Matrix Thermodynamic Uncertainty Relation for Non-Abelian Charge Transport

Thermodynamic uncertainty relations (TURs) bound the precision of currents by entropy production, but quantum transport of noncommuting (non-Abelian) charges challenges standard formulations because different charge components cannot be monitored within a single classical frame. We derive a process-level matrix TUR starting from the operational entropy production $\Sigma = D(\rho'_{SE}\|\rho'_S\!\otimes\!\rho_E)$. Isolating the experimentally accessible bath divergence $D_{\mathrm{bath}}=D(\rho'_E\|\rho_E)$, we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors $\Delta q$: $D_{\mathrm{bath}} \ge B(\Delta q,V,V')$, where the bound depends only on the transported-charge signal $\Delta q$ and the pre/post collision covariance matrices $V$ and $V'$. In the small-fluctuation regime $D_{\mathrm{bath}}\geq\frac12\,\Delta q^{\mathsf T}V^{-1}\Delta q+O(\|\Delta q\|^4)$, while beyond linear response it remains accurate. Numerical strong-coupling qubit collisions illustrate the bound and demonstrate near-saturation across broad parameter ranges using only local measurements on the bath probe.