  Deutsch assumes that the set of possible states of this system are the mixed states, i.e., are represented by the density matrices over the Hilbert space of that system. Deutsch then shows that for any initial state S1, any unitary interaction between the older and younger self, and any unitary development during time travel, there is a consistent solution, i.e., there is at least one pair of states S2 and S3 such that when S1 interacts with S3 it will change to state S2 and S2 will then develop into S3. The states S2, S3 and S4 will typically be not be pure states, i.e., will be non-trivial mixed states, even if S1 is pure. In order to understand how this leads to interpretational problems let us give an example. Consider a system that has a two dimensional Hilbert space with as a basis the states    and  .  Let us suppose that when state    of the young system encounters state    of the older system, they interact and the young system develops into state    and the old system remains in state  .  In obvious notation: