Are right angles special? The optimal primitive reference frame

TL;DR

This paper investigates the optimal primitive quantum reference frames for preserving quantum information, finding that two orthogonal spins are nearly optimal and that slight entanglement can improve preservation efficiency.

Contribution

It identifies the optimal configurations of minimal quantum reference frames for quantum information preservation, including the effects of entanglement and specific spin alignments.

Findings

Two orthogonal spins are optimal for single spin state preservation.

Adding entanglement improves preservation from 22% to 26%.

Optimal spin alignment for entanglement preservation is around 87 degrees.

Abstract

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 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 for spins aligned at an optimal angle of 87 degrees. 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 degrees is selected out non-monotonically, with a peak optimal angle of 96.5 degrees for L = 3 spins.