  But, says Deutsch, in quantum mechanics such a system can also be in any mixture of the states    and  .  Suppose that the older system, prior to the interaction, is in a state S3 which is an equal mixture of 50%  3 and 50%  3. Then the younger system during the interaction will develop into a mixture of 50%  2 and 50%  2, which will then develop into a mixture of 50%  3 and  50%  3, which is consistent! More generally Deutsch uses a fixed point theorem to show that no matter what the unitary development during interaction is, and no matter what the unitary development during time travel is, for any state S1 there is always a state S3 (which typically is not a pure state) which causes S1 to develop into a state S2 which develops into that state S3. Thus quantum mechanics comes to the rescue: it shows in all generality that no constraints on initial states are needed!