Few-shot estimation of entanglement with Bell measurement assistance

Entanglement is fundamental to quantum physics and information processing. In this work, we introduce the Few-Shot Randomized Measurement (FSRM) method, developing an unbiased estimator for mixed-state entanglement from just three experimental shot outcomes. By incorporating the Bell measurement (BM), we supplement the traditional computational-basis measurement to enhance the randomized measurement scheme, which is scalable to n-qubit systems via BMs on qubit pairs. Our approach enables direct estimation of entanglement through random unitary evolution in a photonic system. Compared to the classical shadow method, BM-enhanced FSRM requires no prior knowledge of the local unitaries, offering greater robustness against unitary imperfections. Additionally, we find that utilizing more versatile measurement settings with fewer repeats per setting is more efficient under fixed measurement resources. Our protocol and experimental demonstration represent a significant advancement in the efficient and practical characterization of quantum states.