  Magnons, as fundamental quasiparticles emerged in elementary spin
excitations, hold a big promise for innovating quantum technologies in
information coding and processing. Here we discover subtle roles of
entanglement in a metrological scheme based on an experimentally feasible
cavity magnomechanical system, where the magnons are responsible for sensing a
weak magnetic field whereas the cavity field carries out a precision
measurement of the weak field. By establishing exact relations between the
Fisher information and entanglement, we show that for the weak coupling case
the measurement precision can reach the Heisenberg limit, whereas quantum
criticality enables us to enhance measurement precision for the strong coupling
case. In particular, we also find that the entanglement between magnons and
photons is of crucial importance during the dynamical encoding process, but the
presence of such an entanglement in the measurement process dramatically
reduces the final measurement precision.
