  Studies of quantum metrology have shown that the use of many-body entangled
states can lead to an enhancement in sensitivity when compared to product
states. In this paper, we quantify the metrological advantage of entanglement
in a setting where the quantity to be measured is a linear function of
parameters coupled to each qubit individually. We first generalize the
Heisenberg limit to the measurement of non-local observables in a quantum
network, deriving a bound based on the multi-parameter quantum Fisher
information. We then propose a protocol that can make use of GHZ states or
spin-squeezed states, and show that in the case of GHZ states the procedure is
optimal, i.e., it saturates our bound.
