Time-symmetrized description of nonunitary time asymmetric quantum evolution

We discuss how systems which evolve manifestly asymmetrically in time can be described within the framework of the time-symmetrized quantum mechanics. An obvious case of asymmetry arises when a pure state evolves into a mixed state via effectively non-unitary evolution. A two-state method for finding the intermediate probability in postselected systems under such evolution is developed and the time-symmetry aspects of the method are explicitly considered. A specific feature is the existence of the so-called second scenario in which the state originating from the postselection measurement evolves under different evolution superoperator than the state from the preselection measurement. The evolution of the second scenario is explicitly defined. We illustrate the method with two characteristic examples: the spontaneous deexcitation of atoms and the systems approaching thermal equilibrium. We consider the systems with two energy levels and calculate the time-symmetrized probability of finding the system in excited state, under general preselection and postselection conditions. The consequences of the asymmetry of the time evolution on this probability are discussed. It is demonstrated that the arrow of time can be reconstructed in some special cases of postselected systems, while, for a general system, this is not the case.