  Entanglement is an essential resource in many quantum information tasks and
entanglement witness is a widely used tool for its detection. In experiments
the prepared state generally deviates from the target state due to some noise.
Normally the white noise model is applied to quantifying such derivation and in
the same time reveals the robustness of the witness. However, there may exist
other kind of noise, in which the coherent noise can dramatically "rotate" the
prepared state. In this way, the coherent noise is likely to lead to a failure
of the detection, even though the underlying state is actually entangled. In
this work, we propose an efficient entanglement detection protocol for
$N$-partite Greenberger-Horne-Zeilinger (GHZ)-like states. The protocol can
eliminate the effect of the coherent noise and in the same time feedback the
corresponding noise parameters, which are beneficial to further improvements on
the experiment system. In particular, we consider two experiment-relevant
coherent noise models, one is from the unconscious phase accumulation on $N$
qubits, the other is from the rotation on the control qubit. The protocol
effectively realizes a family of entanglement witnesses by postprocessing the
measurement results from $N+2$ local measurement settings, which only adds one
more setting than the original witness specialized for the GHZ state. Moreover,
by considering the trade-off between the detection efficiency and the
white-noise robustness, we further reduce the number of local measurements to 3
without altering the performance on the coherent noise. Our protocol can
enhance the entanglement detection under coherent noises and act as a benchmark
for the state-of-the-art quantum devices.
