  We study general transformation on the density matrix of two-level system
that keeps the expectation value of observable invariant. We introduce a set of
generators that yields hermiticity and trace preserving general transformation
which casts the transformation into simple form. The general transformation is
in general not factorized and not completely positive. Consequently, either the
parameter of transformation or the density matrix it acts on needs to be
restricted. It can transform the system in the forward and backward direction
with regard to its parameter, not as a semigroup in the time translation
symmetry of dynamical maps. The general transformation can rotate the Bloch
vector circularly or hyperbolically, dilate it or translate it. We apply the
general transformation to study the general symmetry of amplitude damping and
phase damping in two-level system. We generalize the generators to higher level
systems.
