  Projective measurements in quantum theory have a very simple algebraic
definition, but their information theoretic significance is quite elusive. Here
we introduce a simple order relation based on the concentration of Fisher
information, which complements the familiar data-processing order. Under this
order relation, the information theoretic significance of projective
measurements stands out immediately. Notably, projective measurements are
exactly those quantum measurements whose extracted Fisher information is as
concentrated as possible, which we call Fisher-sharp measurements. We also
introduce the concept of sharpness index and show that it is completely
determined by the finest projective measurement among the coarse graining of a
given measurement.
