  The more information a measurement provides about a quantum system's position
statistics, the less information a subsequent measurement can provide about the
system's momentum statistics. This information trade-off is embodied in the
entropic formulation of the uncertainty principle. Traditionally, uncertainty
relations correspond to resolution limits; increasing a detector's position
sensitivity decreases its momentum sensitivity and vice-versa. However, this is
not required in general; for example, position information can instead be
extracted at the cost of noise in momentum. Using random, partial projections
in position followed by strong measurements in momentum, we efficiently
determine the transverse-position and transverse-momentum distributions of an
unknown optical field with a single set of measurements. The momentum
distribution is directly imaged, while the position distribution is recovered
using compressive sensing. At no point do we violate uncertainty relations;
rather, we economize the use of information we obtain.
