Lorentz-invariant look at quantum clock-synchronization protocols based on distributed entanglement

Recent work has raised the possibility that quantum-information-theory techniques can be used to synchronize atomic clocks nonlocally. One of the proposed algorithms for quantum clock synchronization (QCS) requires distribution of entangled pure singlets to the synchronizing parties [R. Jozsa et al., Phys. Rev. Lett. $85$ 2010 (2000)]. Such remote entanglement distribution normally creates a relative phase error in the distributed singlet state, which then needs to be purified asynchronously. We present a relativistic analysis of the QCS protocol that shows that asynchronous entanglement purification is not possible, and, therefore, the proposed QCS scheme remains incomplete. We discuss possible directions of research in quantum-information theory, which may lead to a complete, working QCS protocol.