  Symmetric informationally complete (SIC) POVMs are a class of quantum
measurements which, in addition to being informationally complete, satisfy
three conditions: 1) every POVM element is rank one, 2) the Hilbert-Schmidt
inner product between any two distinct elements is constant, and 3) the trace
of each element is constant. The third condition is often overlooked, since it
may give the impression that it follows trivially from the second. We show that
this condition cannot be removed, as it leads to two distinct values for the
trace of an element of the POVM. This observation has led us to define a
broader class of measurements which we call semi-SIC POVMs. In dimension two we
show that semi-SIC POVMs exist, and we construct the entire family. In higher
dimensions, we characterize key properties and applications of semi-SIC POVMs,
and note that the proof of their existence remains open.
