  The effect of inter-subsystem coupling on the adiabaticity of composite
systems and that of its subsystems is investigated. Similar to the adiabatic
evolution defined for pure states, non-transitional evolution for mixed states
is introduced; conditions for the non-transitional evolution are derived and
discussed. An example that describes two coupled qubits is presented to detail
the general presentation. The effects due to non-adiabatic evolution on the
geometric phase are also presented and discussed.
