Research Article · Preprint Mathematics
Sharp output ranges and bias frontiers for stabilizer purity estimation
- Primary subject
- Mathematics
- Subject subfields
- Quantum Information Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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.
Scope of the work
Projected-purity estimation for odd-prime stabilizer protocols, with an exact two-copy bias/output frontier, an all-finite-copy unbiased endpoint, and higher-copy witness bounds.
Reproduction notes
The supplementary package contains editable source, build instructions and a portable verification program.
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Scott Schubr. (2026). Sharp output ranges and bias frontiers for stabilizer purity estimation (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-output-ranges-bias-frontiers-stabilizer-purity-estimation/v/1
BibTeX
@techreport{luminary_sharp_output_ranges_bias_frontiers_stabilizer_purity_estimation_v1,
title = {Sharp output ranges and bias frontiers for stabilizer purity estimation},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-output-ranges-bias-frontiers-stabilizer-purity-estimation/v/1}
}Loading discussion…