[论文解读] A violation of the Tsirelson bound in the pre-quantum theory of trace dynamics

Rabsan G. Ahmed, Tejinder Singh|arXiv (Cornell University)|Aug 3, 2022

Quantum Mechanics and Applications被引用 8

一句话总结

本文表明，迹动力学的预量子理论违反了Tsirelson界，使得CHSH关联可以超过$2\sqrt{2}$，从而支持了量子力学是更深层、更基本的动力学理论的涌现近似这一观点。这种违反源于在普朗克尺度分辨率下迹动力学的非高斯、非高斯统计，同时通过幺正演化和范数守恒保持了相对论性因果性。

ABSTRACT

The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value $2\sqrt{2}$, known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than $2\sqrt{2}$. We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.

研究动机与目标

• 研究超越量子力学的理论是否可以在尊重相对论因果性的前提下违反Tsirelson界。
• 探讨迹动力学的预量子理论对纠缠系统中非定域关联的影响。
• 证明迹动力学中的CHSH表达式可以超过$2\sqrt{2}$，表明存在一种超量子理论。

提出的方法

• 采用矩阵值拉格朗日动力学形式，使用变换为二维旋量的复标量场。
• 引入产生和湮灭算符，其对易关系为$[a_i, a_j^\dagger] = [b_i, b_j^\dagger] = 1$，以恢复SU(2)代数。
• 通过产生/湮灭算符的双线性组合定义自旋算符$S_i^A$和$S_i^B$，满足$[S_i, S_j] = i\epsilon_{ijk}S_k$。
• 在迹动力学中推导出CHSH表达式$F_{TD}$，基于自旋关联函数的期望值。
• 通过概率的幺正不变性，证明测量结果独立于远处观察者的设置，从而保持了相对论性因果性。
• 通过反对易关系将形式体系扩展至费米子变量，得到与玻色子情况相同的$F_{TD}$表达式。

实验结果

研究问题

• RQ1 像迹动力学这样的预量子理论是否可以在保持相对论因果性的前提下违反Tsirelson界？
• RQ2 在迹动力学框架中，CHSH关联的最大值是多少？与量子和经典界限相比如何？
• RQ3 从迹动力学中涌现出的量子力学是否允许超量子关联？

主要发现

• 迹动力学中的CHSH表达式$F_{TD}$可以超过$2\sqrt{2}$，即量子力学的Tsirelson界，表明违反了这一量子极限。
• 迹动力学中的最大关联值高于$2\sqrt{2}$，表明存在一种超越量子力学的超量子理论。
• 尽管违反了Tsirelson界，该理论仍保持了相对论性因果性，因为测量概率独立于远处设置。
• 迹动力学中自旋算符及其代数结构的推导重现了SU(2)李代数，证实了与量子自旋的一致性。
• 该形式体系在玻色子和费米子统计下均保持一致，两种情况下均得到相同的$F_{TD}$表达式。
• 结果支持了量子力学是迹动力学这一更基本理论的热力学近似这一观点。