  The instability of an atomic clock is characterized by the Allan variance, a
measure widely used to describe the noise of frequency standards. We provide an
explicit method to find the ultimate bound on the Allan variance of an atomic
clock in the most general scenario where N atoms are prepared in an arbitrarily
entangled state and arbitrary measurement and feedback are allowed, including
those exploiting coherences between succeeding interrogation steps. While the
method is rigorous and general, it becomes numerically challenging for large N
and long averaging times.
