The optimal
N qubit states featuring highest sensitivity to small misalignment of Cartesian reference frames are found using the quantum Cram\'er-Rao bound. It is shown that the optimal states are supported on the symmetric subspace and hence are mathematically equivalent to a single spin
J=N∕2. Majorana representation of spin states is used to reveal a beautiful connection between the states optimal for aligning reference frames and the platonic solids.
We consider the smallest possible directional reference frames allowed and determine the best one can ever do in preserving quantum information in various scenarios. We find that for the preservation of a single spin state, two orthogonal spins are optimal primitive reference frames; and in a product state, they do approximately
22 as well as an infinite-sized classical frame. By adding a small amount of entanglement to the reference frame, this can be raised to
2(2/3)5=26. Under the different criterion of entanglement preservation, a very similar optimal reference frame is found; however, this time it is for spins aligned at an optimal angle of ${87}^{\ifmmode^\circ\else\textdegree\fi{}}$. In this case
24 of the negativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that ${90}^{\ifmmode^\circ\else\textdegree\fi{}}$ is selected out nonmonotonically, with a peak optimal angle of ${96.5}^{\ifmmode^\circ\else\textdegree\fi{}}$ for
L=3 spins.
Abstract 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 ϵ beneficial in finding the optimal rotation detection on noncoherent entanglement state.