  A general symmetric-informationally-complete (GSIC)-positive-operator-valued
measure (POVM) is known to provide an optimal quantum state tomography among
minimal IC POVMs with a fixed average purity. In this paper we provide a
general construction of a GSIC POVM by means of a complete orthogonal basis
(COB), also interpreted as a normal quasiprobability representation. A spectral
property of a COB is shown to play a key role in the construction of SIC POVMs
and also for the bound of the mean-square error of the state tomography. In
particular, a necessary and sufficient condition to construct a SIC POVM for
any d is constructively given by the power of traces of a COB. We give three
simple constructions of COBs from which one can systematically obtain GSIC
POVMs.

Quantum Physics