Capacity-Achieving Entanglement Purification Protocol for Pauli Dephasing Channel

Quantum communication enables secure information transmission and entanglement distribution, but these tasks are fundamentally limited by the capacities of quantum channels. While quantum repeaters can mitigate losses and noise, entanglement swapping via a central node is ineffective against the Pauli dephasing channel due to degradation from Bell-state measurements. This suggests that purifying distributed Bell states before entanglement swapping is necessary. Although one-way hashing codes are known to saturate the dephasing channel capacity, no explicit two-way purification protocol has previously been shown to achieve this bound. In this work, we present a two-way entanglement purification protocol with an explicit, scalable circuit that asymptotically achieves the dephasing channel capacity. With each iteration, the fidelity of Bell states increases. At the final round, the residual dephasing error is suppressed doubly-exponentially, scaling as $\mathcal{O}(p^{2^{n}})$, enabling near-perfect Bell pairs for any fixed number of purification rounds $n$. The explicit circuit we propose is versatile and applicable to any number of Bell pairs, offering a practical solution for mitigating decoherence in quantum networks and distributed.