Coherence thresholds for optimal-rate stabilizer purity testing

Abstract

A purity test is exactly one-sided if it can certify mixedness but never does so on a pure state. We ask how many copies must be processed jointly when the available operations are restricted to stabilizer circuits. For every odd prime p and every fixed number of p-level registers, we prove that this minimum block size is p for the optimal total-copy order Θ(ε⁻¹ log(1/δ)). Here ε is the minimum linear impurity to be detected and δ is the allowed probability of missing it. States may be entangled across registers; only classical information passes between blocks. For the upper bound, we construct a p-copy measurement on quadratic-polynomial supports and combine it with a support theorem for mutually unbiased bases. Conditioning on local measurement outcomes extends the construction to entangled inputs while retaining a click probability proportional to impurity. For the lower bound, smaller blocks cannot detect the first-order contribution of a small admixture near a pure stabilizer state. We turn this local obstruction into an Ω(ε⁻² log(1/δ)) bound, even with classical adaptation. In dimensions five and seven, one fixed local circuit suffices, and separate support arguments exclude every shorter universal exact test. Sensitivity formulas and a global Clifford-code construction further quantify the measurement resources.