  We introduce a general model for a network of quantum sensors, and we use
this model to consider the question: When can entanglement between the sensors,
and/or global measurements, enhance the precision with which the network can
measure a set of unknown parameters? We rigorously answer this question by
presenting precise theorems proving that for a broad class of problems there
is, at most, a very limited intrinsic advantage to using entangled states or
global measurements. Moreover, for many estimation problems separable states
and local measurements are optimal, and can achieve the ultimate quantum limit
on the estimation uncertainty. This immediately implies that there are broad
conditions under which simultaneous estimation of multiple parameters cannot
outperform individual, independent estimations. Our results apply to any
situation in which spatially localized sensors are unitarily encoded with
independent parameters, such as when estimating multiple linear or non-linear
optical phase shifts in quantum imaging, or when mapping out the spatial
profile of an unknown magnetic field. We conclude by showing that entangling
the sensors can enhance the estimation precision when the parameters of
interest are global properties of the entire network.
