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Hierarchies of quantum non-Gaussian coherences for bosonic systems: A theoretical study

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Pith's one-line read A single Fock-basis coherence element can certify quantum non-Gaussian coherence, provided it beats hierarchy-specific thresholds defined by what Gaussian squeezing and displacement can generate from restricted core states.

desk verdict Genuine extension of the non-Gaussianity hierarchy program to off-diagonal Fock coherences, but the certification thresholds rest on an unproved conjecture and random sampling, so the numbers are provisional. read the letter →

arxiv 2501.14033 v2 pith:JK36MYFC submitted 2025-01-23 quant-ph

Lukáš Lachman , Beate E. Asenbeck , Ambroise Boyer , Priyanka Giri , Alban Urvoy , Julien Laurat , Radim Filip This is my paper · ORCID
classification quant-ph
keywords quantumnon-GaussiancoherenceFock-statebasisGaussianoperationshierarchybosonicsystemscertificationlossandthermalnoiserelativecriteria
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the quantum non-Gaussian nature of coherence in bosonic systems can be certified one Fock-basis element at a time. Instead of summing all off-diagonal matrix elements into a global coherence quantifier, it isolates a single coherence $C_{m,n}$, the magnitude of the $|m\rangle\langle n|$ density-matrix element, and compares it with the largest value that Gaussian squeezing and displacement can produce from a 'core' state of a given order. Beating the corresponding threshold proves that the state's coherence cannot be mimicked by Gaussian dynamics acting on low-order superpositions of Fock states, i.e. it is genuinely quantum non-Gaussian. The authors construct two hierarchies that order the same coherences differently—one by the highest Fock level $n$, one by the separation $l=n-m$—and show that the two classifications prioritize different states and tolerate different amounts of loss and thermal noise. This matters because single-coherence certification is directly accessible to interferometric measurements and can be tailored to application contexts such as sensing and bosonic error correction.

What carries the argument

The load-bearing object is the coherence measure $C_{m,n}(\rho)=\tfrac12\left[\max_\phi \mathrm{Tr}[\rho X_{m,n}(\phi)]-\min_\phi \mathrm{Tr}[\rho X_{m,n}(\phi)]\right]$ with $X_{m,n}(\phi)=e^{i\phi}|m\rangle\langle n|+e^{-i\phi}|n\rangle\langle m|$, which isolates $|\langle m|\rho|n\rangle|$. Around it the paper builds core Hilbert spaces $H_k$ of Fock states below a given order; free states are Gaussian images $S(\xi)D(\alpha)|\tilde\psi_k\rangle$ of core states. The thresholds $T^{N,k}_{m,n}$ and $T^{L,r}_{k,l}$ are suprema of $C_{m,n}$ over these free states, computed via an eigenvector method for optimizing over core states and polynomial root-finding for the displacement parameter. Relative criteria replace the single threshold with $\min_\lambda[F_{m,n}(\lambda)-\lambda P]$, where $F_{m,n}(\lambda)$ is the maximum of $C_{m,n}+\lambda P$ over free states and $P$ is a diagonal probability; this is the mechanism that relaxes experimental requirements.

What would settle it

Find any core state in the allowed Hilbert space and any squeezing and displacement parameters $\xi,\alpha$ such that $C_{m,n}(S(\xi)D(\alpha)|\tilde\psi\rangle)$ exceeds the quoted $T^{N,k}_{m,n}$ or $T^{L,r}_{k,l}$ for a small case such as $C_{0,4}$; a global numerical optimization producing such a counterexample would refute the claimed bounds.

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Extended reading notes

Core claim

The central claim is that individual Fock-basis coherence elements carry enough information to certify quantum non-Gaussian coherence in a hierarchically ordered way. For each target superposition $(|m\rangle+|n\rangle)/\sqrt{2}$, the paper defines thresholds $T^{N,k}_{m,n}$ and $T^{L,r}_{k,l}$ as the maximum of $C_{m,n}$ over states $S(\xi)D(\alpha)|\tilde\psi\rangle$ obtained by applying Gaussian squeezing and displacement to a core state of order $k$ or $r$—states whose coherence is restricted to lower Fock levels or shorter Fock separations. A measured coherence exceeding such a threshold proves the state is not a Gaussian-transformed core state of that order. The paper derives thresholds for representative cases such as $C_{0,4}$, $C_{3,4}$ and $C_{1,2}$, shows that the two hierarchies give different orderings of the same coherences, and introduces relative criteria that combine $C_{m,n}$ with diagonal probabilities to lower the experimental bar while preserving the hierarchy.

Load-bearing premise

The thresholds are computed by assuming that the maximum coherence occurs for a specific narrow family of core states and Gaussian parameters; if a state outside that family achieves more coherence, every threshold in the paper is too low.

Editorial extensions

If this is right

  • A measured coherence above $T^{N,k}_{m,n}$ certifies that the state is not reachable by squeezing and displacing any order-$k$ core state whose coherence is confined to Fock levels below $n$.
  • The two hierarchies classify the same coherence differently: for $C_{3,4}$ the $N$-hierarchy supports thresholds up to third order while the $L$-hierarchy admits only first order, whereas for $C_{0,4}$ the $L$-hierarchy's second-order threshold is stricter than the $N$-hierarchy's.
  • Adding diagonal probabilities such as $P_n$ or the multi-photon error probability $P_{e,n}$ lowers the coherence value needed for certification while keeping the hierarchical ordering intact.
  • Loss and thermal depths quantify how much imperfection each threshold tolerates; for $C_{3,4}$ the highest $N$-hierarchy threshold permits about seven times smaller loss and thermal noise than the highest $L$-hierarchy threshold.
  • The same construction extends to unbalanced superpositions, multi-state coherences, and multi-mode coherences, which the authors link to dual-rail binomial codes and sensing protocols.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the thresholds rest on an unproved conjecture about where the global maximum occurs, the numerical values should be treated as certified bounds only under that conjecture; a broader optimization could raise them and would make current certifications stricter rather than looser.
  • The context-dependent ordering suggests an application-driven choice: experiments aimed at bosonic error correction or phase sensing should use the $L$-hierarchy, while experiments targeting macroscopic superpositions should use the $N$-hierarchy, and the two should not be compared as if they ranked the same resource.
  • The relative criterion involving $P_{e,n}$ tolerates arbitrarily large loss in the noiseless limit, which implies that photon-number-resolving detection of high-Fock errors can substitute for high transmission in practical certification.
  • A direct falsification test would be to run a global numerical optimizer over core states and Gaussian parameters for a small case such as $C_{0,4}$ and check whether any free state exceeds the quoted $T^{N,k}_{m,n}$ or $T^{L,r}_{k,l}$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces a framework for certifying quantum non-Gaussian (QNG) coherence in the Fock-state basis by comparing individual off-diagonal coherence elements C_{m,n}(ρ) against thresholds derived from Gaussian dynamics acting on restricted 'core states.' Two context-dependent hierarchies are constructed: an N-hierarchy ordered by the larger Fock index n, and an L-hierarchy ordered by the Fock-number difference l = n−m. The paper further develops 'relative criteria' that combine C_{m,n} with diagonal probabilities to relax experimental requirements, and it analyzes the robustness of these criteria under loss and thermal noise using a first-order perturbative model. The central claim is that exceeding a hierarchy-dependent threshold certifies that the tested coherence cannot be produced by any free state of the corresponding order, yielding a structured, application-oriented classification of QNG coherence.

Significance. If the numerical thresholds are valid, the framework is a useful and operational complement to global coherence measures and to stellar-rank classification: it isolates individual off-diagonal elements accessible via interferometric measurements, provides two distinct orderings with clear operational interpretations (macroscopic Fock number versus Fock separation for sensing and error correction), and offers relaxed relative criteria that are experimentally attractive. The paper is clearly written and builds on established techniques such as the Fiurášek eigenvector method. Its main quantitative predictions, however, depend on global maxima of nonconvex functions that are not rigorously established; this makes the thresholds, and every hierarchy comparison and depth estimate built on them, conditional on an unverified conjecture.

major comments (3)
  1. [Sec. IV B, Eqs. (6)–(7); Appendix A] The thresholds T^{N,k}_{m,n} and T^{L,r}_{k,l} are defined as global maxima over all Gaussian evolutions S(ξ)D(α) of all core states in a specified family. The paper explicitly states in Sec. IV B: 'we guess for which parameters the maximum occurs and, then, we use random sampling of the parameters ξ, α, φ, l and θ to verify the estimated maximum,' and Appendix A states that the maximum over ξ and α is obtained 'by random sampling and taking the maximal attempt.' This does not establish a global maximum. Because each threshold is used as an upper bound on all free states, any missed global maximum makes the threshold too low and would cause a Gaussian-generated free state to be falsely certified as having QNG coherence. This is not a peripheral numerical issue: the hierarchy comparisons, the relative criteria in Eqs. (10)–(16), and the depth analyses all inherit these threshold values. The authors should either provide a rigorous global-optimization certificate (e.g., an analytic bound, interval/branch-and-bound method, or a convex relaxation) for each reported threshold, or clearly state the maximizing-family conjecture as such and provide a systematic, dense numerical search with convergence evidence. As written, the central certification claim is conditional on an unverified maximization.
  2. [Sec. III A, Eq. (4); Appendix A, Fig. 6] The claim that thresholds T^{(N)}_{m,n} tend to unity for large N is supported only by selected numerical examples, primarily for m=0 and m=n−1, with N up to 20. This convergence is used to justify the argument that employing thresholds with N≫1 imposes the unattainable condition C_{m,n}=1, which in turn motivates the finite-order hierarchies. Since no proof of generic convergence is given, and the numerical evidence covers a restricted set of (m,n) pairs, the assertion 'we numerically verified this tendency' does not establish the general behavior. If for some (m,n) the limit is below 1, the statement that 'employing the thresholds T^{(N)}_{m,n} with N≫1 for benchmarking implies the condition C_{m,n}=1' would be false, affecting the conceptual foundation of the hierarchy construction. The authors should either provide a proof or clearly state this as a conjecture and delineate the numerical evidence for it.
  3. [Sec. IV B–C, Eqs. (10)–(16)] The relative criteria are derived by computing F_{m,n}(λ) = max_{ξ,α,|ψ̃_k⟩} [C_{m,n} + λ P] and then forming the Legendre-type expression min_λ [F_{m,n}(λ) − λ P]. The validity of the resulting witness requires F_{m,n}(λ) to be the exact global maximum over the entire free-state family. Because the maximization is the same unverified one criticized above, the relative criteria inherit the same false-positive risk. Additionally, the final minimization over λ (and λ1,λ2 in Sec. IV C) is performed numerically without reporting error bars or convergence tests, so the relaxed thresholds in Figs. 4 and 5 are as conditional as the absolute thresholds. The authors should clarify how the numerical minimization is certified and, ideally, provide a rigorous bounding argument for F_{m,n}(λ) before presenting the relative criteria as definitive witnesses.
minor comments (4)
  1. [Eq. (8) and Figs. 3, 4] The model in Eq. (8) is stated to be valid only in the limits 1−η ≪ 1 and n̄ ≪ 1, but the plots in Figs. 3(e)–(h) and 4(d) display parameter ranges extending to 1−η = 0.5. Please clarify whether the reported depth values remain within the model's validity regime or whether some curves are extrapolations beyond it.
  2. [Sec. III D and Fig. 3 caption] There are several apparent typos: in Sec. III D, 'T^{L,1}_{3,1}=0.8' should presumably read 'T^{L,1}_{3,4}=0.8'; the Fig. 3 caption contains 'repectively' and 'T^{l,k}_{3,4}' which should likely be 'T^{L,k}_{3,4}'; and Appendix A's title has 'Hibert space' instead of 'Hilbert space.'
  3. [Sec. II B] The sentence 'As basis states, no Fock state can obey this criterion despite their non-Gaussian nature [6]' is confusing: Fock states have zero coherence C_{m,n}=0, so they trivially fail any positive threshold. Please rephrase to clarify that the statement concerns the inadequacy of using Fock states alone as reference states.
  4. [General notation] The notation for thresholds (t̃_{m,n}, t_{m,n}, T̃_{m,n}, T^{(N)}_{m,n}, T^{N,k}_{m,n}, T^{L,r}_{k,l}) is introduced quickly and used in multiple places. A table summarizing each quantity, its defining free-state family, and its role in the hierarchies would significantly improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity found: the thresholds are defined as maxima over explicitly stated free-state families, and the unproved global-maximum conjecture is a rigor gap, not a circular reduction.

full rationale

We find no circular reduction in the derivation chain. The central thresholds in Eqs. (4)-(7) are defined as genuine maximizations over explicitly enumerated free-state families, namely Gaussian dynamics acting on core states, and the reported threshold values are numerical evaluations of those maxima rather than fitted parameters or renamed inputs. The relative criteria in Eqs. (9)-(16) are constructed via a convex Legendre-type optimization over the same free-state sets, so they inherit the free-state definition without importing the target conclusion. The optimization procedure of Ref. [24] is external to the present authors and is used as a computational tool; the authors' own prior hierarchy, Ref. [6], serves as conceptual motivation and not as a premise that forces the numerical thresholds. The paper's main weakness, which the authors explicitly disclose in Section IV B, is that they conjecture where the global maximum occurs, restrict the search to phi = 0, theta in {0, pi}, and two specific core-state families, and verify only by random sampling. Similarly, Appendix A obtains maxima over xi and alpha by random sampling and taking the maximal attempt. If the true global maxima lie outside the searched families, the thresholds would be too low and free states could be falsely certified. This is a numerical validation and rigor gap, not a circularity, because the definitions do not assume the target answer. The score of 2 reflects the presence of several self-citations that are not load-bearing; the central derivation remains self-contained.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No physical constants are fit to experimental data. The quantitative content is carried by optimization variables and by the choice of free-state families. The ledger records two unproved conjectures, a small-perturbation noise model, and the truncation size used in convergence scans; these are the main assumptions the thresholds rely on.

free parameters (2)
  • λ, λ1, λ2 in relative criteria = scanned over real values; not fitted to data
    Introduced in Eqs. (9) and (14) as weights in convex combinations of coherence and Fock-state probabilities. They are minimization variables in the Legendre-type construction of the criteria, so they are free parameters of the derivation rather than physical constants.
  • Hilbert-space truncation N for convergence tests = N = 10 and N = 20 in Appendix A
    Chosen by hand to demonstrate the tendency T_{m,n}^{(N)} to approach 1; the convergence is not proven for all N, and it motivates the restricted core-state definition that prevents trivial thresholds.
assumptions (4)
  • domain assumption Gaussian operations S(ξ)D(α) and Fock states are treated as free states and free operations in the resource hierarchy.
    Section II B defines thresholds by maximizing over these Gaussian evolutions; this choice determines what counts as quantum non-Gaussian coherence.
  • ad hoc to paper The maximizing states for relative criteria lie in the conjectured family: φ=0, θ∈{0,π}, and core states restricted to two specific superpositions.
    Section IV B states this conjecture after Eq. (12); the derived relative thresholds are valid only if it holds.
  • ad hoc to paper The thresholds T_{m,n}^{(N)} converge to 1 for large N, inferred from selected numerical examples.
    Appendix A demonstrates decaying 1 - T^{(N)} for N=10 to 20 for selected (m,n); this conjecture motivates the restricted core-state definition that prevents trivial C_{m,n}=1 thresholds.
  • domain assumption The noise model in Eq. (8) accurately describes loss and thermalization even when depth analysis considers larger deviations than the stated validity 1-η≪1 and nbar≪1.
    Section III E uses this model to compute loss depth and thermal depth; extrapolation beyond the stated validity is not separately justified.

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Pith. "Pith review of Hierarchies of quantum non-Gaussian coherences for bosonic systems: A theoretical study." pith.science (2026). https://pith.science/paper/JK36MYFC

@misc{pith2026250114033,
  author       = {Pith},
  title        = {Pith review of: Hierarchies of quantum non-Gaussian coherences for bosonic systems: A theoretical study},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JK36MYFC}},
  note         = {Machine review of arXiv:2501.14033}
}
read the original abstract

Quantum coherence in bosonic systems is a fundamental resource for quantum technology applications. In this work, we introduce a framework for analyzing coherence in the Fock-state basis, utilizing context-dependent certification to reveal the quantum non-Gaussian nature of the tested coherence. Rather than relying on global coherence measures, our approach targets specific aspects of coherence, enabling a tailored hierarchical classification. We derive and compare two distinct hierarchies, each representing a different context for coherence in bosonic systems. Motivated by current advancements in optical quantum state engineering, we assess the feasibility and depth of these hierarchies under conditions of loss and thermal noise. The methodology introduced here is versatile and can be extended to multi-state and multi-mode coherences, making it adaptable to a wide range of experimental scenarios and platforms.

Figures

Figures reproduced from arXiv: 2501.14033 by the authors.

Figure 1
Figure 1. FIG. 1. Introduction of non-Gaussian coherence. (a) For [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Context-dependent coherence framework and hierarchical certification (a) Motivation for a context-dependent class of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Absolute non-Gaussian coherence and criteria depth. (a)-(b) Absolute criteria in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Relative criteria. (a) The absolute criterion for the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Analysis of the coherence [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of the thresholds [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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    R.-Z. Liu, Y.-K. Qiao, L. Lachman, Z.-X. Ge, T.-H. Chung, J.-Y. Zhao, H. Li, L. You, R. Filip, and Y. Huo, Experimental Quantum Non-Gaussian Coincidences of Entangled Photons, Phys. Rev. Lett.132, 083601 (2024)

    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.