The optimal N qubit states featuring highest sensitivity to small misalignment of cartesian reference frames are found using the Quantum Cramer-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.
@article{arxiv.0805.3637,
title = {Optimal state for keeping reference frames aligned and the Platonic solids},
author = {Piotr Kolenderski and Rafal Demkowicz-Dobrzanski},
journal= {arXiv preprint arXiv:0805.3637},
year = {2009}
}