Optimal primitive reference frames

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∘.In this case 24% of the egativity is preserved. The classical limit is considered numerically, and indicates under the criterion of entanglement preservation, that 90∘ is selected out nonmonotonically, with a peak optimal angle of 96.5∘ for L=3 spins.