Abstract
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 Σ = D(ρ'_SEρ'_S⊗ρ_E). Isolating the experimentally accessible bath divergence D_bath=D(ρ'_Eρ_E), we prove a fully nonlinear, saturable lower bound valid for arbitrary current vectors Δ q: D_bath ≥ B(Δ q,V,V'), where the bound depends only on the transported-charge signal Δ q and the pre/post collision covariance matrices V and V'. In the small-fluctuation regime D_bath≥1/2Δ q^TV⁻¹Δ q+O(Δ q⁴), 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.