  Entanglement witness is an effective method to detect entanglement in unknown
states without doing full tomography. One of the most widespread schemes of
witnessing entanglement is measuring its fidelity with respect to a pure
entangled state. Recently, a large class of states whose entanglement can not
be detected with the fidelity witness has been discovered in Phys.Rev.Lett
\textbf{124},200502(2020). They are called unfaithful states. In this paper we
propose a new way to detect entanglement by calculating the lower bound of
entanglement using measurement results. Numerical simulation shows our method
can detect entanglement in unfaithful states with a small number of
measurements. Moreover, we generalize our scheme to multipartite states and
show that it can tolerate higher noise than previous entanglement witness
operators with same number of measurement settings.
