  We consider distributed sensing of non-local quantities. We introduce quantum
enhanced protocols to directly measure any (scalar) field with a specific
spatial dependence by placing sensors at appropriate positions and preparing a
spatially distributed entangled quantum state. Our scheme has optimal
Heisenberg scaling and is completely unaffected by noise on other processes
with different spatial dependence than the signal. We consider both Fisher and
Bayesian scenarios, and design states and settings to achieve optimal scaling.
We explicitly demonstrate how to measure coefficients of spatial Taylor and
Fourier series, and show that our approach can offer an exponential advantage
as compared to strategies that do not make use of entanglement between
different sites.
