What Are the Minimal Conditions Required to Define a Symmetric Informationally Complete Generalized Measurement?

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