We consider entanglement purification protocols for multiple copies of qubit states. We use high-dimensional auxiliary entangled systems to learn about the number and positions of errors in the noisy ensemble in an explicit and controlled way, thereby reducing the amount of noise in the ensemble and purifying the remaining states. This allows us to design entanglement purification protocols for any number of copies that work particularly well for a small number of expected errors, i.e., high fidelity of initial states. The main tool is a counter gate with which the required nonlocal information can be transferred into the high-dimensional entangled qudit auxiliary states. We compare our schemes to standard recurrence protocols that operate on pairs of copies, and hashing and breeding protocols that operate on a (asymptotically) large number of copies. Our protocols interpolate between these two regimes, leading to higher achievable fidelity and yield. We illustrate our approach for bipartite qubit states and generalize it to purify multiparty GHZ states.

中文翻译：

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通过纠缠测量计数和定位误差进行纠缠纯化

我们考虑了量子比特状态的多个副本的纠缠纯化协议。我们使用高维辅助纠缠系统以明确且可控的方式了解噪声集成中错误的数量和位置，从而减少集成中的噪声量并净化剩余状态。这使我们能够为任意数量的副本设计纠缠纯化协议，这些协议对于少量预期错误（即初始状态的高保真度）特别有效。主要工具是一个计数器门，通过它可以将所需的非局部信息转换为高维纠缠量子比特辅助状态。我们将我们的方案与在成对副本上运行的标准重复协议进行比较，以及在（渐近）大量副本上运行的散列和繁殖协议。我们的协议在这两种制度之间进行插值，从而实现更高的保真度和产量。我们说明了我们对二分量子比特状态的方法，并将其推广到净化多方 GHZ 状态。