  Given an ensemble of mixed qubit states, it is possible to increase the
purity of the constituent states using a procedure known as state purification.
The reverse operation, which we refer to as dilution, reduces the level of
purity present in the constituent states. In this paper we find asymptotically
optimal procedures for purification and dilution of an ensemble of i.i.d. mixed
qubit states, for some given input and output purities and an asymptotic output
rate. Our solution involves using the statistical tool of local asymptotic
normality, which recasts the qubit problem in terms of attenuation and
amplification of a single displaced Gaussian state. Therefore, to obtain the
qubit solutions, we must first solve the analogous problems in the Gaussian
setup. We provide full solutions to all of the above, for the (global) trace
norm figure of merit.
