Generalized GHZ entanglement states for optimal rotation detection of quantum spin systems using majorana representation

A multitude of engineering and scientific applications require the estimation of an unknown axis' rotation, known as the reference frame alignment problem. These problems can be categorized using the quantum Cramer-Rao bound. Detecting rotations is typically focused on smaller angles, as errors are more significant in this range. Therefore, selecting the optimal combination of measurement techniques based on a quantum entangled state for a particular setup is a critical step. This paper proposes a linear combination of anticoherence and coherence measures based on the generalized GHZ entanglement states. The anticoherence measures, optimizes the spin system's average fidelity in detecting rotated states, while the overlap between states in any spin number is minimal. The optimal quantum rotosensors are characterized by the minimal average fidelity given by the overlap before and after averaging over all directions. The individual subsystems' states have substantial overlap with GHZ states, making the generalized GHZ state the entanglement quantum spin-j state with coherence decay rate epsilon beneficial in finding the optimal rotation detection on noncoherent entanglement state.