Certifying unknown-basis pure-state entanglement dimensionality with few measurement settings

We introduce a basis-independent algorithm that only uses single-copy measurements to certify the Schmidt number of $2n$-qubit bipartite pure states. Assuming the local system dimension is $d = 2^n$, the Schmidt number $χ$ is $O(1)$, and the non-zero Schmidt coefficients are bounded below by $Ω(1)$, our protocol achieves a total sample complexity of $\widetilde{O}(d^2χ^2)$ with only $\widetilde{O}(χ^4)$ measurement settings. This work establishes a measurement-setting-efficient approach to high-dimensional entanglement certification.