SIC-POVMs and the Extended Clifford Group

TL;DR

This paper analyzes the structure of the extended Clifford group and its action on SIC-POVMs, revealing new symmetries and providing explicit fiducial vectors in certain dimensions, which may simplify their construction.

Contribution

It characterizes the extended Clifford group, explores its action on SIC-POVMs, and constructs explicit fiducial vectors in specific dimensions, advancing understanding of their symmetry properties.

Findings

Fiducial vectors are eigenvectors of order 3 Clifford unitaries.

Complete classification of orbits and stability groups in dimensions 2-7.

Explicit fiducial vectors constructed in dimensions 7 and 19.

Abstract

We describe the structure of the extended Clifford Group (defined to be the group consisting of all operators, unitary and anti-unitary, which normalize the generalized Pauli group (or Weyl-Heisenberg group as it is often called)). We also obtain a number of results concerning the structure of the Clifford Group proper (i.e. the group consisting just of the unitary operators which normalize the generalized Pauli group). We then investigate the action of the extended Clifford group operators on symmetric informationally complete POVMs (or SIC-POVMs) covariant relative to the action of the generalized Pauli group. We show that each of the fiducial vectors which has been constructed so far (including all the vectors constructed numerically by Renes et al) is an eigenvector of one of a special class of order 3 Clifford unitaries. This suggests a strengthening of a conjuecture of Zauner's. We give a complete characterization of the orbits and stability groups in dimensions 2-7. Finally, we show that the problem of constructing fiducial vectors may be expected to simplify in the infinite sequence of dimensions 7, 13, 19, 21, 31,... . We illustrate this point by constructing exact expressions for fiducial vectors in dimensions 7 and 19.