Communicating without shared reference frames

We generalize a quantum communication protocol introduced by Bartlett et al. [New J. Phys. 11, 063013 (2009)], in which two parties communicating do not share a classical reference frame, to the case where changes of their reference frames form a one-dimensional noncompact Lie group. Alice sends to Bob the state ${\rho }_R\otimes {\rho }_S$, where ${\rho }_S$ is the state of the system Alice wishes to communicate and ${\rho }_R$ is the state of an ancillary system serving as a token of her reference frame. Because Bob is ignorant of the relationship between his reference frame and Alice's, he will describe the state ${\rho }_R\otimes {\rho }_S$ as an average over all possible reference frames. Bob measures the reference token and applies a correction to the system Alice wished to communicate conditioned on the outcome of the measurement. The recovered state ${\rho }_S^{\prime }$ is decohered with respect to ${\rho }_S$, the amount of decoherence depending on the properties of the reference token ${\rho }_R$. We present an example of this protocol when Alice and Bob do not share a reference frame associated with the one-dimensional translation group and use the fidelity between ${\rho }_S$ and ${\rho }_S^{\prime }$ to quantify the success of the recovery operation.

Figure 1

Communication channel $\mathcal{R}\circ \mathcal{E}$. Alice prepares a state ${\rho }_S$ she wishes to communicate to Bob along with the state $|e\rangle \langle e|$ as a token of her reference frame. As Bob does not know the relation between his reference frame and Alice's, he sees the joint state of the reference token and system as the encoded state $\mathcal{E}\left[{\rho }_S\right]=\mathcal{G}[|e\rangle \langle e|\otimes {\rho }_S]$. Bob then applies the recovery operation to the encoded state and recovers the state ${\rho }_S^{\prime }=\mathcal{R}\circ \mathcal{E}\left[{\rho }_S\right]$.

Figure 2

Fidelity $F({\rho }_S^{\prime },{\rho }_S)$ between the state sent by Alice ${\rho }_S$ and the state recovered by Bob ${\rho }_S^{\prime }$ as a function of $\sqrt{{\sigma }^2+{\delta }^2}$, where $\sigma$ is the width of the reference token in position space and $\delta$ quantifies the accuracy of Bob's measurement of the reference token. It is seen that for a fixed $\sqrt{{\sigma }^2+{\delta }^2}$, states less localized in the position basis (larger $\mathrm{\Delta }$) are better recovered by Bob.

Figure 3

For a reference token prepared in a superposition of two Gaussian states described by Eq. (33), the maximum fidelity $F_{\max }:=max[F({\rho }_S^{\prime },{\rho }_S)\mid \overline{x},\overline{p},\sigma >0]$ and the size of the superposition in momentum space ${\overline{p}}_{\max }/\sigma$ that realizes this maximum is plotted as a function of the width of the in position space $\mathrm{\Delta }/\sigma$ of the state Alice sent to Bob as given in Eq. (31). For all values of $\mathrm{\Delta }/\sigma$ the size of the superposition in momentum space which realizes the maximum fidelity is ${\overline{x}}_{\max }\sigma =0$.