  We introduce a general model for a network of quantum sensors, and we use
this model to consider the question: when do correlations (quantum or
classical) between quantum sensors enhance the precision with which the network
can measure an unknown set of parameters? We rigorously answer this question
for a range of practically important problems. When each sensor in the network
measures a single parameter, we show that correlations between sensors cannot
increase the estimation precision beyond what can be achieved with an
uncorrelated scheme, regardless of the particular details of the estimation
problem in question. We also consider the more general setting whereby each
sensor may be used to measure multiple parameters, e.g., the three spatial
components of a magnetic field. In this case, we show that correlations between
sensors can only provide, at best, a small constant precision enhancement, over
uncorrelated estimation techniques. Finally, we consider optimizing the network
for measuring a single linear function of the unknown parameters, e.g., the
average of all of the parameters. Here quantum correlations between the sensors
can provide a significant precision enhancement over uncorrelated techniques,
and this enhancement factor scales with the number of sensors. To illustrate
the broad implications of this work, we apply our results to a wide range of
estimation problems of practical interest, including multi-mode optical
interferometry, networks of atomic sensors, and networks of clocks. Our
findings shed light on a number of results in the literature, provide a
rigorous general framework for future research on networked quantum sensors,
and have implications for both quantum multi-parameter estimation theory, and
quantum sensing technologies.
