  We consider a version of the extended Clifford Group which is defined in
terms of a finite Galois field in odd prime power dimension. We show that
Neuhauser's result, that with the appropriate choice of phases the standard (or
metaplectic) representation of the discrete symplectic group is faithful also
holds for the anti-unitary operators of the extended group. We also improve on
Neuhauser's result by giving explicit formulae for the (anti-)unitary
corresponding to an arbitrary (anti-)symplectic matrix. We then go on to find
the eigenvalues and the order of an arbitrary (anti-)symplectic matrix. The
fact that in prime power dimension the matrix elements belong to a field means
that this can be done using the same techniques which are used to find the
eigenvalues of a matrix defined over the reals-including the use of an
extension field (the analogue of the complex numbers) when the eigenvalues are
not in the base field. We then give an application of these results to
SIC-POVMs (symmetric informationally complete positive operator valued
measures). We show that in prime dimension our results can be used to find a
natural basis for the eigenspace of the Zauner unitary in which SIC-fiducials
are expected to lie. Finally, we apply our results to the MUB cycling problem.
We show that in odd prime power dimension d, although there is no Clifford
unitary, there is a Clifford anti-unitary which cycles through the full set of
Wootters-Fields MUBs if d=3 (mod 4). Also, irrespective of whether d=1 or 3
(mod 4), the Wootters-Fields MUBs split into two groups of (d+1)/2 bases in
such a way that there is a single Clifford unitary which cycles through each
group separately.

Quantum Physics