  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. 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 fully relativistic analysis of the QCS protocol which shows that
asynchronous entanglement purification is not possible, and, therefore, that
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.
