Abstract: The mixedness of the $N$-qubit quantum states with exchange symmetry has been studied, and the results show that the linear entropy of the single qubit reduced density matrix (RDM), which can describe the mixedness, is completely determined by the expectation values $\langle S_z \rangle $ and $\langle S_\pm \rangle $ for both the pure and the mixed states. The mixedness of the pure states can be used to describe the bipartite entanglement, as an example we have calculated the mixedness of the Dicke state and the spin squeezed Kitagawa--Ueda state. For the mixed states, we determine the mixedness properties of both the ground states and the thermal states in mean-field clusters of spin-1/2 particles interacting via the anisotropy Heisenberg XXZ interaction, and found for the ferromagnetic case ($J < 0)$, the mixedness will approximate to the pairwise entanglement when the anisotropic parameter ${\it\Delta }> {\it\Delta }_{\rm c} $.

Key words: mixedness, linear entropy, symmetric quantum state