  Entanglement is a key resource for quantum information processing. A widely
used tool for detecting entanglement is entanglement witness, where the
measurement of the witness operator is guaranteed to be positive for all
separable states and can be negative for certain entangled states. In reality,
due to the exponentially increasing the Hilbert-space dimension with respective
to the system size, it is very challenging to construct an efficient
entanglement witness for general multipartite entangled states. For $N$-partite
Greenberger-Horne-Zeilinger (GHZ)-like states, the most robust witness scheme
requires $N+1$ local measurement settings and can tolerate up to $1/2$ white
noise. As a comparison, the most efficient witness for GHZ-like states only
needs two local measurement settings and can tolerate up to $1/3$ white noise.
There is a trade-off between the realization efficiency, the number of
measurement settings, and the detection robustness, the maximally tolerable
white noise. In this work, we study this trade-off by proposing a family of
entanglement witnesses with $k$ ($2\le k\le N+1$) local measurement settings.
Considering symmetric local measurements, we calculate the maximal tolerable
noise for any given number of measurement settings. Consequently, we design the
optimal witness with a minimal number of settings for any given level of white
noise. Our theoretical analysis can be applied to other multipartite entangled
states with a strong symmetry. Our witnesses can be easily implemented in
experiment and applied in practical multipartite entanglement detection under
different noise conditions.
