Title: Burau representation, Squier's form, and non-Abelian anyons Show Chinese title

Title: Burau表示，Squier形式，以及非阿贝尔任意子

Abstract: We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the reduced Burau representation of the braid group $B_3$, specialised at $t=e^{i\omega}$ and unitarized by Squier's Hermitian form. Coupled to two non-commuting qubit unitaries $A,B$, the resulting switch admits a closed expression for the single-shot Helstrom success probability and a fixed-order ceiling $p_{\mathrm{fixed}}$, defining the analytic witness gap $\Delta(\omega)=p_{\mathrm{switch}}(\omega)-p_{\mathrm{fixed}}$. The sign change of $\Delta(\omega)$ across the Squier positivity window reveals alternating constructive and destructive interference of causal orders, a hallmark of non-Abelian control, while $\Delta(\omega)>0$ certifies algebraic causal non-separability. Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control that reproduces the characteristic interference pattern expected from a Gedankenexperiment in anyonic statistics. Show Chinese abstract

Abstract: 我们引入了一种频率可调的二维非阿贝尔操作顺序控制，该控制由辫群的约化伯劳表示 $B_3$构成，在 $t=e^{i\omega}$处特化，并通过斯奎尔的厄米形式进行酉化。 与两个非对易的量子比特酉变换 $A,B$耦合，所产生的开关对于单次海洛斯特定成功概率具有闭式表达式和固定顺序上限 $p_{\mathrm{fixed}}$，定义了分析见证间隙 $\Delta(\omega)=p_{\mathrm{switch}}(\omega)-p_{\mathrm{fixed}}$。 在斯奎尔正性窗口中 $\Delta(\omega)$的符号变化揭示了因果顺序的交替建设性和破坏性干涉，这是非阿贝尔控制的特征，而 $\Delta(\omega)>0$证明了代数因果不可分性。 数值模拟证实了增强和抑制两种机制，确立了一种最小的$B_3$辫子控制，该控制能够再现任意子统计中预期的特征干涉图案。