  State estimation is usually analyzed in the situation when copies are in a
product state, either mixed or pure. We investigate here the concept of state
estimation on correlated copies. We analyze state estimation on correlated N
qubit states, which are permutationally invariant. Using a correlated state we
try to estimate as good as possible the direction of the Bloch vector of a
single particle reduced density matrix. We derive the optimal fidelity for all
permutation invariant states. We find the optimal state, which yields the
highest estimation fidelity among the states with the same reduced density
matrix. Interestingly this state is not a product state. We also point out that
states produced by optimal universal cloning machines are the worst form the
point of view of estimating the reduced density matrix.
