Symmetry Breaking as Quantum Gate: Entropy and Weak Mixing Angle

We establish a correspondence between two independent entropic probes -- the variation of R\'{e}nyi mutual information (RMI) across the electroweak symmetry breaking (EWSB) transition and the stabilizer R\'enyi entropy (SRE) -- in tree-level $2\to 2$ elastic scatterings. After angular averaging, the RMI (helicity basis) and the SRE (fixed beam basis) exhibit identical dependence on $\sin^2\theta_W$ within each neutral-current channel. We trace this correspondence to a common physical origin that it's the Yukawa mass insertion acts as a $-\mathrm{i}Y$ quantum gate in chirality space. Minimizing entropies across all processes yields $\sin^2\theta_W$ values matching purely axial vector-like couplings in $Z$ boson exchanged channel.