  Recently a proper genuine multipartite entanglement (GME) measure has been
found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127,
040403 (2021)], but capturing useful entanglement measures for mixed states has
remained an open challenge. So far, it requires not only a full tomography in
experiments, but also huge calculational labor. A leading proposal was made by
G\"uhne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)], who used
expectation values of entanglement witnesses to describe a lower bound
estimation of entanglement. We provide here an extension that also gives
genuine upper bounds of entanglement. This advance requires only the
expectation value of {\em any} Hermitian operator. Moreover, we identify a
class of operators $\A_1$ which not only give good estimates, but also require
a remarkably small number of experimental measurements. In this note we define
our approach and illustrate it by estimating entanglement measures for a number
of pure and mixed states prepared in our recent experiments.
