GHZ states as near-optimal states for reference frame alignment

Let two coordinate systems, in possession of Alice and Bob, be related to each other by an unknown rotation $R\in SO(3)$. Alice is to send identical states $\mathrm{\mid }{\psi }_0\text{ra}$ to Bob who will make measurements on the received state and will determine the rotation $R$. The task of Bob is to estimate these parameters of the rotation $R$ by the best possible measurements. Based on the Quantum Fisher Information, we show that Greenberger-Horne-Zeilinger (GHZ) states are near optimal states for this task. Compared to the optimal states proposed before, the advantage of $GHZ$ states are that they can be more easily prepared experimentally, and more importantly, we show concrete measurements which will allow Bob to determine the rotation $R$. We also study the robustness of these states in keeping their encoded information, against common sources of noises.