  Entanglement plays a fundamental role in quantum physics and information
processing. Here, we develop an unbiased estimator for mixed-state entanglement
in the few-shot scenario and directly estimate it using random unitary
evolution in a photonic system. As a supplement to traditional projective
measurements, we incorporate Bell measurements on qubit-pairs, enriching the
previous randomized measurement scheme, which is no-go in this task with only
local unitary evolution. The scheme is scalable to n-qubits via Bell
measurements on qubit-pairs. The estimator can be derived directly from a few
consecutive outcomes while exhibiting greater robustness to system errors and
noise compared to schemes based on shadow estimation. We find that, under a
fixed measurement resource, the way with more versatile measurement settings
with fewer repeats per setting is more efficient. Our protocol and
demonstration advance the direct characterization of quantum states in
practice.
