Capacity-Achieving Entanglement Purification Protocol for Pauli Dephasing Channel

TL;DR

This work tackles the capacity limits of Pauli dephasing channels by introducing an explicit two-way entanglement-purification protocol. The protocol uses a scalable $N\rightarrow M$ purification circuit based on CNOTs and Hadamard-basis measurements to iteratively purify Bell pairs, with the reverse coherent information (RCI) guiding performance. It achieves asymptotic saturation of the dephasing-channel capacity, with residual dephasing decaying as $\mathcal{O}(p^{2^{n}})$ after $n$ rounds and yield scaling $Y = ((m-1)/m)^n$; as $m$ grows and $n$ remains sublinear in $m$, the average entanglement rate approaches the channel capacity $C = 1 - \mathrm{H}_2(p)$. The results offer a practical, explicit circuit for mitigating decoherence in quantum networks and distributed quantum computing, with potential extensions to other noise models and implementations in repeater-based architectures.

Abstract

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.

Figures (11)

• Figure 1: First stage of the protocol: Alice prepares $m$ copies of $\ket{\phi^+}$ Bell states and sends them to Bob via a Pauli dephasing channel. Both apply CNOT gates on their qubits, with $\mathrm{A_1, A_2}, \ldots, \mathrm{A_m}$ and $\mathrm{B_1, B_2}, \ldots, \mathrm{B_m}$ representing their respective modes. After the CNOTs, $\mathrm{A_m}$ and $\mathrm{B_m}$ are measured in the Hadamard basis.
• Figure 2: Simulation results of the first round of the protocol with increasing number of Bell pairs $m$ shared between Alice and Bob. The black dashed line represents the dephasing channel capacity, while the blue dashed line shows results from Deutsch et al.deutsch1996quantum.
• Figure 3: Second stage of the purification protocol, where blue and yellow boxes represent successful and failed states, respectively. The top half further purifies $m$ copies of the successfully purified states from the first round, while the bottom half purifies $m$ copies of the states that failed in the first round. The last rail serves as the control, with CNOTs applied and the last rail measured. If $j_1 = k_1$, further purification is achieved.
• Figure 4: Schematic representation of measuring a high-dimensional state in terms of individual Bell pairs in the second round of the protocol, with $m=3$ carried over from the first round. In the second round, there are $m=3$ high-dimensional states $\rho_{\mathrm{A_1A_2B_1B_2}}$. CNOTs are applied between Bell pairs that are not entangled with each other. Each high-dimensional state contains two Bell pairs, so measuring one high-dimensional state corresponds to measuring the two Bell pairs within it, $\rho_{\mathrm{A_1B_1}}$ and $\rho_{\mathrm{A_2B_2}}$. Alice and Bob achieve further purification when $\mathrm{A_1}$ and $\mathrm{B_1}$ agree and $\mathrm{A_2}$ and $\mathrm{B_2}$ agree, corresponding to outcomes $(++,++), (++,--), (--,++)$ and $(--,--)$.
• Figure 5: Illustration of the rearrangement of Bell pairs using a CNOT gate in the second stage of the protocol. Black and pink lines indicate how the Bell pairs are grouped together for the CNOT application in the second round.