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Abstract

We determine purity-estimation frontiers under stabilizer operations, bounded unconditional outputs and uniform expectation bias. In every fixed qutrit-register dimension, attaining the physical upper output endpoint requires and suffices to use copies of order the inverse square root of the bias tolerance. A five-block moment polarization, smooth clipping and reciprocal-weighted repair give a log-free upper bound. A likelihood-curvature argument supplies the matching lower order for bounded-real outputs. We also determine exact three-copy single-qutrit frontiers and the nonnegative unbiased copy threshold, dimension squared minus one. Pure-null instrument structure explains the role of outcome information.