The characterization of high-dimensional entanglement plays a crucial role in the field of quantum information science. Conventional entanglement criteria measuring coherent superpositions of multiple basis states face experimental bottlenecks on most physical platforms due to limited multi-channel control. Here, we introduce a practically efficient detection strategy based on randomized product projections. We show that the first-order moments of such projections can be used to estimate entanglement fidelity, thereby enabling practical and efficient certification of the Schmidt number in high-dimensional bipartite systems. By constructing optimal observables, it is sufficient to merely measure a single basis state, substantially reducing experimental overhead. Moreover, we present an algorithm to obtain a lower bound of the Schmidt number with a high confidence level from a limited number of experimental data. Our results open up resource-efficient experimental avenues to detect high-dimensional entanglement and test its implementations in modern information technologies.
@article{arxiv.2501.01088,
title = {Detecting high-dimensional entanglement by randomized product projections},
author = {Jin-Min Liang and Shuheng Liu and Shao-Ming Fei and Qiongyi He},
journal= {arXiv preprint arXiv:2501.01088},
year = {2026}
}