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 and an explicit improvement of their exponential amplification base. A matching polynomial extremizer on the whole density-matrix body identifies the role of operational symmetry in the exact two-copy result.