Interaction with a pre and post selected environment and recoherence

The interaction of an open system $\s$ with a pre- and post-selected environment is studied. In general, under such circumstances $\s$ can not be described in terms of a density matrix, {\it even when $\s$ in not post-selected}. However, a simple description in terms of a two-state (TS) is always available. The two-state of $\s$ evolves in time from an initially `pure' TS to a `mixed' TS and back to a final `pure' TS. This generic process is governed by a modified Liouville equation, which is derived. For a sub-class of observables, which can still be described by an ordinary density matrix, this evolution generates recoherence to a final pure state. In some cases post-selection can even suppress any decoherence.