with no efficiency loophole, by splitting a single photon between two laboratories and experimentally testing whether the choice of measurement in one laboratory really causes a change in the local quantum state in the other laboratory. To this end, we use homodyne measurements with six different measurement settings and quantitatively verify Einstein’s spooky action by violating an Einstein–Podolsky–Rosen-steering inequality by 0.042±0.006. Our experiment also verifies the entanglement of the split single photon even when one side is untrusted.

Introduction

Einstein never accepted orthodox quantum mechanics because he did not believe that its nonlocal collapse of the wavefunction could be real. When he first made this argument in 1927 (ref. 1), he considered just a single particle. The particle’s wavefunction was diffracted through a tiny hole so that it ‘dispersed’ over a large hemispherical area before encountering a screen of that shape covered in photographic film. Since the film only ever registers the particle at one point on the screen, orthodox quantum mechanics must postulate a ‘peculiar mechanism of action at a distance, which prevents the wave... from producing an action in two places on the screen’1. That is, according to the theory, the detection at one point must instantaneously collapse the wavefunction to nothing at all other points.

It was only in 2010, nearly a century after Einstein’s original proposal, that a scheme to rigorously test Einstein’s ‘spooky action at a distance’2 using a single particle (a photon), as in his original conception, was conceived3. In this scheme, Einstein’s 1927 gedankenexperiment is simplified so that the single photon is split into just two wavepackets, one sent to a laboratory supervised by Alice and the other to a distant laboratory supervised by Bob. However, there is a key difference, which enables demonstration of the nonlocal collapse experimentally: rather than simply detecting the presence or absence of the photon, homodyne detection is used. This gives Alice the power to make different measurements, and enables Bob to test (using tomography) whether Alice’s measurement choice affects the way his conditioned state collapses, without having to trust anything outside his own laboratory.

The key role of measurement choice by Alice in demonstrating ‘spooky action at a distance’ was introduced in the famous Einstein–Podolsky–Rosen (EPR) paper4 of 1935. In its most general form5, this phenomenon has been called EPR-steering6, to acknowledge the contribution and terminology of Schrödinger7, who talked of Alice ‘steering’ the state of Bob’s quantum system. From a quantum information perspective, EPR-steering is equivalent to the task of entanglement verification when Bob (and his detectors) can be trusted but Alice (or her detectors) cannot5. This is strictly harder than verifying entanglement with both parties trusted8, but strictly easier than violating a Bell inequality9, where neither party is trusted8.

To demonstrate EPR-steering quantitatively, it is necessary and sufficient to violate an EPR-steering inequality involving Alice’s and Bob’s results6. Such a violation has been shown to be necessary for one-sided device-independent quantum key distribution as well10. Because Alice is not trusted in EPR-steering, a rigorous experiment cannot use postselection on Alice’s side5, 6, 11, 12, 13, 14. Previous experimental tests of nonlocal quantum-state collapse over macroscopic distances, without postselection on Alice’s side, have involved the distribution of entangled states of multiple particles11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27. Experiments demonstrating Bell nonlocality (violating a Bell inequality) for a single photon have involved postselection on both sides, opening the efficiency loophole28, 29; these lines of work would otherwise have demonstrated EPR-steering of a single photon as well. A recent experimental test of entanglement for a single photon via an entanglement witness has no efficiency loophole30; however, it demonstrates a weaker form on nonlocality than EPR-steering5, 6.

While the nonlocal properties of a single particle have spurred much theoretical debate3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 and many fundamental experiments28, 29, 30, 41, 42, 43, 44, 45, it is also recognized that a single photon split between two spatially distant modes is a very flexible entanglement resource for quantum information tasks: they have been teleported43, swapped44 and purified with linear optics45. Spatial-mode entanglement35 more generally has a broad potential ranging from long-distance quantum communication46, 47, quantum computation43, 44, 45, 48, to the simulation of quantum many-body systems in tabletop implementations49.

In this paper, we rigorously demonstrate Einstein’s elusive ‘spooky action at a distance’ for a single particle without opening the efficiency loophole. We note that, unlike ref. 12, we do not claim to have closed the separation loophole. Our work is the one-sided device-independent verification of spatial-mode entanglement for a single photon.

Results

Our gedankenexperiment

First, we explain in detail our simplified version (Fig. 1) of Einstein's original single-particle gedankenexperiment described above, formalized as the task of entanglement verification with only one trusted party, as proposed in ref. 3. That is, assuming only that Bob can reliably probe the quantum state at his location, he can experimentally prove that the choice of measurement by the distant Alice affects his quantum state. This is exactly the ‘spooky action at a distance’ that Einstein found objectionable50.

Figure 1: Our simplified version of Einstein’s original gedankenexperiment.

We start with a pure single photon |1› incident on a beam splitter of reflectivity R. As a result, the state of the single photon becomes spread out between two spatially separated modes A and B:

The transmitted mode is sent to Alice, and the reflected one to Bob. We allow Alice and Bob to use homodyne detection. This allows Bob to do quantum tomography on his state51, 52, and gives Alice the power to do different types of measurement (which is necessary for an EPR-steering test) by controlling her local oscillator (LO) phase θ.

If Alice were simply to detect the presence or absence of a photon, then Bob’s measurement of the same observable will be anticorrelated with Alice's, as in ref. 42. However, this does not prove that Alice’s measurement affected Bob’s local state because such perfect anticorrelations would also arise from a classical mixture of |0›A |1›B and |1›A |0›B, in which Bob’s measurement simply reveals a pre-existing local state for him, |1›B or |0›B. To demonstrate nonlocal quantum-state collapse, measurement choice by Alice is essential5.

Following Alice’s homodyne measurement of the θ-quadrature , yielding result , Bob’s local state is collapsed to

where the proportionality factor is . Thus, by changing her LO phase θ, Alice controls the relative phase of the vacuum and one photon component of Bob’s conditioned state (modulo π, depending on the sign of the she obtains). Because of this, it is convenient for Alice to coarse-grain her result to . It is possible that a more sensitive EPR-steering inequality could be obtained that makes use of a finer-grained binning of Alices results; however, two bins are sufficient for our experiment.

Independently of Alice’s measurement, Bob performs full quantum-state tomography using homodyne detection on his portion of the single photon. This enables him to reconstruct his state, for each value of Alice’s LO setting θ, and coarse-grained result s. Because of the coarse-graining, even under the idealization of the pure state as in Equation (2), Bob’s (normalized) conditioned state will be mixed:

The idealized theoretical prediction for the unconditioned quantum state is

where P(s|θ)=0.5 is the relative frequency for Alice to report sign s given setting θ.

We once again emphasize the intrinsic lack of trust that Bob has with respect to anything that happens in Alice’s laboratory. Neither her honesty nor the efficiency or accuracy of her measurement devices is assumed in an EPR-steering test. On the other hand, Bob does trust his own measurement devices. From the experimental point of view this means that his photoreceivers do not have to be efficient, and that he can post-select on finding his system in a particular subspace. In particular, for our experiment (where there are small two-photon terms), he can restrict his reconstructed state to the qubit subspace spanned by {|0›B, |1›B}. Despite Bob’s lack of trust in Alice, she can convince him that her choice of measurement setting, θ, steers his quantum state , proving that his system has no local quantum description. We now present data showing this effect qualitatively, before our quantitative proof of EPR-steering for this single-photon system.

Bob’s tomography results

In our experiment, the (heralded single-photon) input state to the beam splitter comprises mostly a pure single-photon state |1›, but it has some admixture of the vacuum state |0›, and a (much smaller) admixture of the two-photon state |2›. (For more details on state preparation see the Methods.) Following the beam splitter and Alice’s measurement, Bob reconstructs his conditioned quantum states by separately analysing his homodyne data (taken by scanning his LO phase φ from 0 to 2π) for each value of Alice’s LO phase θ and result s. The reconstructed density matrices in the {|0›B, |1›B} subspace and the corresponding Wigner functions give complementary ways to visualize how Bob’s local quantum state can collapse in consequence of Alice’s measurement.

The results of Bob’s tomography, which take into account inefficiency in Bob’s detection system by using the maximum likelihood method52 during quantum-state reconstruction, are presented in Fig. 2, for the case R=0.50. There is good qualitative agreement between these results and the theoretical predictions for the ideal case in Equations (3) and (4). In particular, there are four features to note.

Figure 2: Bob’s unconditioned and conditioned quantum states for R=0.50.

First, Bob’s unconditioned quantum state is a phase-independent statistical mixture of the vacuum and single-photon components (see Fig. 2a), as in Equation (4). The Wigner functions are rotationally invariant, and the off-diagonal terms in are zero. The vacuum component is slightly greater than the single-photon component because of the less-than-unit efficiency p1=0.857±0.008<1 of single-photon generation (for more details see the Methods).

Second, Bob’s conditioned quantum states are not phase-independent, but rather exhibit coherence between the vacuum and single-photon components (see Fig. 2b–d) as predicted by Equation (3). The Wigner functions are not rotationally invariant (and have a mean field: ), and the off-diagonal terms in are non-zero. Furthermore, the negative dips observed in the conditioned Wigner functions prove the strong nonclassical character of Bob’s local quantum state53.

Third, depending on Alice’s result sε{+,−}, Bob’s local quantum state is collapsed into complementary states (in the sense that they sum to the unconditioned state; compare columns (b) and (c) in Fig. 2). This effect is manifested most clearly by a relative π rotation between the conditioned Wigner functions and the opposite signs of the off-diagonal elements of , as expected from Equation (3).

Finally, Alice can steer Bob’s possible conditioned states by her choice of measurement setting θ, as predicted. Comparing the results in columns (b) and (d) of Fig. 2, it is immediately clear that the conditioned Wigner function is phase-shifted with respect to by an angle . Moreover, we also notice the decrease in the value of the real off-diagonal elements and the emergence of the imaginary off-diagonal elements in the conditioned density matrix as compared with . Naturally, the described EPR-steering effect can be demonstrated for all possible values of Alice’s LO phase θ.

The above results suggest that Bob’s portion of the single photon cannot have a local quantum state before Alice defining her measurement setting θ. However, a proof that this is the case requires much more quantitative analysis, which we now present.

The EPR-steering inequality

From Equation (2), Bob’s portion of the single photon is a qubit (a quantum system spanned by |0›B and |1›B). In the experiment there are small terms with higher photon numbers, but, as explained above, Bob is allowed to restrict to the {|0›B, |1›B} subspace. Here we consider a nonlinear EPR-steering inequality for the qubit subspace. It involves n different measurement settings θj by Alice, and is given by3

Here and f(n) is a monotonically decreasing positive function of the number of measurement settings defined in Eq. (4.15) of ref. 3, under the assumption that . The left-hand-side thus correlates Bob’s tomographic reconstruction with Alice’s announced result s, but makes no assumptions about how Alice generates this result. On the right-hand-side, is Bob’s unconditioned state, while .

For the ideal case considered above, theory predicts a violation of the EPR-steering inequality (5) for n≥2 and any value of R (apart from 0 and 1). (However, experimental imperfections associated with the single-photon source and the inefficiency of Alice’s photoreceivers make it more difficult to obtain a violation. For details of the theoretical predictions, see Supplementary Discussion.) While the inequality is most easily violated for n=∞ (for which f(∞)=2/π≈0.6366) for our experiment it was sufficient to use n=6 (for which f(6)=0.6440). The experimental results in Fig. 3 well match the theoretical predictions calculated using independently measured experimental parameters; see Supplementary Discussion. The EPR-steering inequality is violated for R=0.08, 0.38 and 0.50, but not for R=0.90; it is most violated at R=0.38 by 0.042±0.006.

Figure 3: The left- and right-hand sides of the EPR-steering inequality.

The violation of the EPR-steering inequality by seven s.d.’s is a clear proof that Bob’s quantum state cannot exist independently of Alice, but rather is collapsed by Alice’s measurement. We were able to rigorously demonstrate this for a single particle without opening the efficiency loophole by using the combination of multiple (n=6) measurement settings and highly efficient phase-sensitive homodyne measurements for Alice (ηh=0.96±0.01), coupled with a high single-photon occupation probability (p1=0.857±0.008). Without the close-to-unity values of these parameters, the nonlocal collapse of the single-photon wavefunction could not have been detected, as in the case of ref. 41 (see ref. 3 for a detailed discussion).

Discussion

We have demonstrated, both rigorously and in the easy visualized form of nonclassical Wigner functions, the nonlocality of a single particle using a modern and simplified version of Einstein’s original gedankenexperiment. That is, we demonstrated Einstein’s ‘spooky action at a distance’ in that Bob’s quantum state (of his half of a single photon) was probably dependent on Alice’s choice of measurement (on the other half), and could not have been pre-existing. Quantitatively, we violated a multisetting nonlinear EPR-steering inequality by several s.d.’s (0.042±0.006).

This EPR-steering experiment is a form of entanglement verification, for a single-photon mode-entangled state, which does not require Bob to trust Alice’s devices, or her reported outcomes. It was possible only because we used a high-fidelity single-photon state and very high-efficiency homodyne measurements, to perform the steering measurements on Alice’s side and the tomographic state reconstruction on Bob’s. Our results may open a way to new protocols for one-sided device-independent quantum key distribution10 based on the DLCZ protocol employing single-rail qubits46.

22镑呀，买不起。

Abstract

A single quantum particle can be described by a wavefunction that spreads over arbitrarily large distances; however, it is never detected in two (or more) places. This strange phenomenon is explained in the quantum theory by what Einstein repudiated as ‘spooky action at a distance’: the instantaneous nonlocal collapse of the wavefunction to wherever the particle is detected. Here we demonstrate this single-particle spooky action, with no efficiency loophole, by splitting a single photon between two laboratories and experimentally testing whether the choice of measurement in one laboratory really causes a change in the local quantum state in the other laboratory. To this end, we use homodyne measurements with six different measurement settings and quantitatively verify Einstein’s spooky action by violating an Einstein–Podolsky–Rosen-steering inequality by 0.042±0.006. Our experiment also verifies the entanglement of the split single photon even when one side is untrusted.

Introduction

Einstein never accepted orthodox quantum mechanics because he did not believe that its nonlocal collapse of the wavefunction could be real. When he first made this argument in 1927 (ref. 1), he considered just a single particle. The particle’s wavefunction was diffracted through a tiny hole so that it ‘dispersed’ over a large hemispherical area before encountering a screen of that shape covered in photographic film. Since the film only ever registers the particle at one point on the screen, orthodox quantum mechanics must postulate a ‘peculiar mechanism of action at a distance, which prevents the wave... from producing an action in two places on the screen’1. That is, according to the theory, the detection at one point must instantaneously collapse the wavefunction to nothing at all other points.

It was only in 2010, nearly a century after Einstein’s original proposal, that a scheme to rigorously test Einstein’s ‘spooky action at a distance’2 using a single particle (a photon), as in his original conception, was conceived3. In this scheme, Einstein’s 1927 gedankenexperiment is simplified so that the single photon is split into just two wavepackets, one sent to a laboratory supervised by Alice and the other to a distant laboratory supervised by Bob. However, there is a key difference, which enables demonstration of the nonlocal collapse experimentally: rather than simply detecting the presence or absence of the photon, homodyne detection is used. This gives Alice the power to make different measurements, and enables Bob to test (using tomography) whether Alice’s measurement choice affects the way his conditioned state collapses, without having to trust anything outside his own laboratory.

The key role of measurement choice by Alice in demonstrating ‘spooky action at a distance’ was introduced in the famous Einstein–Podolsky–Rosen (EPR) paper4 of 1935. In its most general form5, this phenomenon has been called EPR-steering6, to acknowledge the contribution and terminology of Schrödinger7, who talked of Alice ‘steering’ the state of Bob’s quantum system. From a quantum information perspective, EPR-steering is equivalent to the task of entanglement verification when Bob (and his detectors) can be trusted but Alice (or her detectors) cannot5. This is strictly harder than verifying entanglement with both parties trusted8, but strictly easier than violating a Bell inequality9, where neither party is trusted8.

To demonstrate EPR-steering quantitatively, it is necessary and sufficient to violate an EPR-steering inequality involving Alice’s and Bob’s results6. Such a violation has been shown to be necessary for one-sided device-independent quantum key distribution as well10. Because Alice is not trusted in EPR-steering, a rigorous experiment cannot use postselection on Alice’s side5, 6, 11, 12, 13, 14. Previous experimental tests of nonlocal quantum-state collapse over macroscopic distances, without postselection on Alice’s side, have involved the distribution of entangled states of multiple particles11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27. Experiments demonstrating Bell nonlocality (violating a Bell inequality) for a single photon have involved postselection on both sides, opening the efficiency loophole28, 29; these lines of work would otherwise have demonstrated EPR-steering of a single photon as well. A recent experimental test of entanglement for a single photon via an entanglement witness has no efficiency loophole30; however, it demonstrates a weaker form on nonlocality than EPR-steering5, 6.

While the nonlocal properties of a single particle have spurred much theoretical debate3, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 and many fundamental experiments28, 29, 30, 41, 42, 43, 44, 45, it is also recognized that a single photon split between two spatially distant modes is a very flexible entanglement resource for quantum information tasks: they have been teleported43, swapped44 and purified with linear optics45. Spatial-mode entanglement35 more generally has a broad potential ranging from long-distance quantum communication46, 47, quantum computation43, 44, 45, 48, to the simulation of quantum many-body systems in tabletop implementations49.

In this paper, we rigorously demonstrate Einstein’s elusive ‘spooky action at a distance’ for a single particle without opening the efficiency loophole. We note that, unlike ref. 12, we do not claim to have closed the separation loophole. Our work is the one-sided device-independent verification of spatial-mode entanglement for a single photon.

Results

Our gedankenexperiment

First, we explain in detail our simplified version (Fig. 1) of Einstein's original single-particle gedankenexperiment described above, formalized as the task of entanglement verification with only one trusted party, as proposed in ref. 3. That is, assuming only that Bob can reliably probe the quantum state at his location, he can experimentally prove that the choice of measurement by the distant Alice affects his quantum state. This is exactly the ‘spooky action at a distance’ that Einstein found objectionable50.

Figure 1: Our simplified version of Einstein’s original gedankenexperiment.

We start with a pure single photon |1› incident on a beam splitter of reflectivity R. As a result, the state of the single photon becomes spread out between two spatially separated modes A and B:

The transmitted mode is sent to Alice, and the reflected one to Bob. We allow Alice and Bob to use homodyne detection. This allows Bob to do quantum tomography on his state51, 52, and gives Alice the power to do different types of measurement (which is necessary for an EPR-steering test) by controlling her local oscillator (LO) phase θ.

If Alice were simply to detect the presence or absence of a photon, then Bob’s measurement of the same observable will be anticorrelated with Alice's, as in ref. 42. However, this does not prove that Alice’s measurement affected Bob’s local state because such perfect anticorrelations would also arise from a classical mixture of |0›A |1›B and |1›A |0›B, in which Bob’s measurement simply reveals a pre-existing local state for him, |1›B or |0›B. To demonstrate nonlocal quantum-state collapse, measurement choice by Alice is essential5.

Following Alice’s homodyne measurement of the θ-quadrature , yielding result , Bob’s local state is collapsed to

where the proportionality factor is . Thus, by changing her LO phase θ, Alice controls the relative phase of the vacuum and one photon component of Bob’s conditioned state (modulo π, depending on the sign of the she obtains). Because of this, it is convenient for Alice to coarse-grain her result to . It is possible that a more sensitive EPR-steering inequality could be obtained that makes use of a finer-grained binning of Alices results; however, two bins are sufficient for our experiment.

Independently of Alice’s measurement, Bob performs full quantum-state tomography using homodyne detection on his portion of the single photon. This enables him to reconstruct his state, for each value of Alice’s LO setting θ, and coarse-grained result s. Because of the coarse-graining, even under the idealization of the pure state as in Equation (2), Bob’s (normalized) conditioned state will be mixed:

The idealized theoretical prediction for the unconditioned quantum state is

where P(s|θ)=0.5 is the relative frequency for Alice to report sign s given setting θ.

We once again emphasize the intrinsic lack of trust that Bob has with respect to anything that happens in Alice’s laboratory. Neither her honesty nor the efficiency or accuracy of her measurement devices is assumed in an EPR-steering test. On the other hand, Bob does trust his own measurement devices. From the experimental point of view this means that his photoreceivers do not have to be efficient, and that he can post-select on finding his system in a particular subspace. In particular, for our experiment (where there are small two-photon terms), he can restrict his reconstructed state to the qubit subspace spanned by {|0›B, |1›B}. Despite Bob’s lack of trust in Alice, she can convince him that her choice of measurement setting, θ, steers his quantum state , proving that his system has no local quantum description. We now present data showing this effect qualitatively, before our quantitative proof of EPR-steering for this single-photon system.

Bob’s tomography results

In our experiment, the (heralded single-photon) input state to the beam splitter comprises mostly a pure single-photon state |1›, but it has some admixture of the vacuum state |0›, and a (much smaller) admixture of the two-photon state |2›. (For more details on state preparation see the Methods.) Following the beam splitter and Alice’s measurement, Bob reconstructs his conditioned quantum states by separately analysing his homodyne data (taken by scanning his LO phase φ from 0 to 2π) for each value of Alice’s LO phase θ and result s. The reconstructed density matrices in the {|0›B, |1›B} subspace and the corresponding Wigner functions give complementary ways to visualize how Bob’s local quantum state can collapse in consequence of Alice’s measurement.

The results of Bob’s tomography, which take into account inefficiency in Bob’s detection system by using the maximum likelihood method52 during quantum-state reconstruction, are presented in Fig. 2, for the case R=0.50. There is good qualitative agreement between these results and the theoretical predictions for the ideal case in Equations (3) and (4). In particular, there are four features to note.

Figure 2: Bob’s unconditioned and conditioned quantum states for R=0.50.

First, Bob’s unconditioned quantum state is a phase-independent statistical mixture of the vacuum and single-photon components (see Fig. 2a), as in Equation (4). The Wigner functions are rotationally invariant, and the off-diagonal terms in are zero. The vacuum component is slightly greater than the single-photon component because of the less-than-unit efficiency p1=0.857±0.008<1 of single-photon generation (for more details see the Methods).

Second, Bob’s conditioned quantum states are not phase-independent, but rather exhibit coherence between the vacuum and single-photon components (see Fig. 2b–d) as predicted by Equation (3). The Wigner functions are not rotationally invariant (and have a mean field: ), and the off-diagonal terms in are non-zero. Furthermore, the negative dips observed in the conditioned Wigner functions prove the strong nonclassical character of Bob’s local quantum state53.

Third, depending on Alice’s result sε{+,−}, Bob’s local quantum state is collapsed into complementary states (in the sense that they sum to the unconditioned state; compare columns (b) and (c) in Fig. 2). This effect is manifested most clearly by a relative π rotation between the conditioned Wigner functions and the opposite signs of the off-diagonal elements of , as expected from Equation (3).

Finally, Alice can steer Bob’s possible conditioned states by her choice of measurement setting θ, as predicted. Comparing the results in columns (b) and (d) of Fig. 2, it is immediately clear that the conditioned Wigner function is phase-shifted with respect to by an angle . Moreover, we also notice the decrease in the value of the real off-diagonal elements and the emergence of the imaginary off-diagonal elements in the conditioned density matrix as compared with . Naturally, the described EPR-steering effect can be demonstrated for all possible values of Alice’s LO phase θ.

The above results suggest that Bob’s portion of the single photon cannot have a local quantum state before Alice defining her measurement setting θ. However, a proof that this is the case requires much more quantitative analysis, which we now present.

The EPR-steering inequality

From Equation (2), Bob’s portion of the single photon is a qubit (a quantum system spanned by |0›B and |1›B). In the experiment there are small terms with higher photon numbers, but, as explained above, Bob is allowed to restrict to the {|0›B, |1›B} subspace. Here we consider a nonlinear EPR-steering inequality for the qubit subspace. It involves n different measurement settings θj by Alice, and is given by3

Here and f(n) is a monotonically decreasing positive function of the number of measurement settings defined in Eq. (4.15) of ref. 3, under the assumption that . The left-hand-side thus correlates Bob’s tomographic reconstruction with Alice’s announced result s, but makes no assumptions about how Alice generates this result. On the right-hand-side, is Bob’s unconditioned state, while .

For the ideal case considered above, theory predicts a violation of the EPR-steering inequality (5) for n≥2 and any value of R (apart from 0 and 1). (However, experimental imperfections associated with the single-photon source and the inefficiency of Alice’s photoreceivers make it more difficult to obtain a violation. For details of the theoretical predictions, see Supplementary Discussion.) While the inequality is most easily violated for n=∞ (for which f(∞)=2/π≈0.6366) for our experiment it was sufficient to use n=6 (for which f(6)=0.6440). The experimental results in Fig. 3 well match the theoretical predictions calculated using independently measured experimental parameters; see Supplementary Discussion. The EPR-steering inequality is violated for R=0.08, 0.38 and 0.50, but not for R=0.90; it is most violated at R=0.38 by 0.042±0.006.

Figure 3: The left- and right-hand sides of the EPR-steering inequality.

The violation of the EPR-steering inequality by seven s.d.’s is a clear proof that Bob’s quantum state cannot exist independently of Alice, but rather is collapsed by Alice’s measurement. We were able to rigorously demonstrate this for a single particle without opening the efficiency loophole by using the combination of multiple (n=6) measurement settings and highly efficient phase-sensitive homodyne measurements for Alice (ηh=0.96±0.01), coupled with a high single-photon occupation probability (p1=0.857±0.008). Without the close-to-unity values of these parameters, the nonlocal collapse of the single-photon wavefunction could not have been detected, as in the case of ref. 41 (see ref. 3 for a detailed discussion).

Discussion

We have demonstrated, both rigorously and in the easy visualized form of nonclassical Wigner functions, the nonlocality of a single particle using a modern and simplified version of Einstein’s original gedankenexperiment. That is, we demonstrated Einstein’s ‘spooky action at a distance’ in that Bob’s quantum state (of his half of a single photon) was probably dependent on Alice’s choice of measurement (on the other half), and could not have been pre-existing. Quantitatively, we violated a multisetting nonlinear EPR-steering inequality by several s.d.’s (0.042±0.006).

This EPR-steering experiment is a form of entanglement verification, for a single-photon mode-entangled state, which does not require Bob to trust Alice’s devices, or her reported outcomes. It was possible only because we used a high-fidelity single-photon state and very high-efficiency homodyne measurements, to perform the steering measurements on Alice’s side and the tomographic state reconstruction on Bob’s. Our results may open a way to new protocols for one-sided device-independent quantum key distribution10 based on the DLCZ protocol employing single-rail qubits46.