  The concatenated Greenberger-Horne-Zeilinger (C-GHZ) state has great
potential application in the future quantum network, for it is robust to the
decoherence in a noisy environment. In the paper, we propose a complete C-GHZ
state analysis protocol with the help of some auxiliary single atoms trapped in
the low-quality cavities. In the protocol, we essentially make the parity check
for the photonic states based on the photonic Faraday rotation effect, and
complete the analysis task combined with the Hadamard operation and single
qubit measurement. The success probability of our protocol can reach 100\% in
principle, and the number of physical qubit encoded in each logic qubit does
not affect the analysis. Our analysis protocol may have its practical
application in future long-distance quantum communication.
