We present two economical one-step error-correction protocols for multipartite polarization-entangled systems in a Greenberger-Horne-Zeilinger state. One uses spatial entanglement to correct errors in the polarization entanglement of an $N$-photon system, resorting to linear optical elements. The other uses frequency entanglement to correct errors in the polarization entanglement of an $N$-photon system. The parties in quantum communication can obtain a maximally entangled state from each $N$-photon system transmitted with one step in these two protocols, and both of their success probabilities are $100\%$, in principle. That is, they both work in a deterministic way, and they do not largely consume the less-entangled photon systems, which is far different from conventional multipartite entanglement purification schemes. These features may make these two protocols more useful for practical applications in long-distance quantum communication.

• Received 18 January 2011

DOI:https://doi.org/10.1103/PhysRevA.83.062316