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

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 fixed-order ceiling $p_{\mathrm{fixed}}^*$ and the witness gaps $\Delta_{\rm sw}(\omega)=p_{\mathrm{switch}}(\omega)-p_{\mathrm{fixed}}^*$ and $\Delta_{\rm test}(\omega)=p_{\mathrm{test}}(\omega)-p_{\mathrm{fixed}}^*$. The non-Abelian mixers can either enhance or suppress the bare switch advantage, which we quantify by the interference contrast $\Delta_{\rm int}(\omega):=\Delta_{\rm test}(\omega)-\Delta_{\rm sw}(\omega)=p_{\rm test}(\omega)-p_{\rm switch}(\omega)$. Across the Squier positivity region, $\Delta_{\rm int}(\omega)$ takes both positive (constructive) and negative (destructive) values, a hallmark of matrix-valued (non-Abelian) order control, while $\Delta_{\rm sw}(\omega)>0$ 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 \emph{Gedankenexperiment} in anyonic statistics.