  An input-output model of a two-level quantum system in the Heisenberg picture
is of bilinear form with constant system matrices, which allows the
introduction of the concepts of controllability and observability in analogy
with those of quantum linear systems. By means of the notions of
controllability and observability, coordinate transformations, which are
rotation matrices, can be constructed explicitly that transform an input-output
model to a new one. The new input-output model enables us to investigate many
interesting properties of the two-level quantum system, such as steady-state
solutions to the Lindblad master equation, quantum decoherence-free (DF)
subspaces, quantum non-demolition (QND) variables, and the realization of
quantum back-action evading (BAE) measurements. The physical system in (Wang,
J. \& Wiseman, H. M. (2001), Feedback-stabilization of an arbitrary pure state
of a two-level atom, Physical Review A 64(6), 063810) is re-studied to
illustrate the results presented in this paper.
