[论文解读] Bohr's Complementarity: Completed with Entanglement

Xiao‐Feng Qian, A. Nick Vamivakas+1 more|arXiv (Cornell University)|Mar 13, 2018

Quantum Mechanics and Applications参考文献 15被引用 29

一句话总结

本文通过引入量子纠缠——以 concurrence 量化——作为缺失的第三种成分，完成了玻尔互补性原理，建立了适用于经典与量子光学场的普遍恒等式 $V^2 + D^2 + C^2 = 1$。该结果将波-粒二象性与量子/经典纠缠统一起来，通过向量空间形式化方法实现了互斥性与完备性，解决了长达九十年的解释性困惑。

ABSTRACT

Ninety years ago in 1927, at an international congress in Como, Italy, Niels Bohr gave an address which is recognized as the first instance in which the term "complementarity", as a physical concept, was spoken publicly [1], revealing Bohr's own thinking about Louis de Broglie's "duality". Bohr had very slowly accepted duality as a principle of physics: close observation of any quantum object will reveal either wave-like or particle-like behavior, one or the other of two fundamental and complementary features. Little disagreement exists today about complementarity's importance and broad applicability in quantum science. Book-length scholarly examinations even provide speculations about the relevance of complementarity in fields as different from physics as biology, psychology and social anthropology, connections which were apparently of interest to Bohr himself (see Jammer [2], Murdoch [3] and Whitaker [4]). Confusion evident in Como following his talk was not eliminated by Bohr's article [1], and complementarity has been subjected to nine decades of repeated examination ever since with no agreed resolution. Semi-popular treatments [5] as well as expert examinations [6-9] show that the topic cannot be avoided, and complementarity retains its central place in the interpretation of quantum mechanics. However, recent approaches by our group [10-13] and others [14-20] to the underlying notion of coherence now allow us to present a universal formulation of complementarity that may signal the end to the confusion. We demonstrate a new relationship that constrains the behavior of an electromagnetic field (quantum or classical) in the fundamental context of two-slit experiments. We show that entanglement is the ingredient needed to complete Bohr's formulation of complementarity, debated for decades because of its incompleteness.

研究动机与目标

• 为解决玻尔互补性原理长期存在的解释性困惑，尽管其在量子力学中具有基础性地位，但其完整表述始终缺失。
• 识别玻尔原始表述中缺失的成分——纠缠，证明其对于实现互补描述中的互斥性与完备性至关重要。
• 证明波粒二象性之间的权衡（可见度 $V$ 与可区分性 $D$）可通过凹度 $C$ 完成，形成对经典与量子系统均有效的普遍恒等式。
• 在统一的向量空间框架下，将经典波物理与量子力学统一起来，表明纠缠并非量子系统独有。
• 提供一个普遍有效、量化的互补性表述，终结自玻尔1927年科莫演讲以来长达90年的争论。

提出的方法

• 将电磁场形式化为空间与其他自由度的叠加：$\vec{E}(r_{\perp},z,t) = A u_a \vec{\phi}_a + B u_b \vec{\phi}_b$，将其视为向量空间中的纯态。
• 从干涉条纹对比度定义可见度 ($V$)，从路径信息定义可区分性 ($D$)，二者均为双缝实验中的标准度量。
• 引入凹度 ($C$) 作为空间自由度与其他自由度之间纠缠的度量，其来源于内积 $\gamma = \langle \vec{\phi}_a^* \cdot \vec{\phi}_b \rangle$。
• 通过归一化及凹度公式 $C = \frac{2\sqrt{(1-|\gamma|^2)I_{ac}I_{bc}}}{I_{ac}+I_{bc}}$ 推导出恒等式 $V^2 + D^2 + C^2 = 1$，适用于归一化场。
• 通过可调偏振与路径控制的实验装置，实现场态的层析重建，对恒等式进行实验验证。
• 将形式化方法扩展至经典光学束与单光子量子态，确认其在不同物理 regime 下的普适性。

实验结果

研究问题

• RQ1 玻尔原始互补性原理中缺失的成分是什么？该缺失导致其无法实现完备性与互斥性？
• RQ2 空间自由度与非空间自由度之间的纠缠能否以一种方式量化，从而完成波行为与粒子行为之间的二象性权衡？
• RQ3 恒等式 $V^2 + D^2 + C^2 = 1$ 是否在具有向量空间结构的经典与量子系统中均普遍成立？
• RQ4 引入纠缠如何解决量子力学中互补性长期存在的解释性困惑？
• RQ5 经典波系统是否能表现出与量子系统相同的基于纠缠的互补性？

主要发现

• 恒等式 $V^2 + D^2 + C^2 = 1$ 作为光学场的普遍向量空间恒等式被推导出来，完成了玻尔互补性原理。
• 在极端情况 $V=1$ 下，场在空间与其他自由度之间实现因子分解，此时 $D=0$ 且 $C=0$，表明为纯粹的波动行为。
• 当 $D=1$ 时，场局域于单个狭缝，$V=0$ 且 $C=0$，表明为纯粹的粒子行为且可分。
• 当 $V=0$ 且 $D=0$ 时，场达到最大纠缠，$C=1$，对应于贝尔态的经典类比。
• 凹度 $C$ 通过公式 $C = \frac{2\sqrt{(1-|\gamma|^2)I_{ac}I_{bc}}}{I_{ac}+I_{bc}}$ 量化，其中 $\gamma$ 为偏振态的重叠。
• 该结果通过经典光学束与单光子态的实验验证，确认了其在两类物理 regime 下的普适性。