NOON state

A NOON state is a quantum-mechanical many-body entangled state:

which represents a superposition of N particles in mode a with zero particles in mode b, and vice versa. Usually, the particles are photons, but in principle any bosonic field can support NOON states.

NOON states are an important concept in quantum metrology and quantum sensing for their ability to make precision phase measurements when used in an optical interferometer. For example, consider the observable

The expectation value of ${\displaystyle A}$ for a system in a NOON state switches between +1 and −1 when the phase changes from 0 to ${\displaystyle \pi /N}$. Moreover, the error in the phase measurement becomes

This is the so-called Heisenberg limit, and gives a quadratic improvement over the standard quantum limit. NOON states are closely related to Schrödinger cat states and GHZ states, and are extremely fragile.

...
Wikipedia