  We previously proposed a mechanism to effectively obtain, after a long time
development, a Hamiltonian being Hermitian with regard to a modified inner
product $I_Q$ that makes a given non-normal Hamiltonian normal by using an
appropriately chosen Hermitian operator $Q$. We studied it for pure states. In
this letter we show that a similar mechanism also works for mixed states by
introducing density matrices to describe them and investigating their
properties explicitly both in the future-not-included and future-included
theories. In particular, in the latter, where not only a past state at the
initial time $T_A$ but also a future state at the final time $T_B$ is given, we
study a couple of candidates for it, and introduce a ``skew density matrix''
composed of both ensembles of the future and past states such that the trace of
the product of it and an operator ${\cal O}$ matches a normalized matrix
element of ${\cal O}$. We argue that the skew density matrix defined with $I_Q$
at the present time $t$ for large $T_B-t$ and large $t-T_A$ approximately
corresponds to another density matrix composed of only an ensemble of past
states and defined with another inner product $I_{Q_J}$ for large $t-T_A$.
