Purity frontiers and the optimal copy exponent for qutrit stabilizer measurements

Abstract

We determine exact output frontiers and the optimal copy power exponent for purity estimation with qutrit stabilizer measurements. For a single qutrit and at most three copies, we give the complete frontiers for nonnegative outputs and for bounded real outputs. The nonnegative feasibility threshold is one thirty-seventh, and the exact minimum bias for unit-interval outputs is two thirty-ninths. A four-outcome Schubr parity refinement attains every finite branch, while code-parity and finite-ensemble witnesses give matching all-protocol bounds. For every fixed qutrit-register dimension, the minimum copy cap attaining the physical upper boundary has inverse-tolerance exponent one half, for both nonnegative and bounded real outputs. A log-free square-root lower bound leaves a single logarithmic factor against the upper construction. The upper construction combines an unbiased polarized third-moment statistic, inverse-binomial calibration, bounded first-success sampling, and a one-sided clipping repair. The lower bound uses one-sided likelihood curvature and an interval-wide deficit bound on a physical Wigner-positive family. At zero bias, the exact minimum nonnegative unbiased copy cap is the Hilbert-space dimension squared minus one, attained by a randomized MUB collision statistic. Supporting results classify three-copy pure-null instruments, give a geometric witness and a coherent-block separation, and establish a larger-prime cyclic spectral obstruction. Established phase-space, moment and Schur–Weyl ingredients are credited explicitly.