Sharp output ranges and bias frontiers for stabilizer purity estimation

Abstract

We determine the exact tradeoff between uniform expectation bias and the upper output endpoint for projected-purity estimation by odd-prime stabilizer protocols using at most two identical input copies. If an isotropic Weyl subgroup leaves logical dimension d, the least upper endpoint at bias tolerance ε is max{d−(2d+1)ε,1−ε} for d>1, and 1-ε for d=1. An explicit common-basis collision statistic attains every branch. In particular, the optimal bias with outputs in [0,1] is (d-1)/(2d+1). The proof combines the two-copy Clifford commutant with positive responses at a nonphysical Wigner phase point. The same response argument gives the sharp upper endpoint d for exactly unbiased estimation at every finite copy cap. At larger caps we derive a continuum of Chebyshev witness bounds, an improved finite-cap constant, and the exact leading asymptotic and maximizing location within this witness family. A matching polynomial extremizer on the whole density-matrix body identifies the role of operational symmetry in the exact two-copy result.