Properties of the extended Clifford group with applications to SIC-POVMs and MUBs

TL;DR

This paper studies the extended Clifford group over finite Galois fields in odd prime power dimensions, providing explicit formulas for (anti-)unitaries, analyzing eigenvalues, and applying results to SIC-POVMs and MUB cycling problems.

Contribution

It offers explicit formulas for (anti-)unitaries associated with (anti-)symplectic matrices and applies these to SIC-POVMs and MUBs in prime power dimensions, advancing understanding of their symmetries.

Findings

Explicit formulas for (anti-)unitaries corresponding to (anti-)symplectic matrices.

In prime dimension, a natural basis for SIC-POVM eigenspaces is identified.

Existence of Clifford anti-unitaries cycling Wootters-Fields MUBs in certain dimensions.

Abstract

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.