Quantum discord, one of the famous quantum correlations, has been recently generalized to multipartite systems by Radhakrishnan et al. Here we give analytical solutions of the quantum discord for a family of N -qubit quantum states. For the bipartite system, we derive a zero quantum discord which will remain unchanged under the phase damping channel. For multiparitite systems, it is found that the quantum discord can be classified into three categories and the quantum discord for odd-partite systems can exhibit freezing under the phase damping channel, while the freezing does not exist in the even-partite systems.
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