  Local pure states are an important resource for quantum computing. The
problem of distilling local pure states from mixed ones can be cast in an
information theoretic paradigm. The bipartite version of this problem where
local purity must be distilled from an arbitrary quantum state shared between
two parties, Alice and Bob, is closely related to the problem of separating
quantum and classical correlations in the state and in particular, to a measure
of classical correlations called the one-way distillable common randomness. In
Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation
is derived when many copies of a bipartite quantum state are shared between
Alice and Bob, and the parties are allowed unlimited use of a unidirectional
dephasing channel. In the present paper, we extend this result to the setting
in which the use of the channel is bounded. We demonstrate that in the case of
a classical-quantum system, the expression for the local purity distilled is
efficiently computable and provide examples with their tradeoff curves.
