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We consider entanglement purification protocols for multiple copies of qubitstates. We use high-dimensional auxiliary entangled systems to learn aboutnumber and positions of errors in the noisy ensemble in an explicit andcontrolled way, thereby reducing the amount of noise in the ensemble andpurifying the remaining states. This allows us to design entanglementpurification protocols for any number of copies that work particularly well fora small number of expected errors, i.e. high fidelity of initial states. Themain tool is a counter gate with which the required non-local information canbe transferred into the high-dimensional entangled qudit auxiliary states. Wecompare our schemes to standard recurrence protocols that operate on pairs ofcopies, and hashing and breeding protocols that operate on a (asymptotically)large number of copies. Our protocols interpolate between these two regimes,leading to a higher achievable fidelity and yield. We illustrate our approachfor bipartite qubit states, and generalize it to purify multi-party GHZ states.

Entanglement purification by counting and locating errors with entangling measurements