  One might wonder why Deutsch appeals to mixed states: will superpositions of states    and    not suffice? Unfortunately such an idea does not work. Suppose again that the initial state is  1. One might suggest that that if state S3 is  1/√2 3  +  1/√2 3 one will obtain a consistent development. For one might think that when initial state  1 encounters the superposition  1/√2 3  +  1/√2 3, it will develop into superposition  1/√2 2  +  1/√2 2, and that this in turn will develop into  1/√2 3  +  1/√2 3, as desired. However this is not correct. For initial state  1 when it encounters  1/√2 3 +  1/√2 3, will develop into the entangled state  1/√2 24  +  1/√2 24. In so far as one can speak of the state of the young system after this interaction, it is in the mixture of 50%  2 and  50%  2, not in the superposition  1/√2 2 +  1/√2 2. So Deutsch does need his recourse to mixed states.