  The concept of the {\em half density matrix} is proposed. It unifies the
quantum states which are described by density matrices and physical processes
which are described by completely positive maps. With the help of the
half-density-matrix representation of Hermitian linear map, we show that every
positive map which is not completely positive is a {\em difference} of two
completely positive maps. A necessary and sufficient condition for a positive
map which is not completely positive is also presented, which is illustrated by
some examples.
