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 negativity 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.

Figure 1

Quantum Pythagorean theorem: getting $\sqrt{2}$ with spins up to $L=25$. The vertical axis is a rescaled probability to account for the increase in data points as we increase $L$. The horizontal axis is $J/L$ where $J$ is the measured total angular momentum of the two orthogonal spins.

Figure 2

Classical limit: optimal angle of inclination ${\beta }_{\mathrm{opt}}$ for the two spins in the QRF as a function of $L$. A peak occurs of ${\beta }_{\mathrm{opt}}={96.5}^{°}$ at $L=3$. Numerical analysis for large $L$ implies that $\mathcal{N}(\beta )$ approaches a step function with zeros at $\beta =0$ and $\beta ={180}^{°}$, and that asymptotically ${\beta }_{\mathrm{opt}}\to {90}^{°}$.

Figure 3

Negativity in the classical limit: preservation of negativity increases rapidly with increasing spin $L$. States ${\sigma }^{k_c}$ from sectors with angular momentum close to the classical value dominate the negativity.