General symmetry in the reduced dynamics of two-level system

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.