  The correlated projection superoperator techniques provide a better
understanding about how correlations lead to strong non-Markovian effects in
open quantum systems. Their superoperators are independent of initial state,
which may not be suitable for some cases. To improve this, we develop a new
approach, that is extending the composite system before use the correlated
projection superoperator techniques. Such approach allows choosing different
superoperators for different initial states. We apply these techniques to a
simple model to illustrate the general approach. The numerical simulations of
the full Schr\"{o}dinger equation of the model reveal the power and efficiency
of the method.
