  We consider the task of distilling local purity from a noisy quantum state
$\rho^{ABC}$, wherein we provide a protocol for three parties, Alice, Bob and
Charlie, to distill local purity (at a rate $P$) from many independent copies
of a given quantum state $\rho^{ABC}$. The three parties have access to their
respective subsystems of $\rho^{ABC}$, and are provided with pure ancilla
catalytically, i.e., with the promise of returning them unaltered after the end
of the protocol. In addition, Alice and Bob can communicate with Charlie using
a one-way multiple-access dephasing channel of link rates $R_1$ and $R_2$,
respectively. The objective of the protocol is to minimize the usage of the
dephasing channel (in terms of rates $R_1$ and $R_2$) while maximizing the
asymptotic purity that can be jointly distilled from $\rho^{ABC}$. To achieve
this, we employ ideas from distributed measurement compression protocols, and
in turn, characterize a set of sufficient conditions on $(P,R_1,R_2)$ in terms
of quantum information theoretic quantities such that $P$ amount of purity can
be distilled using rates $R_1$ and $R_2$. Finally, we also incorporate the
technique of asymptotic algebraic structured coding, and provide a unified
approach of characterizing the performance limits.
