Symmetric informationally complete measurements of arbitrary rank

There has been much interest in so-called SIC-POVMs, i.e., rank 1 symmetric informationally complete positive operator valued measures. In this paper we discuss the larger class of POVMs that are symmetric and informationally complete, but not necessarily rank 1. This class of POVMs is of some independent interest. In particular it includes a POVM that is closely related to the discrete Wigner function. However, it is interesting mainly because of the light it casts on the problem of constructing rank 1, symmetric, informationally complete POVMs. In this connection we derive an extremal condition alternative to the one derived by Renes et al.