  The physical problem behind informationally complete (IC) measurements is
determining an unknown state statistically by measurement outcomes, known as
state tomography. It is of central importance in quantum information processing
such as channel estimating, device testing, quantum key distribution, etc.
However, constructing such measurements with good properties is a long-standing
problem. In this work, we investigate projective realizations of IC
measurements. Conditions of informational completeness are presented with
proofs first. Then the projective realizations of IC measurements, including
proposing the first general construction of minimal projective IC measurements
(MPICM) in no prime power dimensional systems, as well as determining an
unknown state in $C^{n}$ via a single projective measurement with some kinds of
optimalities in a larger system, are investigated. Finally, The results can be
extended to local state tomography. Some discussions on employing several kinds
of optimalities are also provided.
