Detecting quantum non-Gaussianity with a single quadrature

Clara Wassner; Francesco Arzani; Jack Davis; Sacha Cerf; Ulysse Chabaud

REVIEW 3 major objections 5 minor 1 cited by

Detecting quantum non-Gaussianity with a single quadrature

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A single quadrature measurement can certify quantum non-Gaussianity at any stellar rank, without tomographic reconstruction.

desk verdict Strong idea and mostly sound existence proofs, but the quantitative state-independent bounds have an off-by-one energy error and an overstrong uniformity claim that need revision. read the letter →

arxiv 2507.23005 v3 pith:JEBDZKST submitted 2025-07-30 quant-ph

Clara Wassner , Jack Davis , Sacha Cerf , Ulysse Chabaud , Francesco Arzani This is my paper · ORCID

classification quant-ph MSC 81P4581V80

keywords quantumnon-GaussianitystellarrankhomodynedetectionquadraturedistributionzerosHudsontheoremwitnessesenergy-boundedGaussianstatesbosonic

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 a single quadrature measurement—the standard homodyne detection of one field angle—can certify quantum non-Gaussianity at any level of the stellar-rank hierarchy, not just detect it. The core observation, taken from a companion result, is that a zero in a quadrature probability distribution is a signature of non-Gaussianity. The authors build a windowed witness: measure the probability of finding the quadrature outcome in a small interval around the zero, and compare it to the minimum possible value for any Gaussian state with bounded mean energy. They prove that for every energy-bounded state with a zero, some window size makes the measured value strictly lower than that Gaussian floor, and that states with k zeros certify stellar rank at least k. If correct, this drastically simplifies experimental non-Gaussianity certification: no tomographic reconstruction, no Wigner-negativity measurement, and no non-Gaussian measurement is needed.

What carries the argument

The load-bearing object is the windowed quadrature witness $\hat{W}_{\theta,x,\eta} = \int_{x-\eta/2}^{x+\eta/2} |q\rangle\langle q|_{\hat q_\theta} \, dq$, an unnormalized POVM element that describes the probability of a homodyne outcome in a bin of width $\eta$ centered at $x$. Its threshold $w^E_{\theta,x,\eta}$ is the infimum of that probability over all Gaussian states with mean energy at most $E$, computed numerically from closed-form quadrature distributions of displaced-squeezed states. The argument couples this witness to a Hudson theorem for wavefunctions: an energy-bounded non-Gaussian state has a quadrature wavefunction with a complex zero, and when that zero is real it is a zero of the measured distribution. Completeness and soundness are proved with two ingredients: compactness of the sets $\mathcal{S}^E_r$ in trace norm, which makes the Gaussian infimum attainable and strictly positive on any fixed window, and a Taylor expansion showing that the target's window probability decays as $\eta^2$ while the Gaussian floor decays only linearly.

What would settle it

A single counterexample would settle the completeness claim: find any state in $\mathcal{S}^E$ whose quadrature distribution vanishes at $x$ but whose window probability never drops below the Gaussian floor $w^E_{\theta,x,\eta}$ for any window size; a direct numerical search over rank-finite, energy-bounded states could be carried out today.

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Extended reading notes

Core claim

The central claim is that the statistics of one quadrature—the distribution of outcomes of a standard homodyne measurement at a fixed angle—can witness non-Gaussianity at every level of the stellar-rank hierarchy, provided the state has a known energy bound and its quadrature distribution has a zero. Concretely, for any state $\hat\rho \in \mathcal{S}^E$ whose quadrature distribution satisfies $p_{\hat\rho,\theta}(x)=0$, there is a window size $\eta$ such that the window probability $\operatorname{Tr}(\hat\rho \hat{W}_{\theta,x,\eta})$ lies strictly below the threshold $w^E_{\theta,x,\eta} = \inf_{\hat\sigma\in \mathcal{S}^E_0}\operatorname{Tr}(\hat\sigma \hat{W}_{\theta,x,\eta})$, so the measured state cannot be any mixture of energy-bounded Gaussians. If the distribution has $k$ separate zeros, wrapping windows around all of them certifies stellar rank at least $k$; Fock states $|n\rangle$ have $n$ zeros in every quadrature, and cat states have infinitely many real zeros along suitable quadratures, giving arbitrarily high rank certification from a single quadrature angle. The key proof ingredients are a Hudson-type theorem from the companion paper (a zero in the wavefunction is a signature of non-Gaussianity), a compactness result for the sets $\mathcal{S}^E_r$, and a scaling separation: the target's window probability falls as $\eta^2$ while the Gaussian floor falls only linearly.

Load-bearing premise

The state-independent claim that a coarse-grained bin search always finds a witness relies on the companion paper's Hudson-type theorem and exponential-energy bounds; if those fail, the general guarantee fails, although the explicit Fock and cat-state examples survive.

Editorial extensions

If this is right

A single fixed homodyne angle suffices to certify that a Fock state $|n\rangle$ has stellar rank $n$, and a cat state can have its stellar rank certified as arbitrarily high, from measurement data alone.
The certification comes with an explicit energy promise: a violation of the Gaussian floor proves the state is non-Gaussian or has mean energy larger than $E$, so experiments must know or bound $E$ in advance.
The sample complexity is $\log(1/\delta)/(2\epsilon^2)$ for confidence $1-\delta$, with no dependence on the stellar rank being certified.
The size of the violation is a lower bound on the trace distance from the set $\mathcal{S}^E_0$ of energy-bounded Gaussian states, giving the witness an operational meaning beyond a yes/no test.
Adding a second quadrature angle makes the witness roughly a hundred times more robust to loss in the single-photon example, and similar multi-angle gains appear for stellar-rank witnesses.

Reading between the lines

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

The same windowed-witness construction should carry over to heterodyne detection, replacing one-dimensional quadrature bins with two-dimensional bins around Husimi-function zeros; the paper identifies this direction but leaves it open.
An automated certification protocol could be built by coarse-graining each measured quadrature into bins, testing every bin against the Gaussian floor, and then spending fresh samples only on the most promising bin; the paper sketches this for a two-quadrature Fock example but does not prove general guarantees for the data-driven variant.
Because the Gaussian floor decreases exponentially with the energy bound $E$, the protocol is most practical for low-energy states; extending it to higher energies may require the photon-number-mixed witness that removes the energy promise at the cost of a second quadrature.
The numerical observation that robustness tracks low energy, high stellar rank, and many zeros near the origin suggests a target-state design principle for future non-Gaussian state preparation, a connection the paper does not pursue.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth a serious look. It gives a clean way to certify quantum non-Gaussianity and even stellar rank from a single quadrature distribution, using zeros of the quadrature wavefunction. The key ingredients—compactness of energy-bounded finite-rank state sets, and the window-projector witness with thresholds from energy-bounded Gaussian states—are genuinely new relative to the prior single-marginal result [24]. The sample complexity bound is simple but correct, and the numerical exploration of noise robustness is honest, including where multi-quadrature measurements help.

The main soft spot is in the quantitative claims. The proof of Lemma 11 uses the energy constraint σ²+μ²+σ_p²+μ_p² ≤ 2E, but with the paper's quadrature normalization the mean photon number is (σ²+μ²+σ_p²+μ_p²−1)/2, so the constraint should be ≤ 2E+1. The derived lower bound on the Gaussian threshold is therefore not justified as stated. The error is repairable—replace 2E by 2E+1 and re-derive—and it does not affect the existence Theorems 4 and 5, whose proofs rely instead on compactness. But Section C.2's state-independent window-size guarantee and the heuristic bin-search protocol in H use the explicit bound, so those quantitative guarantees need revision. Relatedly, the proof of Theorem 5 asserts a uniform constant c2>0 for all η>0; that is stronger than what is needed and, as written, false for large η. The theorem only needs small-η behavior, which compactness supplies, so this too is fixable.

The reliance on the companion paper [1] for the Hudson-type theorem and exponential-energy bounds is reasonable and clearly flagged; the explicit Fock and cat state results stand alone. The numerical thresholds are computed without a global optimality certificate, which is a limitation but not a fatal one for a witnessing proposal.

Who is this for? Experimental groups doing homodyne-based state certification and theorists working on continuous-variable non-Gaussianity witnesses. It deserves a serious referee; the central idea is likely right, and the flaws are quantitative and localized. I would send it to review, with a request to fix the energy constraint and tighten the uniformity claim.

Referee Report

3 major / 5 minor

Summary. This manuscript proposes and analyzes a protocol for certifying quantum non-Gaussianity of bosonic states from samples of a single quadrature distribution. The witness is a projector onto a finite interval of quadrature outcomes; the threshold is the minimal possible expectation value over energy-bounded Gaussian states. The central theoretical results are Theorem 4 (soundness: the threshold is strictly positive for sufficiently small windows) and Theorem 5 (completeness: an energy-bounded state whose quadrature density has a zero at x violates the threshold for sufficiently small windows), both resting on Lemma 6, which asserts compactness of energy- and stellar-rank-bounded state sets. The paper also gives a sample-complexity bound, state-independent window-size bounds under an exponential energy assumption, numerical evaluations for Fock and cat states, and an extension to multiple quadratures that improves robustness.

Significance. If the results hold, the paper offers a genuinely minimal measurement setting for a practically relevant certification task: ordinary homodyne detection at one fixed angle, with no tomographic completeness, can certify non-Gaussianity and, for states with many real quadrature zeros, high stellar rank. The compactness lemma and the explicit sample-complexity theorem are valuable in their own right, and the numerical data, including the multi-quadrature robustness analysis, are informative. The reliance on the companion paper [1] for the Hudson-type theorem is clearly identified; the self-contained part of the argument is elegant and, modulo the proof gaps discussed below, supports the main existence claims.

major comments (3)

[C.2, Lemma 11] The energy constraint used in the proof is misstated. The authors write ⟨n⟩ = (σ²+μ²+σ_p²+μ_p²)/2, leading to σ²+μ²+σ_p²+μ_p² ≤ 2E in Eq. (45). For the quadrature normalization q=(a+a†)/√2, p=−i(a−a†)/√2, the correct identity is q²+p²=2n+1, so the constraint is σ²+μ²+σ_p²+μ_p² ≤ 2E+1. As a consequence, the derivation of the extremal variances σ²_± and the lower bound g(x0,E) in Eqs. (49)-(51) is not justified. In particular, from the paper's own formula σ²_-=E−√(E²−1/4) one gets σ²_-≈1/(8E) for large E, which is incompatible with the asserted bound σ²_-≥1/8 used in the exponent. The state-independent window condition (52) and the binning guarantee in §H depend directly on this lemma, so the quantitative state-independent claim is unsupported as written. This is repairable by re-deriving the bound under the correct energy constraint, and it does not by itself invalidate the existence statements in Theorems 4 and 5.
[C.1, proof of Theorem 5] The proof of Theorem 5 asserts a constant c2>0 such that min_{σ∈S_E^0} min_{q∈[x−η/2,x+η/2]} pσ(q) ≥ c2 for all η>0. This statement is false: for fixed x and unbounded η the interval contains points arbitrarily far from any energy-bounded Gaussian mean, so the minimum tends to zero as η→∞. The comparison wE(η) ≥ c2η is only needed for small η, and that version can be obtained from compactness together with the positivity of inf_{σ∈S_E^0} pσ(x). The proof should be restated with the η-dependence made explicit; as written, it is a gap in the proof of the completeness theorem.
[C.2, Theorem 10] The state-independent upper bound relies on Eq. (32), Lemma 9, and Theorem 1 of the companion paper [1], none of which is proved or reproduced here. Since the coarse-grained search guarantee in §C.2 and §H is explicitly advertised as state-independent, the present manuscript is not self-contained on this point. I do not regard this as an error, but the dependency should be stated prominently and the needed companion statements should be collected in an appendix, or the section should be explicitly marked as conditional on [1].

minor comments (5)

[Throughout] The manuscript contains several unresolved LaTeX cross-references (e.g., '??1', '??4', '??5', '??10', '??11', '??13'); these should be converted to proper equation/theorem numbers.
[C.2, Eq. (45)] The notation 'Varρ(ˆq2)' appears to mean Varρ(ˆq)=⟨ˆq²⟩−μ²; similarly for p. Please correct the notation.
[C.2, Lemma 11 proof] The intermediate inequality 'σ²_-≥...≥1/8' is numerically inconsistent with the preceding formula σ²_-=E−√(E²−1/4); this should be corrected together with the energy-constraint issue raised in Major Comment 1.
[E.1] Section E.1 states that soundness/completeness 'can be established' for the soft witness without giving a proof; since this is presented as a generalization, either provide the proof or clearly label the statement as a conjecture.
[Figure 2 caption] The phrase 'The insets schematically display the zeros and their quadratures' is vague; a brief legend defining the markers would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the witness theorems are derived from compactness and Gaussian minimization rather than from the target claim.

full rationale

Walking the derivation chain, I find no step where a prediction is equivalent to its input by construction. The central objects are a fixed quadrature-window observable W_{θ,x,η} (Eq. 2) and a Gaussian threshold w^E defined as an infimum over S^E_0 (Eq. 3). Theorems 4 and 5 are proved by comparing the O(η²) decay of a target quadrature density at a zero with an O(η) lower bound on the Gaussian window probability; the lower bound comes from compactness of S^E_r (Lemma 6) and explicit Gaussian-density minimization, not from the target state. The numerical thresholds in Section D are optimizations over the Gaussian set, and the protocol deliberately uses separate batches for selecting the bin and evaluating the witness, so there is no fitted parameter disguised as a prediction. The companion-paper Hudson theorem and exponential-energy bounds [1] are cited to justify the prevalence of zeros and the state-independent bin-size guarantee in Section C.2; this is a genuine dependency on a related same-group result, but it is not circular because the cited theorem's assumptions do not contain the present witness conclusion and the explicit Fock/cat demonstrations are self-contained. I also note two apparent mathematical gaps—the missing −1/2 in the ⟨n⟩ relation in Lemma 11 and the claimed uniform c2 in Eq. (29)—but these are correctness risks, not circularity. Overall, no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The witness operators are operational constructions built from quadrature projectors; the only external dependencies are known theorems and the companion paper's results.

free parameters (4)

Energy bound E
Input energy promise in the witness threshold w^E; must be known from experimental conditions, not fitted to measurement data.
Window size η = small, chosen to satisfy existence bounds
Width of the quadrature window around the zero; the soundness and completeness theorems guarantee existence of a sufficiently small η, and in practice it is chosen heuristically or via the state-independent bounds.
Zero location x and quadrature angle θ = chosen from prior knowledge of the prepared state
The protocol requires knowing (or finding) a quadrature where the target distribution vanishes; for phase-invariant states any angle works, but for generic states a search over angles is needed.
Auxiliary parameters t, α, r in Theorem 10 bound = optimized in the bound f(x0,s,S)
Free parameters appearing in the derivation of the state-independent probability bound; they can be chosen to tighten the bound but do not affect the core witness logic.

assumptions (5)

domain assumption Hudson theorem for wavefunctions and the exponential-energy bounds of companion paper [1] (Theorem 1, Eq. (32), Lemma 9)
Used to motivate zeros as signatures of non-Gaussianity and to derive the state-independent window-size guarantee in Theorem 10. The theorem is proven in a companion paper by the same authors and is not included in this submission.
standard math The set of states with bounded average energy is compact in trace norm (Holevo [42])
Used as fact (ii) in the proof of Lemma 6 to show compactness of S^E.
standard math The set of states with stellar rank at most r is closed (Chabaud et al. [34])
Used as fact (iv) in the proof of Lemma 6.
standard math The closed convex hull of a compact set in a Banach space is compact (Aliprantis [43])
Used as fact (v) in the proof of Lemma 6.
domain assumption Quadrature distributions of energy-bounded Gaussian states are uniformly bounded below on compact windows
Used in the soundness proof (section C.1) to assert a positive constant c2 in the lower bound on w^E; this follows from energy bounds on Gaussian parameters but is stated without detailed proof.

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Cite this review

Pith. "Pith review of Detecting quantum non-Gaussianity with a single quadrature." pith.science (2026). https://pith.science/paper/JEBDZKST

@misc{pith2026250723005,
  author       = {Pith},
  title        = {Pith review of: Detecting quantum non-Gaussianity with a single quadrature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JEBDZKST}},
  note         = {Machine review of arXiv:2507.23005}
}

read the original abstract

Full reconstruction of quantum states from measurement samples is often a prohibitively complex task, both in terms of the experimental setup and the scaling of the sample size with the system. This motivates the relatively easier task of certifying application-specific quantities using measurements that are not tomographically complete, i.e. that provide only partial information about the state related to the application of interest. Here, we focus on simplifying the measurements needed to certify non-Gaussianity in bosonic systems, a resource related to quantum advantage in various information processing tasks. We show that the statistics of a single quadrature measurement, corresponding to standard homodyne detection in quantum optics, can witness arbitrary degrees of non-Gaussianity as quantified by stellar rank. Our results are based on a version of Hudson's theorem for wavefunctions, proved in a companion paper [arXiv:2507.23468], revealing that the zeros in a homodyne distribution are signatures of quantum non-Gaussianity and higher stellar ranks. The validity of our witnesses is supported by a technical result showing that sets of states with bounded energy and finite stellar rank are compact. We provide an analysis of sample complexity, noise robustness, and experimental prospects. Our work drastically simplifies the setup required to detect quantum non-Gaussianity in bosonic quantum states.

Figures

Figures reproduced from arXiv: 2507.23005 by the authors.

**Figure 3.** Non-Gaussian witness violation with a Fock [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Numerical example for the extended witness operator [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: Non-Gaussianity witness for Fock state |1⟩ versus all Gaussian states with ⟨nˆ⟩ ≤ 1. We consider witness operators Wˆ ij = Wˆ θ,xij ,η as defined in Eq. (75) with xij = (xi, pj ) spanning a two dimensional grid (10 bins per quadrature), a constant width η = (η, η), so …

**Figure 6.** Figure 6: Convergence of the non-Gaussianity witness with measurement number for the optimal bin pair identified in Fig. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the complex zeros of the wavefunction
quant-ph 2025-07 conditional novelty 7.0 of 10

For energy-bounded bosonic states, a pure state is non-Gaussian exactly when its complex-extended position wavefunction has at least one zero, and the number of zeros equals the stellar rank.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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