NOON state (two-mode, N=2)

Classical (coherent-state) interferometry estimates a phase φ with uncertainty scaling as 1/√N in the number of probe particles — the standard quantum limit or shot-noise limit. NOON states were introduced to reach the fundamentally better 1/N Heisenberg scaling by concentrating N particles into a single maximally path-entangled superposition rather than spreading them across N independent particles.

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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.

TakesNothingWhat joins here

Nothing goes in, and that is what this entry is: a pipeline starts here. There is no upstream to choose — what you pick is what comes after.

Nothing in the Atlas meets this end.

Returns1 qubitWhat joins here

Returns a 1-qubit register a next stage can take. This is the end that joins.

14 entries meet this end. Named below.

Prepares a state. Nothing goes in, and what comes out is a register another stage can take. See all 13 →

Fits after this 14

Modes: 2 (a, b)
Particle number: N (N=2 shown)
Phase scaling: e^{iNφ} (Heisenberg limit Δφ=1/N)
State: (|N,0⟩+|0,N⟩)/√2
Standard quantum limit: Δφ = 1/√N
Heisenberg limit: Δφ = 1/N

H on NOON qubit (|N,0⟩/|0,N⟩)
RZ(Nφ) on NOON qubit (|N,0⟩/|0,N⟩)

|N,0⟩ (before phase)50%

|0,N⟩ (before phase)50%

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Nobody has reviewed this record for gaps yet.

FrameworkNative

noon_state.py

import numpy as np
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

def noon_qubit_embedding(N: int, phi: float = 0.0) -> QuantumCircuit:
    # Effective 2-level embedding: |0> == |N,0>, |1> == |0,N>.
    qc = QuantumCircuit(1)
    qc.h(0)              # (|N,0> + |0,N>) / sqrt(2)
    qc.rz(N * phi, 0)     # N-fold phase accumulation e^{i N phi}
    return qc

N = 2
sv0 = Statevector.from_instruction(noon_qubit_embedding(N, phi=0.0))
print("Amplitudes at phi=0:", sv0.data)  # equal superposition

phi = np.pi / (2 * N)
sv = Statevector.from_instruction(noon_qubit_embedding(N, phi))
relative_phase = np.angle(sv.data[1]) - np.angle(sv.data[0])
print(f"Relative phase at phi={phi:.4f}: {relative_phase:.4f}  (expected N*phi = {N * phi:.4f})")

FINAL_CIRCUIT = noon_qubit_embedding(N, phi)

Where the map uses this

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State you can prepare 5 of 34 processes

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Prepare an input state hands one back
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Minimise the objective over the parameters hands one back