Symmetric informationally complete–positive operator valued measures and the extended Clifford group

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 conjecture 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

On Weyl-Heisenberg orbits of equiangular lines

“…The argument Legendre fiducial vectors are almost flat and are discussed by Appleby [2] in specific dimensions. Since the real Legendre fiducial vectors have very similar properties to the argument Legendre ones and also all of the coordinates of such vectors have the same argument, we have also investigated this class of fiducial vectors.…”

“…If such a maximum set exists then we must have α = 1/ √ d + 1. To the best of our knowledge, such maximum sets are presented in [2,[10][11][12][13]17] for 2 ≤ d ≤ 10 and d ∈ {12, 19}. In addition, it is claimed in [7] (with reference to private communication with Markus Grassl) that such sets also exist for d ∈ {11, 13, 15}.…”

“…It is proved [9] that there are at most To take it one step further, we say a vector is almost flat if it is flat except for one coordinate. Appleby [2] constructed fiducial vectors in dimensions 7 and 19 which are almost flat. Using an eigenvalue argument, Roy [14] proved that for almost flat fiducial vectors in C d , the absolute values of the coordinates are determined in terms of d. We provide an alternative proof of this fact here: …”

“…It is widely believed that for every d there exists a fiducial vector in C d (for example see [2,7,9,11,13,17]). The set of d 2 equiangular lines in C d has been discussed in different contexts.…”

“…Using the new characterization, we show that the existence of almost flat fiducial vectors implies the existence of certain cyclic difference sets. The fiducial vectors in dimensions 7 and 19 constructed by Appleby [2] are examples of the argument Legendre fiducial vectors. We prove that the construction of these vectors does not generalize to other prime dimensions (except for possibly a set with density zero).…”