  We show that estimation of multiple independent field strengths of a local
Hamiltonian, i.e., one formed by a sum of single-party terms, necessitates the
utility of a highly symmetric genuine multiparty entangled state, viz. the
Greenberger-Horne-Zeilinger (GHZ) state, as the input probe. This feature
however depends on the choice of the weight matrix considered, to define a
figure of merit in the multiparameter estimation. We obtain this result by
providing a lower bound on the precision of multiparameter estimation,
optimized over input probes, for an arbitrary weight matrix. We find that the
bound can be expressed in terms of eigenvalues of the arbitrary multiparty
local encoding Hamiltonian. We show that there exists a weight matrix for which
this bound is attainable only when the probe is the GHZ state, up to a relative
phase. In particular, no pure product state can achieve this lower bound.
Indeed, the gap - in precision - between genuinely multiparty entangled and
product states acting as probes, increases with increasing number of parties.
Finally we also prove that using a probe that is in a mixed state provides a
precision lower than that for the GHZ state. To emphasize the importance of the
weight matrix considered, we also prove that the choice of identity operator as
the same - thereby ignoring the covariances in the precision matrix - does not
require the use of genuine multiparty entanglement in input probes for
attaining the best precision, and the optimal probe can be a pure product.
