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REVIEW 4 major objections 7 minor 93 references

Impact of photon addition and subtraction on nonclassical and phase properties of a displaced Fock state

T0 review · 4 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Photon addition strengthens nonclassicality across a family of engineered light states.

desk verdict A competent extension of the authors' earlier PASDFS work with new higher-order witnesses and phase analysis, but missing coefficient definitions and unstated numerical truncation make the plotted claims non-reproducible as written. read the letter →

arxiv 1908.04086 v1 pith:DX3MW5IU submitted 2019-08-12 quant-ph physics.optics

Priya Malpani , Kishore Thapliyal , Nasir Alam , Anirban Pathak , V. Narayanan , Subhashish Banerjee This is my paper · ORCID
classification quant-phphysics.optics MSC 81V80 PACS 42.50.-p42.50.Dv42.50.Ct03.65.-w
keywords photonadditionsubtractiondisplacedFockstatenonclassicalitywitnesseshigher-ordersub-PoissonianstatisticsquantumphaseQfunctionengineering
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 studies what happens when one first adds $k$ photons to, then subtracts $q$ photons from, a displaced Fock state, namely $|\psi(k,q,n,\alpha)\rangle = \mathcal{N}\hat{a}^q\hat{a}^{\dagger k}\hat{D}(\alpha)|n\rangle$. Its central claim is that photon addition generally improves the nonclassicality of this family of states, while photon subtraction extends that improvement to large displacement amplitudes, and increasing the Fock index $n$ extends it to small amplitudes. It also claims that higher-order moment-based witnesses detect nonclassicality in parameter regions where the corresponding lower-order witnesses fail, and that higher-order sub-Poissonian photon statistics appear only for odd orders. These results matter because the family reduces in different limits to coherent, Fock, photon-added coherent, photon-subtracted coherent, and displaced Fock states, so the trends give a practical guide for choosing operations when engineering nonclassical light.

What carries the argument

The load-bearing object is the photon-added-then-subtracted displaced Fock state $$|\psi(k,q,n,\$\alpha$)\rangle = \mathcal{N}\,\hat{a}^q\,\hat{a}^{\dagger k}\,\hat{D}(\$\alpha$)|n\rangle,$$ with normalization $\mathcal{N}$, together with the moment formula $$\langle \hat{a}^{\dagger t}\hat{a}^j\rangle = \mathcal{N}^2 \sum_{m=0}^\infty C_m^*\, C_{m-j+t}\, \frac{\sqrt{(m+k-q)!\,(m+k-q-j+t)!}}{(m+k-q-j)!}.$$ All the nonclassicality witnesses are constructed from such moments, so this formula is the central engine of the paper: once it is fixed, each criterion is just a different combination of moments. The phase distribution and the $Q$ function are built from the same expansion coefficients $C_m$, so a single analytic expression drives all the reported results.

What would settle it

Take one of the plotted witnesses, say the antibunching parameter $d(1)$ or Agarwal-Tara's $A_3$, for a state with large $\alpha$ (for example $\alpha = 4$) and recompute it while increasing the truncation cutoff until the value stops changing. If the sign of the witness flips between a low cutoff and the converged value, the reported nonclassicality regions for large displacement are an artifact of truncation rather than a property of the state.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes a parameter-by-parameter map of how photon addition, photon subtraction, displacement, and the initial Fock level shape nonclassicality and phase properties. Using moments of the bosonic operators, it evaluates six criteria: Klyshko's, Agarwal-Tara's, Vogel's, antibunching, Hong-Mandel squeezing, and sub-Poissonian statistics, together with the phase distribution, phase fluctuation parameter $U$, and the $Q$ function. The consistent pattern is that photon addition deepens the nonclassicality witnesses in most regimes; photon subtraction is the operation that induces squeezing and is the most effective at altering phase properties; and the Fock parameter behaves oppositely to addition and subtraction, helping at small displacement and hurting at large. A further claim is that the higher-order versions of the criteria are more sensitive, catching nonclassicality that the lower-order versions miss, with the exception that higher-order sub-Poissonian statistics is never negative for even orders.

Load-bearing premise

The numerical claims assume that truncating the infinite sums in Eqs. (3) and (19) at a finite photon number gives converged values, but the paper does not report the truncation order or a convergence check, and for large displacement $\alpha$ the neglected high-photon terms can be substantial.

Editorial extensions

If this is right

  • The same witness hierarchy can certify single-photon sources: when lower-order antibunching or sub-Poissonian statistics gives no signal, a higher-order criterion still records nonclassicality, and the reported depth grows with the order.
  • Photon subtraction is the operation to reach for when squeezing is the target, and adding several photons lets squeezing survive at large displacement amplitudes, trading away the squeezing seen at small amplitudes.
  • Because the state reduces to coherent, Fock, photon-added coherent, photon-subtracted coherent, and displaced Fock states, the reported trends carry over to those known states as limiting cases.
  • The phase-distribution plots indicate that subtraction changes phase properties more than addition, while the Fock parameter shifts the phase behaviour in the opposite direction, so the three operations are not interchangeable knobs.
  • The phase fluctuation parameter $U$ detects antibunching in the same cases where Vogel's criterion does, giving an independent, phase-based route to the same nonclassicality signal.

Reading between the lines

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

  • The odd-order-only higher-order sub-Poissonian pattern suggests a combinatorial constraint in the Stirling-number expansion of the witness: if that constraint holds for all orders, then even-order sub-Poissonian statistics can never certify nonclassicality for any state in this family, not just for the plotted cases.
  • A practical state-engineering protocol could exploit the reported trade-off by choosing $k$, $q$, and $n$ according to the target displacement: more subtraction for bright fields, more Fock level for dim fields, with photon addition as the baseline improvement.
  • Because the $Q$ function shows zeros and non-Gaussian deformation at specific parameters, a heterodyne measurement of $Q$ could serve as a direct experimental test of the predicted parameter regions without full tomography.
  • The claim that higher-order criteria catch what lower-order criteria miss suggests a noise-robust detection strategy: in the presence of losses that wash out low-order moments, the higher-order witnesses may still give a nonclassicality signal.
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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

4 major / 7 minor

Summary. The paper studies the family of photon-added-then-subtracted displaced Fock states |ψ(k,q,n,α)⟩ = N a^q a†^k D(α)|n⟩. Analytic formulas are given for the moments ⟨a†^t a^j⟩, the phase distribution, and the Q function; these are used to evaluate a battery of nonclassicality witnesses—lower- and higher-order antibunching, higher-order sub-Poissonian photon statistics, Hong–Mandel-type squeezing, Klyshko's, Agarwal–Tara's, and Vogel's criteria—whose dependence on α is plotted for various (k,q,n). Phase properties (phase distribution and Carruthers–Nieto fluctuation parameter U) and the Q function are also studied. The main conclusions are that photon addition generally deepens nonclassicality; photon subtraction extends nonclassicality to larger displacement α at the cost of the small-α depth; higher-order criteria can detect nonclassicality where lower-order ones fail, with an exception noted for HOSPS; HOSPS appears only in odd orders; and subtraction modifies phase properties more strongly than addition.

Significance. If the numerical results are reproducible, this is a useful systematic survey of a broad and experimentally relevant state family that unifies coherent, Fock, displaced Fock, PACS, and photon-subtracted states in various limits. The computations are parameter-free (no fitting), and the comparative phenomenology—especially the complementarity between photon addition at small α and photon subtraction at large α, and the added reach of higher-order criteria—is a plausible guide for continuous-variable state engineering. The analytic framework is standard and the criteria are applied in the conventional way. The principal weaknesses are completeness and reproducibility: the central coefficient C_m(α,n,k,q) is never defined, the summation limits are omitted, and the truncation of the infinite sums used for all figures is undocumented. These gaps currently prevent verification of the paper's headline claims.

major comments (4)
  1. [§2, Eqs. (2)–(3)] The coefficient C_m(α,n,k,q) in the expansion (2) is never defined, and the summation ranges in Eqs. (2), (3), (14), and (19) are not specified. For k < q the Fock index m+k−q is negative for m < q−k, and the factorials appearing in Eq. (3) are undefined unless the sum is restricted so that both m+k−q−j and m−j+t are non-negative. As printed, the central moment formula (3) is ill-defined for states such as |ψ(1,2,1,α)⟩ used in Fig. 1, and the plotted results are not reproducible as written. Please define C_m(α,n,k,q) explicitly (e.g., as the displaced-Fock coefficient C_m(α,n) times the factorial factors arising from a^q a†^k) and state the allowed summation limits for each moment order.
  2. [§3, Figs. 1–4, 6, 7] The infinite sums in Eq. (3) (and in Eq. (19) for the Q function) are truncated for numerical evaluation, but the truncation order and any convergence criterion are not reported. This is quantitatively consequential: a displaced Fock state has its photon-number distribution peaked near m ≈ |α|² with width O(|α|), and the higher-order witnesses weight the tail by powers m^l. Since α ranges up to about 4–5 in Figs. 1 and 2, an insufficient cutoff could shift zero crossings or change the sign of witnesses such as d(l−1), D_h(l−1), and S(l). Because the headline claims of Section 3 are read from the signs and depths of these curves, please report the cutoff used, add a convergence check showing that the qualitative conclusions are unchanged when the cutoff is doubled, and state the number of terms retained for each figure.
  3. [§3.2 and Abstract] The abstract and Conclusions state that higher-order sub-Poissonian photon statistics 'is only observed for the odd orders.' This is presented as a general statement but is supported only by the representative curves of Fig. 1(d) for selected parameter values. If a theorem guarantees non-negativity of the HOSPS witness for even orders, please provide a citation or a short proof; otherwise, restrict the claim to the parameter ranges actually studied and describe it as a numerical observation.
  4. [§3.5, Eq. (9)] Equation (9) as printed, A3 = det m(3)/det μ(3) − det m(3) < 0, is not the standard Agarwal–Tara witness: the two terms have incompatible scalings, and for a coherent state both determinants vanish, leaving the ratio indeterminate rather than giving the boundary value A3 = 0. The usual definition is A3 = det m(3) − det μ(3) < 0; please correct Eq. (9) accordingly and confirm that Fig. 4(a) was generated with the corrected expression.
minor comments (7)
  1. [Throughout] There are several typographical errors, including 'a nd' in the title, 'nonclasscial' in the Conclusions, and 'Here, We will establish' in Section 5; the manuscript needs a careful proofread.
  2. [Figs. 1 and 2 captions] The figure captions are garbled and incomplete: the Fig. 1 caption mixes the antibunching panels (a)–(b) with the HOSPS panels (c)–(d), and the Fig. 2 caption does not clearly identify all three panels. Please rewrite the captions to identify each panel and the plotted quantity with its formula number.
  3. [§3.4] The sentence 'The Klyshko's nonclassicality witness is positive for some photon numbers only if k+n>q' presumably should read 'negative' rather than 'positive'; please clarify the intended condition and state it precisely.
  4. [§3.3, Eq. (7)] The upper limit 'r/2' of the second sum in Eq. (7) should be written as floor(r/2), with an explicit statement that the sum over i runs over integers, to remove ambiguity for odd r.
  5. [§4.2] The three parameter combinations for which the phase-fluctuation parameter U dips below its coherent-state value are not identified; please list them explicitly rather than referring to Fig. 6 without a specification.
  6. [§5] The claim that non-Gaussianity is established via the Q function is qualitative: a Gaussian state always has a positive Q function, so please state the specific criterion used (e.g., presence of zeros, or departure from a fitted Gaussian) and acknowledge the qualitative nature of the non-Gaussianity statement.
  7. [Abstract vs. §3.2] The abstract's statement that 'higher-order nonclassicality criteria are found to detect nonclassicality even in the cases when corresponding lower-order criteria failed to do so' should be qualified, since §3.2 reports the opposite behavior for HOSPS (higher-order HOSPS fails where lower-order succeeds for certain α).

Circularity Check

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No circularity: the paper evaluates standard nonclassicality witnesses on an explicitly defined state; no fitted parameter is disguised as a prediction.

full rationale

The paper's derivation chain is self-contained at the level claimed. It defines the photon-added-then-subtracted displaced Fock state in Eq. (2), computes expectation values of bosonic operator moments in Eq. (3) directly from that state, and then evaluates established, external nonclassicality criteria (antibunching, sub-Poissonian statistics, squeezing, Klyshko, Agarwal-Tara, Vogel, phase distribution, and Q function). There are no fitted parameters, no data subsets used to predict closely related quantities, and no result that is assumed in the inputs. The comparative claims about photon addition, photon subtraction, Fock parameter, and displacement parameter are obtained by evaluating the same criteria for different parameter values, not by construction from the definitions. Self-citations appear, such as the use of higher-order sub-Poissonian and squeezing criteria from [35, 36], but these supply standard formulas and context; the numerical witnesses plotted in Figs. 1-7 are computed from the present state definitions, so the citations are not load-bearing reductions. The lack of an explicit closed form for C_m(α,n,k,q) and the unreported truncation of the infinite sums in Eqs. (3) and (19) are reproducibility concerns, not circularity. The paper therefore does not exhibit a circular derivation or a fitted-input-called-prediction structure.

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

No free parameters are fitted. The paper relies on standard quantum-optics definitions and criteria, plus an unstated numerical truncation. No new entities are introduced.

assumptions (4)
  • domain assumption The state |psi(k,q,n,alpha)> is defined by applying a^q a^dag^k to a displaced Fock state, with photon addition before subtraction.
    This operator ordering is stated in Eq. (2) and underlies all subsequent moments and witnesses.
  • standard math Negativity of the Glauber-Sudarshan P function is the operational definition of nonclassicality.
    Invoked in Section 3, with references [2,3].
  • domain assumption Moments-based criteria (antibunching, HOSPS, squeezing, Klyshko, Agarwal-Tara, Vogel) are valid witnesses of nonclassicality.
    These are cited from the literature and applied directly without independent derivation.
  • ad hoc to paper Numerical evaluations truncate the infinite sums in Eq. (3) and Eq. (19) at a finite photon number with assumed convergence.
    No truncation or tolerance is reported in the text or figures, yet all plotted results depend on it.

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Pith. "Pith review of Impact of photon addition and subtraction on nonclassical and phase properties of a displaced Fock state." pith.science (2026). https://pith.science/paper/DX3MW5IU

@misc{pith2026190804086,
  author       = {Pith},
  title        = {Pith review of: Impact of photon addition and subtraction on nonclassical and phase properties of a displaced Fock state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DX3MW5IU}},
  note         = {Machine review of arXiv:1908.04086}
}
abstract

Various nonclassical and quantum phase properties of photon added then subtracted displaced Fock state have been examined systematically and rigorously. Higher-order moments of the relevant bosonic operators are computed to test the nonclassicality of the state of interest, which reduces to various quantum states (having applications in quantum optics, metrology and information processing) in different limits ranging from the coherent (classical) state to the Fock (most nonclassical) states. The nonclassical features are discussed using Klyshko's, Vogel's, and Agarwal-Tara's criteria as well as the criteria of lower- and higher-order antibunching, sub-Poissonian photon statistics and squeezing. In addition, phase distribution function and quantum phase fluctuation have been studied. These properties are examined for various combinations of number of photon addition and/or subtraction and Fock parameter. The examination has revealed that photon addition generally improves nonclassicality, and this advantage enhances for the large (small) values of displacement parameter using photon subtraction (Fock parameter). The higher-order sub-Poissonian photon statistics is only observed for the odd orders. In general, higher-order nonclassicality criteria are found to detect nonclassicality even in the cases when corresponding lower-order criteria failed to do so. Photon subtraction is observed to induce squeezing, but only large number of photon addition can be used to probe squeezing for large values of displacement parameter. Further, photon subtraction is found to alter the phase properties more than photon addition, while Fock parameter has an opposite effect of the photon addition/subtraction. Finally, nonclassicality and non-Gaussianity is also established using $Q$ function.

Figures

Figures reproduced from arXiv: 1908.04086 by the authors.

Figure 1
Figure 1. (Color online) For PASDFS the lower-and higher-or [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. (Color online) Dependence of the Hong-Mandel-typ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. (Color online) Illustration of Klyshko’s paramet [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: (Color online) Nonclassicality reflected through [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: (Color online) Polar plot of phase distribution fu [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: (Color online) Variation of phase fluctuation para [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: (Color online) Q function for PASDFS |ψ (k, q, n, α)i with (a) k = q = n = 1, (b) k = 2, q = n = 1, and (c) q = 2, k = n = 1 with α = 1 5 √ 2 exp (ιπ/4). (d) Similarly, Q function of PASDFS with q = 2, k = n = 1 and α = √ 2 exp (ιπ/4). Q function for |ψ (k, q, n, α)i w…

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