Thermodynamic uncertainty relations (TURs) bound current precision by entropy production, but transport of noncommuting (non-Abelian) charges lacks a single joint classical record of all current components. We derive an operational, process-level matrix TUR for this setting from the entropy production $\mathrm{\Sigma }=D({\rho }_{SE}^{\prime }\parallel {\rho }_S^{\prime }\otimes {\rho }_E)$. Isolating the bath divergence $D_{\mathrm{bath}}=D({\rho }_E^{\prime }\parallel {\rho }_E)$, we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors $\mathrm{\Delta }q$: $D_{\mathrm{bath}}\ge B(\mathrm{\Delta }q,V,V^{\prime })$, where the bound depends only on the transported-charge signal $\mathrm{\Delta }q$ and on the pre- and postcollision covariance matrices $V$ and $V^{\prime }$ reconstructed from separate probe measurement settings. In the small-fluctuation regime $D_{\mathrm{bath}}\ge \frac{1}{2}\mathrm{\Delta }{q}^{\mathsf{T}}{V}^{-1}\mathrm{\Delta }q+{O(\parallel \mathrm{\Delta }q\parallel }^4)$, while beyond linear response the full integral bound remains valid. Numerical strong-coupling qubit collisions illustrate the bound and its near-saturation using only local measurements on the bath probe.

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• Entropy production
• Onsager relations
• Quantum thermodynamics
• Quantum transport
• Uncertainty relation