NOON state is an entangled quantum state of two modes in which $N$ photons are in one mode and zero in the other, in an equal superposition with the reverse. It is written as:
$$\lvert N\!:\!0\rangle\!\rangle = \frac{1}{\sqrt{2}}\left(\lvert N\rangle_a\lvert 0\rangle_b + \lvert 0\rangle_a\lvert N\rangle_b\right)$$
NOON states arise naturally in quantum metrology and quantum lithography. Their key property is phase sensitivity that scales as $1/N$ — the Heisenberg limit — compared to $1/\sqrt{N}$ for classical light (the shot-noise limit). This $\sqrt{N}$-fold improvement in phase sensitivity is the central metrological advantage of NOON states.
When a NOON state accumulates a phase shift $\phi$ in mode $a$ (e.g. from a delay in one arm of an interferometer), the $N$-photon Fock state picks up a phase $N\phi$ rather than just $\phi$.
$$\frac{1}{\sqrt{2}}\left(e^{iN\phi}\lvert N\rangle_a\lvert 0\rangle_b + \lvert 0\rangle_a\lvert N\rangle_b\right)$$
This $N$-fold amplification of the phase allows interferometric measurements with Heisenberg-limited precision.
NOON states are extremely fragile. A single photon loss from the $N$-photon component collapses the superposition and destroys all phase information. This decoherence sensitivity grows worse as $N$ increases, making large-$N$ NOON states very difficult to generate and maintain experimentally. Current experimental demonstrations are limited to small $N$ (typically $N \leq 5$).