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 $\mathrm{\Sigma }=D({\rho }_{SE}^{\mathrm{\prime }}\mathrm{\parallel }{\rho }_S^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}\otimes \text{\hspace{0.17em}⁣}{\rho }_E)$. Isolating the experimentally accessible bath divergence $D_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}=D({\rho }_E^{\mathrm{\prime }}\mathrm{\parallel }{\rho }_E)$, we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors $\mathrm{\Delta }q$: $D_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}\ge B(\mathrm{\Delta }q,V,V^{\mathrm{\prime }})$, where the bound depends only on the transported-charge signal $\mathrm{\Delta }q$ and the pre/post collision covariance matrices $V$ and $V^{\mathrm{\prime }}$. In the small-fluctuation regime $D_{\mathrm{b}\mathrm{a}\mathrm{t}\mathrm{h}}\ge \frac{1}{2}\mathrm{\Delta }{q}^{\mathsf{T}}{V}^{-1}\mathrm{\Delta }q+O(\mathrm{\parallel }\mathrm{\Delta }q{\mathrm{\parallel }}^4)$, 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.