Classical-Input Simulation of Quantum Superchannels: From Linear to Quadratic Error with One More Outcome

One additional retained measurement outcome can change the optimal approximation order in classical-input simulation of a quantum superchannel. A classical-input realization first measures the external input and then applies a deterministic conditional process. The budget counts the complete retained label, including any shared random choice used at a later stage. We construct a fixed-dimensional family of state-input superchannels for which increasing the budget from four outcomes to five changes the optimal approximation error from linear to quadratic just above the exact realization threshold. The error is measured in the trace norm of the process Choi operator, and the optimization allows arbitrary measurement effects and deterministic conditional processes. The quadratic order is optimal over all finite measurements, and five is the smallest outcome budget that attains it. The same approximation orders hold in diamond distance for the associated joint-input channels. A correlated splitting of one effect cancels the linear defect with only a second-order change in the measurement obtained by merging the split outcomes. Throughout the family, the output channels share a four-outcome mother measurement and the process Choi operator is separable across the external-input cut. An independent theorem establishes local linear stability under stochastic postprocessing for rank-one target POVMs with distinct support rays, provided that the unknown POVM has no more outcomes than the target.