Certifying entanglement dimensionality by random Pauli sampling

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 $\text{caO}(\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n){\chi }^2)$, a substantial improvement over the $\text{caO}(2^n\chi )$ 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.