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From classical probability densities to quantum states: quantization of Gaussians for arbitrary orderings

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Pith's one-line read By quantizing a Gaussian phase-space density with $s$-ordered creation-annihilation operator orderings, the paper shows a valid quantum state results exactly when the inverse variance $\lambda \le (1+s)^{-1}$, and for antinormal ordering…

desk verdict The s-ordered Gaussian critical line is real and clean; the paper overreaches when it extends the result to arbitrary distributions on a false Fock-diagonality premise. read the letter →

arxiv 2411.14043 v2 pith:6VHFEOA3 submitted 2024-11-21 quant-ph hep-thmath-phmath.MP

Giorgio Lo Giudice , Lorenzo Leone , Fedele Lizzi This is my paper · ORCID
classification quant-phhep-thmath-phmath.MP MSC 81S1081S3081R30
keywords phase-spacequantizationCahill-Glauberorderings-orderedGaussianstatesantinormalWignerfunctionFockbasisquantum-classicalcorrespondence
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

The paper asks when a classical Gaussian probability density can be promoted to a genuine quantum density matrix through quantization, and shows the answer depends on the operator ordering chosen. For the $s$-ordered family that interpolates between normal, symmetric, and antinormal ordering of creation and annihilation operators, a centered Gaussian of inverse variance $\lambda$ becomes a valid quantum state exactly when $\lambda \le \lambda_c(s)=(1+s)^{-1}$. On the critical line the state is pure; above it the map still returns a Hermitian, trace-one operator, but with negative eigenvalues. The endpoint is antinormal ordering ($s=-1$), where $\lambda_c$ diverges and even the fully localized $\delta$-function is mapped to the vacuum state $|0\rangle\langle0|$, so no amount of classical sharpness destroys the quantum correspondence.

What carries the argument

The engine is the $s$-ordered Cahill-Glauber quantizer $\hat{\Omega}_s(f)=\int d^2\xi\,\tilde f(\xi)\,e^{\xi a^\dagger}e^{-\bar\xi a}e^{-\frac{1-s}{2}|\xi|^2}$, which realizes normal ($s=1$), symmetric ($s=0$), and antinormal ($s=-1$) ordering of annihilation and creation operators. Applied to the Gaussian, its Fourier transform $\tilde\rho_\lambda(\xi)=\pi^{-1}e^{-|\xi|^2/2\lambda}$ combines with the quantizer's Gaussian kernel; the angular integral enforces $n=m$, so the Fock-basis matrix is diagonal, and the remaining radial integrals sum to the geometric formula for $\rho_{nn}$. The critical line $\lambda_c(s)=(1+s)^{-1}$ is where the base of the geometric series changes sign, separating positive from non-positive spectra.

What would settle it

Take a Gaussian with nonzero mean or unequal variances, for example $\rho(q,p)\propto\exp[-q^2/(2\sigma_q^2)-p^2/(2\sigma_p^2)]$ with $\sigma_q^2+\sigma_p^2<1+s$ or a nonzero mean, compute its $s$-ordered quantization in the Fock basis, and check whether off-diagonal matrix elements vanish and whether positivity fails exactly at the claimed bound. Any non-vanishing off-diagonal element, or a positive operator outside the bound, would refute the general criterion; the centered radial Gaussian formula would remain unaffected.

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Extended reading notes

Core claim

The paper's central result is that the $s$-ordered Cahill-Glauber quantization of the Gaussian $\rho_\lambda(z)=\frac{\lambda}{\pi}e^{-2\lambda|z|^2}$ is diagonal in the Fock basis, with entries $\rho_{nn}=\frac{2\lambda}{1+(1-s)\lambda}\left(\frac{1-(1+s)\lambda}{1+(1-s)\lambda}\right)^n$ and $\rho_{nm}=0$ for $n\neq m$. This operator is a valid density matrix, positive and of trace one, if and only if $\lambda\le(1+s)^{-1}$ for $s>-1$; for $s=-1$ every $\lambda$, including the $\lambda\to\infty$ delta limit, gives a valid state, with $\hat{\Omega}_{-1}(\delta)=|0\rangle\langle0|$. The Weyl symmetric case is recovered at $s=0$, with critical value $\lambda_c=1$, matching the Heisenberg uncertainty principle; the paper further derives ordering-dependent necessary conditions $\Delta_q^2+\Delta_p^2\ge 1+s$ for arbitrary phase-space densities under the assumption that the quantized operator is diagonal in the Fock basis.

Load-bearing premise

The appendices assume that every classical probability density turns into an operator that is diagonal in the harmonic-oscillator basis after quantization; this is true for the centered radial Gaussian, but not for shifted or squeezed densities, so the paper's general variance-sum criterion for arbitrary densities rests on that assumption.

Editorial extensions

If this is right

  • For symmetric Weyl ordering ($s=0$), a Gaussian maps to a valid state only for $\lambda\le1$, i.e. phase-space variance $(2\lambda)^{-1}\ge1/2$; at $\lambda=1$ it becomes the coherent-state projector $|0\rangle\langle0|$, and sharper Gaussians are unphysical.
  • For every $s>-1$ there is a sharpness ceiling: inverse variances above $(1+s)^{-1}$ produce a Hermitian, normalized but non-positive operator, so the classical density has no quantum counterpart.
  • For antinormal ordering ($s=-1$) no such ceiling exists: the $\delta$-function is mapped to the vacuum state, so perfectly localized classical data remain quantizable.
  • The temperature mapping shifts with $s$: a thermal Wigner function with inverse temperature $\beta=(1+s)^{-1}$ corresponds exactly to the critical density, so negative-temperature classical Gaussians are precisely those with $\lambda>\lambda_c(s)$.
  • For arbitrary densities the paper argues $\Delta_q^2+\Delta_p^2\ge1+s$ is necessary for positivity, reducing to $\Delta q\,\Delta p\ge(1+s)/2$ when the position and momentum variances are equal.

Reading between the lines

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

  • This suggests a practical probe: prepare a classical phase-space distribution with controlled variance and measure whether the quantized state has negative eigenvalues; the boundary should move with the ordering parameter $s$ according to $\lambda_c(s)=(1+s)^{-1}$.
  • Because the Fock-diagonal proof exploits the radial symmetry of the Gaussian, anisotropic or displaced Gaussians may develop off-diagonal Fock elements; an immediate extension would check whether the same threshold or a modified variance inequality governs those cases.
  • The antinormal map sends the delta function to the vacuum state, indicating that the correspondence cannot preserve all classical information at extreme localization; multiple classical densities must collapse onto the same low-rank quantum image.
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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 / 3 minor

Summary. The paper studies the s-ordered Cahill-Glauber quantization of the centered Gaussian ρ_λ(z) = (λ/π)exp(-2λ|z|²), together with the γ-ordered Weyl quantization of the same family. The explicit Fock-basis calculation yields ρ_nn = (2λ)/(1+(1-s)λ) [(1-(1+s)λ)/(1+(1-s)λ)]^n, from which the authors derive the critical line λ_c(s) = (1+s)^{-1}; for s = -1 the critical value diverges and the δ-function maps to |0⟩⟨0|. The paper further claims general necessary conditions for arbitrary probability densities, Δ_q²+Δ_p² ≥ 1 for Weyl quantization and Δ_q²+Δ_p² ≥ 1+s for s-ordering, based on Appendices B.2 and C.4.

Significance. The Gaussian-specific result is correct, and the computations in Appendices B.1 and C.2 are explicit and checkable. The identification of the antinormal ordering as the one for which every classical distribution, including a delta, has a quantum image is a useful and concrete observation, and it can be supported independently by the antinormal P-representation. The general uncertainty-type statements, however, are not established by the arguments given: they rest on a false Fock-diagonality assumption for non-radial distributions. The paper's value therefore lies in the Gaussian critical line, not in the claimed general theorems, and the latter need to be either proved by covariance-matrix methods or removed/restricted.

major comments (3)
  1. [Appendices B.2 and C.4; main text after Eq. (13) and after Eq. (22)] The blanket assertion in Appendix B.2 (repeated in C.4) that the Weyl or s-ordered quantization of an arbitrary probability density is diagonal in the Fock basis is false. The argument in Appendix C.2 works because the Gaussian (19) is radial: the angular integral forces k = 0. For a displaced Gaussian, or for a centered anisotropic Gaussian, the quantized operator has off-diagonal Fock elements (the latter is a squeezed state). Consequently the vanishing conditions (33), the covariance matrix (37), and the main-text statements "if Δ_q²+Δ_p² < 1 then Ω̂(ρ) is not positive" and its s-ordered analogue are not proven for general densities. These claims should be proved by a covariance-matrix/Gaussification argument that does not presuppose Fock diagonality, or explicitly restricted to centered radially symmetric distributions.
  2. [Appendix C.4, Eqs. (53)-(62)] The derivation of the s-dependent uncertainty condition contains a factor-1/2 error: q̂² = (a² + a†² + 2N + 1)/2, so Δ = tr(q̂²Ω_s(f)) equals (1/2)tr({a,a†}Ω_s(f)), not tr({a,a†}Ω_s(f)) as written in Eq. (53). Using the corrected expression together with Ω^{-1}_{-s}({a,a†}) = 2z\bar z - s reproduces the final inequality Δ_q²+Δ_p² ≥ 1+s, but the intermediate displayed inequality (61) with 1/2+s is not consistent with Eq. (62). Please correct the intermediate step so that the displayed derivation is coherent.
  3. [Main text after Eq. (21)] Because the general condition in Appendix C.4 is not established, the sentence claiming that for the antinormal ordering any classical probability distribution can have a quantum analogue does not follow from the preceding derivation. The statement is nevertheless true: for s = -1 the quantizer (17) gives Ω_{-1}(f) = ∫ d²z f(z)|z⟩⟨z|, which is positive for every nonnegative f. The paper should present this direct argument rather than deriving the statement from the faulty general condition.
minor comments (3)
  1. [Summary and conclusions] The final paragraph says "only for λ ≥ λc we get a quantum state", which contradicts Eq. (20), Fig. 1, and the earlier correct statement λ ≤ λ_c(s). Please correct the inequality direction in the conclusions.
  2. [Throughout] The manuscript contains numerous typos and grammatical slips ('Guassian', 'Summry', 'annihalation', 'perfomed', 'alings', 'eiegenvalues', 'for the reminder', missing spaces in expressions such as 'values ofλ < λc'); a careful proofreading pass is needed.
  3. [Appendix C.4] The notation Ω^{-1}_{-s} is confusing because the dequantizer is not the inverse operator in the usual sense but the quantizer with s replaced by -s; please clarify this notation at first use.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the Gaussian critical line λ_c(s)=(1+s)^{-1} follows from explicit Fock-basis integrals with no fitted parameters; the flagged weakness is a false general-diagonality premise in Apps. B.2/C.4, which is a correctness issue, not circularity.

full rationale

The central claim, Eq. (20), is derived self-containedly: the Gaussian's Fourier transform Eq. (43) is inserted into the Cahill-Glauber map Eq. (42), the angular integral forces the Fock matrix to be diagonal for the radial Gaussian, and the radial integral yields ρ_nn. Positivity of these eigenvalues gives λ ≤ λ_c(s)=(1+s)^{-1}, and the δ→|0⟩⟨0| limit for s=-1 is the direct λ→∞ limit of the same formula. No parameter is fitted to data and no 'prediction' is a renamed input. The Weyl case is the same calculation at s=0. The temperature-shift remark in App. C.3 is a direct comparison of Fourier transforms, not a fit or a self-citation. The only self-citation, Ref. [38] (Lizzi & Vitale), is background on star-product bases and is not load-bearing. The manuscript's general variance conditions in Apps. B.2 and C.4 rely on the premise that the quantization of an arbitrary probability density is diagonal in the Fock basis ('Let us start with a probability distribution ρ(q,p). Its Weyl-symmetric quantization, i.e. Ω(f), is diagonal in the Fock basis'), which is unproven and in fact false for non-radial or displaced distributions; this is a missing-support/correctness flaw in the general statement, but it is not circular because the Gaussian theorem does not depend on that premise. No equation reduces to its own input by construction.

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

The central calculation depends only on standard quantum mechanics, the cited quantization maps, and standard Gaussian-state facts. The only questionable load-bearing premise is the Fock-diagonality assumption for general distributions, which is not required for the Gaussian-specific results. lambda and s are inputs, not fitted parameters.

assumptions (5)
  • standard math Standard single-mode continuous-variable quantum mechanics: Hilbert space L2(R), canonical commutation relations [q,p]=i, Fock basis |n> with N |n> = n |n>.
    The whole calculation is carried out in this framework; no alternative interpretation is used.
  • domain assumption Weyl quantization map Eq. (10) and Cahill-Glauber s-ordered map Eq. (17) are valid quantizers/dequantizers satisfying the trace orthogonality Eq. (8).
    The paper takes these maps from the literature [1,13,37,38]; they define what 'quantization of a Gaussian' means.
  • standard math Baker-Campbell-Hausdorff formula e^{A+B} = e^{-[A,B]/2} e^A e^B for operators with c-number commutator.
    Used in derivations of the Fock-basis matrix elements in Appendices B.1 and C.2.
  • standard math Covariance matrix positivity criterion V + iOmega >= 0 characterizes valid Gaussian states (Eq. (23)).
    Imported from Gaussian quantum information theory [26,27,36]; used in Appendices B.2 and C.4 to convert operator positivity into a variance condition.
  • ad hoc to paper The quantization of an arbitrary probability density is diagonal in the Fock basis.
    Assumed without proof in Appendices B.2 and C.4 to derive general necessary conditions for positivity. It is false for non-radial distributions, so the general claims are not established.

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Pith. "Pith review of From classical probability densities to quantum states: quantization of Gaussians for arbitrary orderings." pith.science (2026). https://pith.science/paper/6VHFEOA3

@misc{pith2026241114043,
  author       = {Pith},
  title        = {Pith review of: From classical probability densities to quantum states: quantization of Gaussians for arbitrary orderings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VHFEOA3}},
  note         = {Machine review of arXiv:2411.14043}
}
abstract

The primary focus of this work is to investigate how the most emblematic classical probability density, namely a Gaussian, can be mapped to a valid quantum states. To explore this issue, we consider a Gaussian whose squared variance depends on a parameter $\lambda$. Specifically, depending on the value of $\lambda$, we study what happens in the classical-quantum correspondence as we change the indeterminacy of the classical particle. Furthermore, finding a correspondence between a classical state and a quantum state is not a trivial task. Quantum observables, described by Hermitian operators, do not generally commute, so a precise ordering must be introduced to resolve this ambiguity. In this work, we study two different arbitrary orderings: the first is an arbitrary ordering of the position and momentum observables; the second, which is the main focus of the present work, is an arbitrary ordering of the annihilation and creation operators. In this latter case, we find the interesting result that even a $\delta$-function, which in general has no quantum correspondence, can be mapped into a valid quantum state for a particular ordering, specifically the antinormal one (all creation operators are to the right of all annihilation operators in the product). This means that the Gaussian probability density corresponds to a valid quantum state, regardless of how localized classical particles are in phase space.

Figures

Figures reproduced from arXiv: 2411.14043 by the authors.

Figure 1
Figure 1. The curve represents the critical line of the (inverse square) variance λc(s) = (1 + s) −1 . In particular, Gaussian probability densities with λ > λc(s) get mapped from the quantization map Ωˆ s to non-positive (yet hermitian and normalized) operators, here represented by the gray region. Conversely, for λ < λc(s) they get mapped to positive operators, i.e. valid quantum states. ordering of a and a † yields a valid… view at source ↗

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