Measurement-device-independent Schmidt number certification of all entangled states

Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilisation. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: it cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell-nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with Schmidt number exceeding $r$, there exists a semiquantum nonlocal game that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.