  Keeping in view the exchange symmetry aspects in studies on spin squeezing of
multiqubit states, we show that Ising type Hamiltonian does not retain the
exchange symmetry of initially symmetric multiqubit states. Specifically, we
show that all N-qubit states (N greater than or equal to 5) obeying permutation
symmetry lose their symmetry after evolution with one dimensional Ising chain
with nearest neighbor interactions. Among 4-qubit symmetric pure states, only W
type states retain their symmetry under time evolution with Ising Hamiltonian.
