Superoperator
In using the SIC-POVM elements, an interesting superoperator can be constructed, the likes of which map . This operator is most useful in considering the relation of SIC-POVMs with spherical t-designs. Consider the map
This operator acts on a SIC-POVM element in a way very similar to identity, in that
But since elements of a SIC-POVM can completely and uniquely determine any quantum state, this linear operator can be applied to the decomposition of any state, resulting in the ability to write the following:
- where
From here, the left inverse can be calculated to be, and so with the knowledge that
- ,
an expression for a state can be created in terms of a quasi-probability distribution, as follows:
where is the Dirac notation for the density operator viewed in the Hilbert space . This shows that the appropriate quasi-probability distribution (termed as such because it may yield negative results) representation of the state is given by
Read more about this topic: SIC-POVM, Properties