  We investigate theoretically the ultimate resolution that can be achieved
with passive remote sensing in the microwave regime used e.g.~on board of
satellites observing Earth, such as the Soil Moisture and Ocean Salinity (SMOS)
mission. We give a fully quantum mechanical analysis of the problem, starting
from thermal distributions of microscopic currents on the surface to be imaged
that lead to a mixture of coherent states of the electromagnetic field which
are then measured with an array of receivers. We derive the optimal detection
modes and measurement schemes that allow one to saturate the quantum
Cram\'er-Rao bound for the chosen parameters that determine the distribution of
the microscopic currents. For parameters comparable to those of SMOS, a quantum
enhancement of the spatial resolution by more than a factor of 20 should be
possible with a single measurement and a single detector, and a resolution down
to the order of 1 meter and less than a 1/10 Kelvin for the theoretically
possible maximum number of measurements.
