GHZ states as near-optimal states for reference frame alignment

Abstract

Let two coordinate systems, in possession of Alice and Bob, be related to each other by an unknown rotation \(R\in \hbox {SO}(3)\). Alice is to send identical states \(|\psi _0\rangle \) 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. We show concrete measurements which will allow Bob to determine the rotation R. This shows that GHZ states, as superposition of macroscopically distinct states, are useful in yet another context in quantum information, namely in communicating the information of a whole coordinate system between two parties where no prior information is available on the relative orientation of the two frames.