NOON state (two-mode, N=2)

The N00N state on two bosonic modes $a,b$ with $N$ total particles is

$|\mathrm{NOON}\rangle = \frac{|N\rangle_a|0\rangle_b + |0\rangle_a|N\rangle_b}{\sqrt2},$

an equal superposition of "all $N$ particles in mode $a$" and "all $N$ particles in mode $b$." Because this superposition lives entirely within the two-dimensional subspace spanned by $\{|N,0\rangle,|0,N\rangle\}$, it can be tracked with a single effective qubit for circuit-level bookkeeping: $|0\rangle_{\text{eff}}\equiv|N,0\rangle$, $|1\rangle_{\text{eff}}\equiv|0,N\rangle$, giving $|\mathrm{NOON}\rangle \cong |+\rangle_{\text{eff}} = H|0\rangle_{\text{eff}}$, formally identical to the Hadamard-basis state elsewhere in this catalog. This embedding is exact for tracking amplitudes and relative phase within the N00N manifold, but it does not capture the full bosonic Fock space or the nonlinear-optical resources (or heralded multi-photon interference) actually required to generate $|N,0\rangle+|0,N\rangle$ physically for $N>2$; the $N=2$ case alone is producible with linear optics via Hong–Ou–Mandel-style two-photon bunching at a 50:50 beamsplitter fed with one photon per input port.

Phase sensitivity. Passing mode $a$ through a phase shift $\phi$ (relative to $b$) evolves the state to

$|\mathrm{NOON}(\phi)\rangle = \frac{e^{iN\phi}|N,0\rangle + |0,N\rangle}{\sqrt2},$

since each of the $N$ particles picks up phase $\phi$ independently, giving $N\phi$ total — an $N$-fold compression of the interference fringe compared to a single-particle probe, where the fringe would oscillate as $\cos\phi$ rather than $\cos(N\phi)$. Standard phase-estimation error propagation gives $\Delta\phi = 1/N$ for an ideal N00N-state measurement (the Heisenberg limit), compared to $\Delta\phi = 1/\sqrt N$ for $N$ independent uncorrelated probes at the same total resource count (the standard quantum limit) — the quadratic advantage in $N$ that motivates using entangled probe states for quantum-enhanced sensing.

Fragility. The metrological advantage is purchased at a steep cost in robustness: losing even a single one of the $N$ particles collapses the coherence between the $|N,0\rangle$ and $|0,N\rangle$ branches entirely (an environment that "sees" which branch the lost particle came from performs a which-path measurement), so N00N-state sensitivity degrades catastrophically under loss, unlike less fragile metrological resource states such as spin-squeezed or Dicke states. This tradeoff between Heisenberg-limited scaling and loss-sensitivity is the central practical constraint on N00N-state interferometry.

Where this shows up. N00N states are the textbook example motivating quantum-enhanced interferometry and quantum radar/lidar proposals, and the $N=2$ case connects directly to two-photon Hong–Ou–Mandel interference — the same bosonic bunching effect that underlies photonic SWAP-test and boson-sampling architectures.