Automatic hermiticity for mixed states

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 O matches a normalized matrix element of 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-QJ for large t - T-A.