We introduce a Pauli-measurement-based algorithm to certify the Schmidt number of n-qubit pure states. Our protocol achieves an average-case sample complexity of \caO(poly(n)χ2), a substantial improvement over the \caO(2nχ) worst-case bound. By utilizing local pseudorandom unitaries, we ensure the worst case can be transformed into the average-case with high probability. This work establishes a scalable approach to high-dimensional entanglement certification and introduces a proof framework for random Pauli sampling.
@article{arxiv.2601.11040,
title = {Certifying entanglement dimensionality by random Pauli sampling},
author = {Changhao Yi},
journal= {arXiv preprint arXiv:2601.11040},
year = {2026}
}