# NOON state
A **NOON state** is a maximally path-entangled quantum state of N particles in a two-mode interferometer, written \( |\psi\rangle = (|N,0\rangle + |0,N\rangle)/\sqrt{2} \), in which all N particles travel together in one arm or all N travel in the other. Kok and coauthors gave the state its name in a 2002 review of quantum interferometry, reserving "High NOON state" for large N[1](https://ar5iv.labs.arxiv.org/html/quant-ph/0202133); the term "high-N00N state" itself was introduced by Dowling's group, first appearing as a footnote in a follow-on paper in quantum metrology[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf). Because every photon in the state accumulates the same phase, the state is the canonical resource for reaching the [Heisenberg limit](https://www.edgepedia.org/heisenberg-limit) in optical metrology, though it decoheres easily when photons are lost[3](https://repository.lsu.edu/cgi/viewcontent.cgi?article=4203&context=physics_astronomy_pubs).
| Key fact | Detail |
|---|---|
| Definition | \( (|N,0\rangle + |0,N\rangle)/\sqrt{2} \) over two interferometer arms; named by Kok et al. in 2002[1](https://ar5iv.labs.arxiv.org/html/quant-ph/0202133) |
| Phase response | Signal \( \cos(N\varphi) \), N times faster fringes than a coherent state; phase uncertainty \( \Delta\varphi = 1/N \), the Heisenberg limit[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf) |
| Quantum Fisher information | \( F_Q = N^2 \) for an ideal state, a √N sensitivity gain over a classical probe of the same energy[4](https://ar5iv.labs.arxiv.org/html/1611.08700) • [5](https://www.nature.com/articles/s41534-020-00320-y) |
| Loss fragility | With 3 dB loss in one arm, the visibility of an N = 3 NOON state drops by a factor of \( 2^3 = 8 \); the advantage disappears with loss, and below about 90% efficiency generalized Holland–Burnett probes surpass NOON states[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf) • [5](https://www.nature.com/articles/s41534-020-00320-y) |
| Photonic demonstrations | Up to N = 5 photons by 2013 (N = 5 by mixing quantum and classical light, Science 2010)[6](https://www.nature.com/articles/srep02684) • [7](https://www.science.org/doi/10.1126/science.1188172) |
| Largest N overall | N = 9 phonons in a single trapped 171Yb+ ion, generated deterministically[4](https://ar5iv.labs.arxiv.org/html/1611.08700) |
| Applications | Quantum lithography with λ/(2N) features[8](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.2733), quantum metrology benchmarks, phase sensing[9](https://iopscience.iop.org/article/10.1088/1751-8113/47/42/424006) |
## Mathematical form and phase response
In [Dirac notation](https://www.edgepedia.org/bra-ket-notation) the NOON state for N photons in the two arms a and b of an interferometer is
\[ |\psi_{\mathrm{NOON}}\rangle = \frac{1}{\sqrt{2}}\left(|N,0\rangle + e^{iN\varphi_S}|0,N\rangle\right), \]
where the relative phase acquired in one arm enters as \( e^{iN\varphi_S} \) because each of the N photons picks up the phase independently, so the total phase is linearly proportional to N[4](https://ar5iv.labs.arxiv.org/html/1611.08700). Recombining the arms produces a signal \( M_{\mathrm{NOON}}(\varphi) = I_A \cos(N\varphi) \), which oscillates N times as fast as the coherent-state signal \( \cos(\varphi) \); the fringe spacing shrinks from the Rayleigh diffraction limit by a factor of N, an effect called *super-resolution*, and the slope at the axis crossing is N times larger[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf).
The same N-fold phase accumulation underlies the *super-sensitivity*. Phase uncertainty is bounded by the uncertainty in the photon-number difference between the arms divided by the slope of the mean signal: for N photons the maximal photon-number-difference uncertainty is \( \Delta(n_1 - n_2) = N \), giving a minimal phase uncertainty \( \Delta\varphi = 1/N \), the Heisenberg limit, and the N-photon state that achieves it is the NOON state[6](https://www.nature.com/articles/srep02684). Maximally correlated NOON-type states reach \( \Delta\varphi_{HL} = 1/N \) even for small N, whereas entangled or dual-Fock schemes approach this bound only asymptotically[1](https://ar5iv.labs.arxiv.org/html/quant-ph/0202133).
## Quantum metrology and the Heisenberg limit
Metrology quantifies the information a probe carries about a phase through the quantum Fisher information Q, which sets the quantum [Cramér–Rao bound](https://www.edgepedia.org/cramer-rao-bound) \( \Delta\varphi \ge 1/\sqrt{Q} \). A probe of N entangled photons provides up to a \( \sqrt{N} \) enhancement in phase sensitivity compared with a classical probe of the same energy[5](https://www.nature.com/articles/s41534-020-00320-y); for an ideal NOON state the [Fisher information](https://www.edgepedia.org/fisher-information) is \( F_Q = N^2 \), so \( \Delta\varphi = 1/N \)[4](https://ar5iv.labs.arxiv.org/html/1611.08700). This is the same N-fold gain in precision as the GHZ-state approach in quantum information science, and optical NOON-state metrology experiments have surpassed the shot-noise limit[9](https://iopscience.iop.org/article/10.1088/1751-8113/47/42/424006).
Two qualifications apply. In single-fringe interferometry, where the measurement is confined to one output fringe, the NOON state does not achieve Heisenberg-limited scaling for N > 5; its Fisher information scales as \( N^{3/2} \), the geometric mean of the shot-noise and Heisenberg scalings, and it reaches the output uncertainty limit only for N = 2 and N = 4[6](https://www.nature.com/articles/srep02684). The ideal \( N^2 \) Fisher information also assumes perfect preparation, which the measured form \( F_Q = N^2 C_P^2/(P_{N,0}+P_{0,N}) \) makes explicit through the contrast and population factors[4](https://ar5iv.labs.arxiv.org/html/1611.08700).
## Generation schemes
**Hong–Ou–Mandel interference.** The two-photon NOON state can be generated with a Hong–Ou–Mandel interferometer and two single-photon inputs: the two indistinguishable photons bunch, exiting the beam splitter together in one arm or the other. Beyond two photons, though, a beam splitter with a 50/50 ratio alone cannot generate path-entangled states.[1](https://ar5iv.labs.arxiv.org/html/quant-ph/0202133) Perfect HOM interference requires the photons to be indistinguishable in spatial, temporal, spectral, and polarization modes, and any partial distinguishability limits preparation of the desired optimized NOON-type state[10](https://arxiv.org/html/2402.19079).
**Mixing quantum and classical light.** In 2010, high-NOON states with N = 5 photons were generated by multiphoton interference of quantum down-converted light with a classical coherent state, in an approach described as inherently scalable, producing super-resolving phase measurements with up to five entangled photons[7](https://www.science.org/doi/10.1126/science.1188172).
**Heralded down-conversion.** Heralded photon-number states are prepared by measuring one beam of parametric down-conversion with a photon-number-resolving detector, which projects the other beam onto a known photon-number state[5](https://www.nature.com/articles/s41534-020-00320-y). Using high-gain parametric down-conversion sources and photon-number-resolving detectors, heralded entangled probes of sizes up to N = 8, corresponding to measured 16-photon coincidences, have been prepared for interferometry[5](https://www.nature.com/articles/s41534-020-00320-y). Earlier post-selected work had reported \( |3,0\rangle + |0,3\rangle \) states by Mitchell et al. and \( |4,0\rangle + |0,4\rangle \) states by Walther et al.[11](https://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/boyd_spie_05.pdf).
**Three-mode and matter-wave states.** A 2025 photonic experiment prepared the input state \( |1,1,1,0\rangle \) by pumping a periodically poled potassium titanyl phosphate (ppKTP) crystal with a 775 nm pulsed picosecond laser via type-II spontaneous parametric down-conversion, then heralded a three-mode NOON state[12](https://arxiv.org/html/2512.08458). Beyond photons, a deterministic method has generated NOON states up to N = 9 phonons in two radial modes of a single trapped 171Yb+ ion, with the measured phase-uncertainty bound from N = 2 to N = 9 violating the classical bound and reaching the Heisenberg limit[4](https://ar5iv.labs.arxiv.org/html/1611.08700).
## Decoherence and loss fragility
The NOON state's weakness is loss. Its Heisenberg-scaling advantage is lost in the presence of loss, whether from absorption inside the interferometer, such as absorption in a sample, or external losses in state preparation and detection[5](https://www.nature.com/articles/s41534-020-00320-y). The mechanism is exponential, sometimes called a *super-Beer's Law*: with a loss parameter corresponding to 3 dB, that is 50% loss in one arm, the visibility of an N = 3 NOON state drops by a factor of \( 2^3 = 8 \), while a classical coherent state drops only by a factor of 2[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf). NOON states tend to decohere easily when photons are lost from the system[3](https://repository.lsu.edu/cgi/viewcontent.cgi?article=4203&context=physics_astronomy_pubs).
The practical consequence is a threshold. For loss-free interferometry, with efficiency \( \eta_s = 1 \), the optimal probe state maximizing quantum Fisher information is the NOON state; for efficiencies below about 90%, generalized Holland–Burnett probes surpass the NOON state in Fisher information[5](https://www.nature.com/articles/s41534-020-00320-y). Under loss, even optimal states can surpass the standard quantum limit only by a numerical factor, so \( \Delta\varphi = O(N^{-1/2}) \) is the best scaling possible[13](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf). Probes designed for lossy conditions retain an advantage: the NIST heralded-photon-number experiment achieved quantum-enhanced phase sensitivity even in the presence of significant optical loss, unlike NOON-state probes[5](https://www.nature.com/articles/s41534-020-00320-y).
## Comparison with other quantum-enhanced resources
NOON states give Heisenberg-limited phase uncertainty \( O(N^{-1}) \), much more advantageous than the best phase sensitivity reachable with classical systems, \( 1/\sqrt{N} \), but they are extremely fragile and can perform worse than classical states under loss[13](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf). Twin-Fock and two-mode squeezed states also achieve \( O(N^{-1}) \) scaling without loss and are more robust, though their sensitivity degrades to \( O(N^{-1/2}) \) under loss; with a twin-[Fock state](https://www.edgepedia.org/fock-state) and 95% quantum efficiency an 8 dB improvement over the standard quantum limit is achievable, against a theoretical lossy bound of 13 dB, while a two-mode squeezed state yields 4.4 dB[13](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf).
Other routes avoid deliberate NOON preparation altogether. A passive linear-optical interferometer fed with only uncorrelated single-photon inputs, coupled with simple single-mode disjoint photodetection, can significantly beat the shot-noise limit by exploiting spontaneously generated entanglement[14](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.170802). The overall picture is that NOON states are optimal in the lossless, low-flux regime, while twin-Fock and squeezed probes are often more practical choices when loss is significant[5](https://www.nature.com/articles/s41534-020-00320-y) • [13](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf).
## Applications
**Quantum lithography.** Boto and coauthors showed in 2000 that an N-photon entangled state can write features of minimum size \( \lambda/(2N) \) in an N-photon-absorbing substrate, allowing a factor of \( N^2 \) more elements on a semiconductor chip, where classical optical lithography is diffraction limited to features of \( \lambda/2 \) or greater[8](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.2733). The same \( \cos(N\varphi) \) fringe compression that gives super-resolution in metrology gives the sub-diffraction exposure pattern[2](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf).
Implementation has lagged the proposal. As of 2005 there had been no direct laboratory demonstrations of the quantum lithography process; the two obstacles were intense NOON-state sources and N-photon-sensitive recording media, although the polymer PMMA was shown to serve as an N = 4 resist with a demonstrated spatial resolution of 425 nm[11](https://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/boyd_spie_05.pdf). In sensing and phase estimation, NOON states serve chiefly as the benchmark against which loss-tolerant probes are measured, since they set the lossless Fisher-information optimum[5](https://www.nature.com/articles/s41534-020-00320-y).
## Recent developments
Post-2023 work has concentrated on heralded and multi-mode generation rather than pushing the photonic N record. In 2025 a heralded three-mode two-photon NOON state was generated with fidelity \( 0.823 \pm 0.018 \) with respect to the ideal state and certified as genuine multipartite entanglement, with a nominal success probability of 0.25 (measured \( 0.237 \pm 0.009 \)); before this, a three-mode NOON state had only been reported with full post-selection[12](https://arxiv.org/html/2512.08458). A 2024 experimental study quantified how mode imperfections limit HOM-based preparation, showing that photons must be indistinguishable in all degrees of freedom for the scheme to work[10](https://arxiv.org/html/2402.19079). A 2024 theoretical analysis consolidated the practical hierarchy, establishing twin-Fock and two-mode squeezed states as the loss-robust alternatives to NOON states[13](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf). The largest photonic-scale entangled probes remain the heralded N = 8 states of the 2020 NIST-affiliated experiment, which are generalized photon-number probes rather than strict NOON states[5](https://www.nature.com/articles/s41534-020-00320-y).
## References
1. [A Quantum Rosetta Stone for Interferometry, Kok et al. (2002)](https://ar5iv.labs.arxiv.org/html/quant-ph/0202133)
2. [Quantum Optical Metrology: The Lowdown on High-N00N States, J. P. Dowling (2008)](http://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/Dowling08.pdf)
3. [Strategies for choosing path-entangled number states for optimal robust quantum-optical metrology in the presence of loss, LSU repository](https://repository.lsu.edu/cgi/viewcontent.cgi?article=4203&context=physics_astronomy_pubs)
4. [Experimental Preparation of High NOON States for Phonons (2016)](https://ar5iv.labs.arxiv.org/html/1611.08700)
5. [Quantum-enhanced interferometry with large heralded photon-number states, npj Quantum Information (2020)](https://www.nature.com/articles/s41534-020-00320-y)
6. [Optimal multi-photon phase sensing with a single interference fringe, Scientific Reports (2013)](https://www.nature.com/articles/srep02684)
7. [High-NOON States by Mixing Quantum and Classical Light, Science (2010)](https://www.science.org/doi/10.1126/science.1188172)
8. [Quantum Interferometric Optical Lithography, Boto et al., Phys. Rev. Lett. 85, 2733 (2000)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.85.2733)
9. [Quantum metrology from a quantum information science perspective, J. Phys. A 47, 424006 (2014)](https://iopscience.iop.org/article/10.1088/1751-8113/47/42/424006)
10. [Experimental investigation of a multi-photon Heisenberg-limited interferometric scheme: the effect of imperfections (2024)](https://arxiv.org/html/2402.19079)
11. [Progress in Quantum Lithography, SPIE (2005)](https://www.hajim.rochester.edu/optics/sites/boyd/archive/Quantum-Imaging/Assets/papers/boyd_spie_05.pdf)
12. [Heralded generation of a three-mode NOON state (2025)](https://arxiv.org/html/2512.08458)
13. [Sub-shot-noise interferometry with two-mode quantum states, Physical Review A (2024)](https://www.lcf.institutoptique.fr/sites/default/files/2024_PRA_Subshot_Noise_Interferometry_0.pdf)
14. [Linear Optical Quantum Metrology with Single Photons, Phys. Rev. Lett. 114, 170802 (2015)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.170802)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Quantum optics and photonics › Quantum imaging and quantum sensing › Quantum-enhanced interferometry*
*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*
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