  Interferometry with NOON quantum states can provide unbiased phase estimation
with a sensitivity scaling as $\Delta \theta \sim 1/N_T$ given a prior
knowledge that the true phase shift $\theta$ lies in the interval $-\pi \leq
\theta \leq \pi$. The protocol requires a total of $N_T = 2^{p}-1$ particles
(unequally) distributed among $p$ independent measurements and overcomes basic
difficulties present in previously proposed approaches. We demonstrate the
possibility to obtain a phase sensitivity beating the classical shot-noise
limit using published probabilities retrieved experimentally for the creation
of Schr\"odinger cat quantum states containing up to N=6 beryllium ions.
