I have a silly question, and I'm going to shamelessly use HN to ask it.

In Kitaev's construction of the high purity approximation to a magic state, he starts with the assumption that we start with a state which can be represented as the tensor product of n mixed states which are "close enough". I don't understand where this separability property comes from. My (very) naive assumption would be that there is some big joint state which you have a piece of, and the information that I have about this piece are n of its partial traces, which are indeed n copies of the "poor man's" magic state.

Can I know more than that? There's lots of stuff in the preimage of these partial traces. Why am I allowed to assert that I have the nicest one?