Optimal measurements in phase estimation: simple examples

We identify optimal measurement strategies for phase estimation in different scenarios in which the interferometer acts on two-mode symmetric states. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed, this set reduces to merely two possible measurements. For two-qubit symmetric Werner state, we find the optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. We also determine the optimal measurements for a wide class of symmetric N-qubit Werner-like states. Finally, for a pure symmetric state of N qubits, we find under which conditions the estimation from the full N-body correlation and from the population imbalance is optimal.