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 ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$, where ${\ensuremath{\rho}}_{S}$ is the state of the system Alice wishes to communicate and ${\ensuremath{\rho}}_{R}$ 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 ${\ensuremath{\rho}}_{R}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{S}$ 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 ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ is decohered with respect to ${\ensuremath{\rho}}_{S}$, the amount of decoherence depending on the properties of the reference token ${\ensuremath{\rho}}_{R}$. 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 ${\ensuremath{\rho}}_{S}$ and ${\ensuremath{\rho}}_{S}^{\ensuremath{'}}$ to quantify the success of the recovery operation.

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