  We study the simultaneous estimation of multiple phases as a discretised
model for the imaging of a phase object. We identify quantum probe states that
provide an enhancement compared to the best quantum scheme for the estimation
of each individual phase separately, as well as improvements over classical
strategies. Our strategy provides an advantage in the variance of the
estimation over individual quantum estimation schemes that scales as O(d) where
d is the number of phases. Finally, we study the attainability of this limit
using realistic probes and photon-number-resolving detectors. This is a problem
in which an intrinsic advantage is derived from the estimation of multiple
parameters simultaneously.
