Multi-Party Quantum Purity Distillation with Bounded Classical Communication

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.