Changing quantum reference frames

We consider the process of changing reference frames in the case where the reference frames are quantum systems. We find that, as part of this process, decoherence is necessarily induced on any quantum system described relative to these frames. We explore this process with examples involving reference frames for phase and orientation. Quantifying the effect of changing quantum reference frames provides a theoretical description for this process in quantum experiments, and serves as a first step in developing a relativity principle for theories in which all objects including reference frames are necessarily quantum.

Figure 1

The procedure (17) on states of the form given by (19) is the same map as ${\mathcal{E}}_{{|h\rangle }_B}\circ \mathcal{R}$ on an encoding ${\mathcal{E}}_{{|\psi \left(a\right)\rangle }_A}\left({\rho }_S\right)$. The change of quantum reference frame map results in description of the state relative to a different reference frame without measuring against a background classical frame, whereas the reencoding map ${\mathcal{E}}_{{|h\rangle }_B}\circ \mathcal{R}$ changes a reference frame using a background reference frame for the intermediate state ${\rho }_S^{\prime }$.

Figure 2

(Color online) Plotted are the state overlaps $D_{s_A}{\left|\langle g|\psi \left(0\right)\rangle \right|}^2$ for reference frame $A$ in a $\mathrm{U}\left(1\right)$ phase eigenstate [(34), “PE”] and coherent state [(38), “CS”], for choices of average photon number ${\langle n\rangle }_A$. This indicates the distribution of unitaries in the decoherence maps ${\mathcal{F}}_S^{\left(A\right)}$. For small average photon number, the decoherence for the coherent states has a narrower peak than the phase eigenstate, but the phase eigenstate becomes more narrowly peaked by ${\langle n\rangle }_A=5$. For calculations, the summations for the coherent-state overlap were truncated at the 21st terms, accounting for 99.99% of the support.

Figure 3

Plots of integrand of decoherence map (46) for fiducial state Cartesian reference frames with $s_A=1,4,8$. The fiducial-state overlap depends only on the rotation angle $\omega$ and not on the axis of the rotation $\mathbf{n}=(sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta )$. The mixing is symmetric about the identity, identified by $\omega =0$. The first plot shows the details of the plot across $\omega$, and the second plot shows the full range for small $\omega$, indicating that overlap function becomes extremely peaked near $\omega =0$ even for small $s_A$ since the density of states approaches zero as $\omega \to 0$.

Figure 4

(Color online) Overlap functions in $\beta$ for the $\mathrm{SU}\left(2\right)$ coherent-state decoherence map (52), where $\beta =0$ to $\pi$ is the standard polar angle. The overlap function depends on $\beta$ and is unconstrained in $\alpha$ and $\gamma$: they are rotationally symmetric on the two-sphere about the north pole (the identity), indicating that there is mixing around circles of constant latitude. The distribution becomes more tightly peaked near $\beta =0$ as $s_A$ increases.