Effect of inter-subsystem couplings on the evolution of composite systems

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.