Burau representation, Squier's form, and non-Abelian anyons

Abstract: We introduce a frequency-tunable, two-dimensional non-Abelian control over operation order built 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 and a fixed-order ceiling $p_{\rm fixed}^*$, yielding an explicit, analytic witness gap $\Delta(\omega)=p_{\rm switch}(\omega)-p_{\rm fixed}^*$. We prove that $\Delta(\omega)>0$ is achievable, thereby certifying causal non-separability by purely algebraic means, and confirm this behaviour numerically. Conceptually, this furnishes a minimal non-Abelian $B_3$ control for a Gedankenexperiment in anyonic statistics.