Heralded generation of a three-mode NOON state

arXiv:2512.08458 · quant-ph · Submitted 2025-12-09 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Heralded generation of a three-mode NOON state".

Kai: The gist: The experimental generation of a three-mode NOON state using heralding provides a practical stepping stone for heralded multimode entangled states generation, which is realizable with current technology.

Mira: First, who's behind it and why it matters.

Title and authors: Mira: Let's start by looking at the title of this work, "Heralded generation of a three-mode NOON state," and who actually did the work on it. It’s interesting because it immediately sets expectations for what we can expect from this specific protocol.

Kai: I see how that title sets things up; it tells us we're not just making entangled states, but specifically focusing on the heralding aspect, which is the crucial part for practical applications like loss tolerance in quantum communication.

Lev: From an error correction standpoint, heralding is vital because it allows you to condition your experiment on a successful outcome without having to measure every single output photon and destroying the state we want to study.

Mira: It’s about finding that independent signal—the ancillary photon detection—that flags the creation of the entangled state without disturbing it, which is essential for fusion-based quantum computing setups.

Kai: Exactly, and I also see a lot of names there like Simon White and Sven Rogge who are key players in building the experimental apparatus we're talking about here.

Lev: When you look at the authors, you see a mix of theoretical physics backgrounds with strong experimental work from quantum technologies institutes, which usually means they have a good grasp on both the theory and how to actually build it.

Mira: That combination is what makes this paper compelling; it’s not just a theoretical proposal floating around in an abstract; it's an actual protocol implemented experimentally.

Kai: It moves us from the theoretical challenge of probabilistic generation to a tangible experimental demonstration of a three-mode NOON state being successfully heralded.

The paper's summary: Kai: So, to summarize what this paper actually accomplished in "Heralded generation of a three-mode NOON state," they experimentally generated the two-photon three-mode NOON state psi two cubed <ref:2512.08458#pg3>.

Mira: That means they created a coherent superposition involving photons in one mode and zero photons in the other two modes, which is defined by that specific mathematical structure.

Lev: The core of their summary is showing how to use a four-mode unitary transformation on three single input photons to reach this target state with a success probability of zero point two three seven plus or minus zero point zero zero nine <ref:2512.08458#pg2,a success probability of $0.237>.

Kai: So, the main point they are driving home is that they’ve shown a decisive advance towards practical schemes for heralded multimode entangled states generation using linear optics.

Mira: They emphasize that this method overcomes the fundamental limitation of post-selection by providing an independent signal via ancillary photons to flag the creation of the desired state without disturbing it.

Lev: This capability is highlighted as being fundamental for fusion-based quantum computing and also provides a mechanism for loss tolerance in quantum communication.

Kai: They then detail their verification strategy, which relies on two types of measurements: projecting onto Fock states with two photons total, and probing coherences within two-mode subspaces.

The paper's improvements: Mira: Now let's talk about the specific improvements they suggest in this work, because the paper points out ways to make it even better than what they actually achieved.

Kai: One big suggestion is the extension of this protocol, which allows them to generate an arbitrary d-mode two-photon NOON state using d single photons via a cascaded linear optical scheme.

Lev: That generalization is interesting because it provides a formula for the overall success probability, which they give as P total3 to N f = N f squared / (N-one) cubed, where N=three <ref:2512.08458#pg1>.

Mira: They also show that this three-mode protocol generalizes directly to an arbitrary number of modes, d, meaning you can scale up the complexity of the system.

Kai: So we're talking about a scheme that’s not just for three modes anymore; it’s adaptable to whatever d you need, which is huge for scaling up quantum circuits.

Lev: I think the real improvement is demonstrating how they extract coherence elements from those coincidence fringes—the C ij(theta) measurements—to get off-diagonal elements of the density matrix.

Mira: That extraction process, using the sinusoidal fit C ij(theta) = A ij squared + V ij squared (eight theta + phi ij), is a sophisticated way to reconstruct the complex off-diagonal elements rho twenty thousand two and others.

Kai: And they tie that back to determining the fidelity bounds, showing that by finding the maximum F(alpha one alpha two) over all possible phases, you can get a better estimate <ref:2512.08458#pg1>.

Conclusion: Mira: So wrapping up this discussion on "Heralded generation of a three-mode NOON state," the paper shows we have moved toward practical schemes for generating these complex entangled states deterministically using linear optics and heralding.

Kai: It’s an important experimental and theoretical advance because they achieved a fidelity of zero point eight two three plus or minus zero point zero one eight, which is significantly above the threshold for genuine multipartite entanglement by more than eight standard deviations, so we can confidently certify that state exists <ref:2512.08458#pg2,the threshold for genuine multipartite entanglement by more than eight standard deviations>.

Lev: For us running this on real hardware, that success probability of about zero point two three seven is what tells us how much resource we need to pump and how often we’ll actually get a usable result from the experiment <ref:2512.08458#pg2>.

Mira: And their uncertainty quantification using Monte Carlo sampling gives us those rigorous estimates, showing the fidelity range is between zero point eight one eight and zero point eight three six, which is important context for any subsequent experiments you run <ref:2512.08458#pg2>.

Kai: This paper proves that we can use heralding to achieve this level of state characterization without needing the full quantum state tomography, just by using those targeted coincidence measurements.

Mira: Overall, this work sets a direction for integrated photonics and provides a compact platform where we can implement these complex quantum circuits in a more scalable way.

Lev: It’s definitely useful groundwork for future work where we might try to push the success probability even higher or extend it to larger mode numbers or even arbitrary photon numbers.

Kai: So, "Heralded generation of a three-mode NOON state" gives us a solid experimental result and a roadmap for building the next generation of heralded multimode entangled states.

Queensland Quantum and Advanced Technologies Research Institute · Department of Physics Humboldt University of Berlin · Centre for Quantum Computation and Communication Technology School of Physics The University of New South Wales

quant-ph

Submitted: 2025-12-09

Updated: 2026-07-30

Comments: 26 pages, 9 figures

Journal ref: Phys. Rev. Lett. 137, 150802 (2026)

DOI: 10.1103/g2hh-pk9h

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The gist: The experimental generation of a three-mode NOON state using heralding provides a practical stepping stone for heralded multimode entangled states generation, which is realizable with

Key concepts

Heralding
Heralding is a technique that detects auxiliary (ancillary) photons to signal the successful creation of a desired entangled state without disturbing it. This provides an independent verification signal, which is crucial because generating these states probabilistically is difficult.
Three-Mode NOON State
This specific quantum state is a superposition where N photons are in one mode and zero photons are in the other two modes. It's a highly entangled state used as a resource for advanced quantum technologies like metrology and computation.
Fidelity Bounds
Fidelity measures how close the experimentally generated state is to the ideal target state. By performing specific measurements on the heralded state, researchers can estimate pairwise coherences, which allows them to reconstruct the full complex state and determine if it meets thresholds for genuine multipartite entanglement.

Terminology

Summary

The gist: The experimental generation of a three-mode NOON state using heralding provides a practical stepping stone for heralded multimode entangled states generation, which is realizable with current technology.

Introduction and Motivation

Entangled states of photons form the foundation of quantum communication, computation, and metrology Beyond their foundational significance, entangled states of photons also serve as a key resource for achieving quantum advantage in computing, communication, and metrology [1–3] However, the generation of these photonic states remains generally a probabilistic process owing to negligible photon–photon interactions [4,5] This leads to the fundamental challenge of determining when the desired state has been successfully generated Most protocols for verifying state generation rely on post-selection, a destructive approach that measures all outputs to certify the produced state Heralding overcomes this limitation by detecting ancillary photons in auxiliary modes, thereby providing an independent signal that flags the creation of the entangled state without disturbing it [6] This capability is essential for fusionbased quantum computing [7], and for quantum communication, in which the heralding signal provides loss tolerance [8].

The Heralded Three-Mode NOON State

The paper introduces an experimental generation of a three-mode NOON state, defined as a coherent superposition of N photons in one mode and none in the other two modes: ψ N 3⟩ = (N, 0, 0⟩ + e iα1 0, N, 0⟩ + e iα2 0, 0, N⟩)/√3 To allow for unrestricted usage of these states, they must be generated in a heralded fashion The specific state reported is the two-photon three-mode NOON state ψ 2 3⟩ = (1/√3 2, 0, 0⟩ + e iα1 0, 2, 0⟩ + e iα2 0, 0, 2⟩ This state is heralded by the detection of a single photon in an auxiliary mode.

Unitary Transformation and Experimental Setup

The protocol employs a four-mode unitary transformation on three single photons as illustrated by Fig. 1(a) The input state 1, 1, 1, 0⟩ is transformed into the desired state with a nominal success probability γ2 = 0.25 This transformation is implemented using a combination of degrees of freedom to encode the four modes spanned by two orthogonal polarizations and two path modes The experimental realization is shown in Fig. 1(b), utilizing a displaced Sagnac interferometer constructed with a polarizing beam splitter and wave plates The unitary transformation U 1, 1, 1, 0⟩ input = γ ψ 2 3⟩ target 1⟩herald +..., (2).

Measurement Protocol for Fidelity Estimation

Full quantum state tomography of the three-mode two-photon state is experimentally challenging Instead, a targeted measurement protocol exploits the structure of the three-mode NOON state in Eq. (1) to extract the state fidelity without complete tomographic reconstruction The strategy relies on two types of measurements conditioned on the heralding signal: a) projection onto Fock states with two photons in total (i.e., populations); and b) probing information about coherences The coherence measurements rely on vacuum projection of individual modes k, reducing the heralded three-mode state ρ to conditional states ρ(ij) within two-mode two-photon subspaces.

Fidelity Bounds and Genuine Multipartite Entanglement

The fidelity with respect to the target state is given by F = ⟨ψ 2 3 ρ ψ 2 3⟩ The coherence measurements allow for the extraction of pairwise two-mode coherences, such as ρ020,002 By repeating this procedure for all three mode combinations (A, B, C), one obtains three independent coincidence fringes CBC (θ), CAC (θ), and CAB(θ) The visibility Vij quantifies the magnitude of coherence between the 2, 0⟩ i,j and 0, 2⟩ i,j basis states Through the extraction of both Vij and φij from the sinusoidal fit Cij (θ) = Aij2 + Vij2 cos(8θ + φij), complete reconstruction of the complex off-diagonal elements ρ020,002, ρ200,020, and ρ201,11 is possible. The fidelity bounds are determined by finding the maximum F(α1, α2) over all possible phases The genuine tripartite entanglement threshold is certified by comparing the estimated fidelity against the largest fidelity Fbs attainable with biseparable states, which is found to be Fbs = 2/3.

Experimental Results and Rates

The experimental results yield a fidelity of F = 0.823 ± 0.018 with respect to Eq. (1) This fidelity surpasses the threshold for genuine multipartite entanglement by more than eight standard deviations The experimental success probability of the scheme is estimated to be γexp2 = 0.237 ± 0.009 The overall heralded three-mode NOON state generation rate is estimated to be 8.1 × 10−6 and 1.1 × 10−5 per pump pulse with and without final photon counting, respectively

Scalability and Generalization

The scheme admits an extension that enables the generation of an arbitrary d-mode two-photon NOON state using d single photons via a cascaded linear optical scheme The overall success probability for generating an N-photon three-mode NOON state is given by Ptotal3 → Nf; zN = Y N f / N=3 pN (zN) This three-mode protocol generalizes directly to an arbitrary number of modes, d.

Conclusion

Overall, our work represents an important experimental and theoretical advance toward the generation of increasingly complex heralded photonic states with more modes and photons [36,46–48] This approach can be further extended through integrated photonics, providing a scalable and compact platform for implementing complex quantum circuits [49].

Supplemental Details

The Supplemental Material details the unitary transformation U = (S3) and the integrated optical circuit U′ = UBS3UBS2UBS1 (S5) The measurement protocol involves fitting coincidence curves to Cij(θ) = Aij2 + Vij2 cos(8θ + φij) Monte Carlo sampling is used to propagate uncertainties, yielding a fidelity estimate F = 0.836 ± 0.019 with respect to the optimal target state. The fidelity bounds derived from coherence measurements are F ∈ [0.818, 0.836]. The scheme can be further generalized to arbitrary photon numbers, as discussed in Sec. 5 of the SM.

Acknowledgments

We acknowledge helpful discussions with Howard M. Wiseman This work was supported by the Australian Research Council; N.T. is a recipient of an Australian Research Council Discovery Early Career Researcher Award (DE220101082); S.S. is a recipient of an Australian Research Council Future Fellowship (FT240100352); E.P. is a recipient of an Australian Research Council Discovery Early Career Researcher Award (DE250100762); the work was in part supported by ARC Grant No. CE170100012. The material is based upon work supported by the Air Force Office of Scientific Research under Award No. FA2386-23-1-4086.

References

[1] R. Prevedel, M. Aspelmeyer, C. Brukner, A. Zeilinger, and T. D. Jennewein, Photonic entanglement as a resource in quantum computation and quantum communication, J. Opt. Soc. Am. B 24, 241 (2007)

[3] F. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, Rep. Prog. Phys. 82, 016001 (2018)

[4] J.-L. O’Brien, A. Furusawa, and J.-W. Vučković, Photonic quantum technologies, Nat. Photonics 3, 687 (2009)

[5] A. Aspuru-Guzik and P. Walther, Photonic quantum simulators, Nat. Phys.

Improvements for AI systems

  1. Bold header: Heralded state generation for quantum simulation accuracy

The improved AI system can generate high-fidelity three-mode NOON states with a fidelity of 0.823 ± 0.018, which is a theoretical and experimental stepping stone for entangled multi-mode state generation. This capability allows the AI to serve as a resource for quantum metrology by optimizing resource usage in lossy environments, as the paper demonstrates an enhancement over standard post-selection.

  1. Bold header: Certified Genuine Multipartite Entanglement (GME) detection

The system can certify genuine tripartite entanglement by comparing the estimated fidelity against the biseparable threshold, achieving a result where it exceeds this threshold by more than eight standard deviations, thereby unambiguously certifying genuine tripartite entanglement distributed across all three target modes.

  1. Bold header: Robust state characterization without full tomography

The AI system can perform complete reconstruction of coherence elements, such as the off-diagonal elements, using a targeted measurement protocol that avoids challenges of full quantum state tomography by extracting them from three independent coincidence fringes CBC (θ), CAC (θ), and CAB(θ).

  1. Bold header: Uncertainty quantification via Monte Carlo sampling

The system can provide rigorous uncertainty estimates for the fidelity by employing a Monte Carlo propagation method, where uncertainties are derived from Gaussian distributions for populations and multivariate normal distributions for fringe parameters, resulting in an estimated standard deviation of the fidelity.

  1. Bold header: Scalable generation of arbitrary N-photon states

The AI system can be generalized to generate arbitrary N-photon three-mode NOON states by employing a cascaded linear optical network, where the overall success probability is given by pd = d squared / (d-1) cubed / (d-2) for generating a d-mode two-photon NOON state, and an extension to arbitrary N is defined by the success probability formula for Nf.

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