  The characterization of the evolution of a quantum system is one of the main
tasks to accomplish to achieve quantum information processing. The standard
quantum process tomography (SQPT) has the unique property that it can be
applied without introducing any additional quantum resources. In present work,
we shall focus on the following two topics about the SQPT. At first, in the
SQPT protocol for a $d$-dimensional system, one should encounter a problem in
solving of a set of $d^4$ linear equations in order to get the matrix
containing the complete information about the unknown quantum channel. Until
now, the general form of the solution is unknown. And a long existed conviction
is that the solutions are not unique. Here, we shall develop a self-consistent
scheme, in which bounded linear operators are presented by vectors, to
construct the set of linear equations. With the famous Cramer's rule for the
set of linear equations, we are able to give the general form of the solution
and prove that it is unique. In second, the central idea of the SQPT is to
prepare a set of linearly independent inputs and measuring the outputs via the
quantum state tomography (QST). Letting the inputs and the measurements be
prepared by two sets of the rank-one positive-operator-valued measures [POVMs],
where each POVM is supposed to be linearly independent and informationally
complete (IC), we observe that SQPT now is equivalent to deciding a unknown
state with a set of product IC-POVM in the $d^2$-dimensional Hilbert space.
Following the general linear state tomography theory, we show that the product
symmetric IC-POVM should minimize the mean-square Hilbert-Schmidt distance
between the estimator and the true states. So, an optimal SQPT can be realized
by preparing both the inputs and the measurements as the symmetric IC-POVM.
