  Generalized symmetric informationally complete (SIC) measurements are SIC
measurements that are not necessarily rank one. They are interesting originally
because of their connection with rank-one SICs. Here we reveal several merits
of generalized SICs in connection with quantum state tomography and Lie algebra
that are interesting in their own right. These properties uniquely characterize
generalized SICs among minimal IC measurements although, on the face of it,
they bear little resemblance to the original definition. In particular, we show
that in quantum state tomography generalized SICs are optimal among minimal IC
measurements with given average purity of measurement outcomes. Besides its
significance to the current study, this result may help understand tomographic
efficiencies of minimal IC measurements under the influence of noise. When
minimal IC measurements are taken as bases for the Lie algebra of the unitary
group, generalized SICs are uniquely characterized by the antisymmetry of the
associated structure constants.
