  A quantum control landscape is defined as the physical objective as a
function of the control variables. In this paper the control landscapes for
two-level open quantum systems, whose evolution is described by general
completely positive trace preserving maps (i.e., Kraus maps), are investigated
in details. The objective function, which is the expectation value of a target
system operator, is defined on the Stiefel manifold representing the space of
Kraus maps. Three practically important properties of the objective function
are found: (a) the absence of local maxima or minima (i.e., false traps); (b)
the existence of multi-dimensional sub-manifolds of optimal solutions
corresponding to the global maximum and minimum; and (c) the connectivity of
each level set. All of the critical values and their associated critical
sub-manifolds are explicitly found for any initial system state. Away from the
absolute extrema there are no local maxima or minima, and only saddles may
exist, whose number and the explicit structure of the corresponding critical
sub-manifolds are determined by the initial system state. There are no saddles
for pure initial states, one saddle for a completely mixed initial state, and
two saddles for other initial states. In general, the landscape analysis of
critical points and optimal manifolds is relevant to the problem of explaining
the relative ease of obtaining good optimal control outcomes in the laboratory,
even in the presence of the environment.
