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REVIEW 2 major objections 6 minor 30 references

Ring Optimized M-APSK Modulation for Discrete Modulated CV-QKD

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Optimizing ring radii and probabilities extends 16-APSK CV-QKD reach by about 15 percent.

desk verdict A clean, transparent in-model optimization study for M-APSK constellations in DM-CV-QKD; the ~15% distance gain is real within the chosen security model, but its external validity rests on an unchecked bound-tightness assumption. read the letter →

arxiv 2608.01723 v1 pith:B2TLHQVH submitted 2026-08-03 quant-ph

Seonguk Kim , Jun Heo This is my paper · ORCID
classification quant-ph MSC 81P94 PACS 03.67.Dd
keywords continuous-variablequantumkeydistributiondiscretemodulationM-APSKconstellationshapingfinite-sizesecretrateGrammatrixfidelity
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 argues that the fixed ring spacings and fixed selection probabilities used in conventional multi-ring APSK modulation for discrete-modulated CV-QKD are not optimal. By treating ring radius ratio, ring probability, and modulation variance as joint optimization variables under a fixed finite-size secret key rate model, the authors find constellations whose average state sits closer to the ideal Gaussian thermal state, raising the correlation parameter Z* that enters the security bound. The concrete payoff is reach: at the 1e-5 bits/use threshold, 16-APSK (4+12) extends from about 40.5 km to 46.6 km, a roughly 15% gain, with smaller gains of about 10.9% and 6.9% for 32- and 64-APSK. The gain shrinks as constellation size grows because larger APSK sets already approximate Gaussian modulation. If the paper is right, simple geometric and probabilistic shaping of existing APSK formats can buy a real distance extension without changing the protocol, detector, or channel model.

What carries the argument

The Gram matrix G = V†V built from weighted coherent states, whose nonzero eigenvalues coincide with those of the average state τ = VV†, supplies the spectral decomposition needed to evaluate the fractional powers τ^{1/2} and τ^{-1/2} appearing in the correlation bound Z* from [15]. A grid search over (V_A, r, p) then maximizes the finite-size key rate at each distance, and the fidelity F(τ_D, τ_G) between the discrete average state and the Gaussian thermal state acts as a structural diagnostic connecting the geometric change to the key-rate gain.

What would settle it

Recompute the threshold distance for the r,p-optimized 16-APSK constellation (r = [0.41,1], p = [0.71,0.29], V_A = 1.30) using a composable finite-size security proof such as those cited as [28] or [29], and compare it against the conventional binomial constellation; if the optimized constellation no longer beats the binomial, the 15% gain is an artifact of the [15] bound.

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Extended reading notes

Core claim

For each transmission distance, the paper jointly searches the modulation variance V_A, ring radius ratio r, and ring probability p to maximize the finite-size secret key rate, using the same rate model and channel parameters as the conventional multi-ring M-APSK baseline [17]. The optimized 16-APSK (4+12) constellation—inner ring radius ratio pulled inward from 0.50 to 0.41 and outer-ring probability raised from 0.25 to 0.29—increases the threshold distance at K = 1e-5 bits/use from 40.5 km to 46.6 km. The average state of the optimized constellation has higher fidelity to the Gaussian thermal state and a higher Z* value, and the paper uses this as evidence that the gain is structural: the

Load-bearing premise

The ranking of constellations by the inherited security model is taken as correct: the Z* lower bound from [15] is assumed to be equally tight for the optimized states, and reconciliation efficiency is fixed, so if the bound is looser for the new constellations the distance gain is an artifact of the model rather than a real improvement.

Editorial extensions

If this is right

  • The binomial ring probability is not generally optimal: it happens to be close to a good probability shaping for 16-APSK, but for larger multi-ring APSK the optimized probabilities differ more markedly.
  • Smaller constellations have the most to gain from radius-probability optimization, because their average state has a larger structural gap from Gaussian modulation.
  • The Gram matrix method reduces the eigen decomposition from a large truncated Fock-space matrix to an M×M Gram matrix, making joint constellation optimization feasible for multi-ring APSK.
  • The optimized patterns—inner rings shifted inward and ring probabilities more evenly distributed—suggest a practical design rule for APSK modulation in discrete-modulated CV-QKD.

Reading between the lines

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

  • Because the result relies on the [15] lower bound with fixed reconciliation efficiency, the 15% gain is an in-model result; under the newer composable finite-size security analyses cited in the paper ([26]–[29]), the relative gain could shrink, grow, or reverse. Testing the optimized constellation under those proofs is a natural next step.
  • Fidelity to the Gaussian thermal state may serve as a fast pre-screening proxy for constellation search, since it separated structures more cleanly than Z* in the paper's figures; however, the paper does not prove a monotone link, so this should be validated before relying on it.
  • The same r,p-optimization recipe could be applied to other multi-amplitude formats such as probabilistically shaped QAM or 128-APSK under the same model; the expected gain should scale with their distance from Gaussian modulation.
  • The optimized constellations change only transmitted amplitudes and probabilities, so the predicted 16-APSK threshold of 46.6 km is a concrete, testable prediction for an experimental implementation with existing APSK transmitters.
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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

2 major / 6 minor

Summary. The paper proposes a numerical optimization of ring radius ratios and ring probabilities for multi-ring M-APSK constellations in discrete-modulated continuous-variable QKD, using a Gram-matrix spectral decomposition of the average state to compute the Z* correlation parameter in the finite-size secret key rate model of Ref. [17]. The joint search over (V_A, r, p) is reported to extend the maximum transmission distance at K_th = 1e-5 bits/use by about 15% for 16-APSK (46.6 km vs 40.5 km), with smaller gains of 10.93% for 32-APSK and 6.94% for 64-APSK. The fidelity between the discrete average state and the Gaussian thermal state is used as a structural diagnostic.

Significance. The Gram-matrix method in Sec. 2.3 is correct and standard, and the fidelity formula in Eq. (25) is appropriate. The internal comparison is fair: the baseline and proposed constellations are evaluated with the same security model and simulation parameters. If the ranking is robust across security models, the paper offers a simple, practical constellation-shaping technique (two scalar parameters per ring) for DM-CV-QKD, with a clear and physically sensible trend that gains shrink as the constellation approaches the Gaussian average state. However, the central 15% claim rests on the inherited lower-bound security model from Ref. [15]; the paper does not validate the tightness of this bound for the optimized constellations, which is the main correctness risk.

major comments (2)
  1. [§2.1, Eq. (2); §4.1, Table 2] The distance-gain claim is computed with the Z* lower bound of Ref. [15] in the finite-size model of Ref. [17]. Since Z* is a lower bound, the key-rate comparison is meaningful only if the bound's slack is approximately uniform for the conventional and r,p-optimized constellations. The paper provides no tightness check and explicitly bypasses the composable security analyses [26–29]. If the bound is looser for the optimized structures, the reported gain could be an artifact. Please validate the ranking for at least one optimized constellation (e.g., 16-APSK) against the finite-size security proof of Ref. [28] or [29], or provide an independent estimate of the bound's slack.
  2. [§3.1, Eq. (20)] The improvement direction is guaranteed by construction: the conventional binomial constellation is a feasible point in the joint (V_A, r, p) grid, so the optimum of Eq. (20) cannot be lower than the conventional rate. The paper should state this explicitly and frame the result as the magnitude of the achievable gain rather than as evidence that the optimized structure is intrinsically superior. This is important for calibrating the reader's expectations, though it does not invalidate the optimized parameters.
minor comments (6)
  1. [§2.2, Eq. (2)] The operator τ^{-1/2} is not defined for low-rank τ. Please state explicitly that the inverse is taken on the support of τ (pseudo-inverse), consistent with the later description in Sec. 2.3.
  2. [§2.3, Eq. (14)] In Eq. (14), if some λ_j are zero the expression is undefined. Clarify that the spectral decomposition and the eigenvectors are restricted to the nonzero eigenvalues.
  3. [§4.1 and Table 2] The fine-grid result in §4.1 gives 46.39 km and 14.49% improvement, while Table 2 reports 46.6 km and 15.06% using a coarser distance step. The discrepancy is likely due to the 0.1-km distance grid in §4.2; the fine-grid number should be the headline, and the grid-resolution sensitivity should be reported.
  4. [Figs. 3 and 5] Figure 3 and 5 are difficult to read in the text: the curves are not labeled in the caption, and the reference Gaussian state should be specified with the same V_A as the discrete states. Please add clear legends and ensure the figures are legible in print.
  5. [§4.2, Table 1] The V_A values in Table 1 are stated at the maximum-distance threshold, but the exact operating point (e.g., optimum at the threshold crossing) should be defined. Also clarify whether these are the grid-search values with the given steps.
  6. [Throughout] There are several typographical and grammatical issues: 'a t' in the Fig. 4 caption, inconsistent hyphenation of 'multi-ring', missing articles, and Ref. [20] lists an arXiv number with year 2026; please verify the reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported distance gain is the value of the optimized key-rate objective, not an input, and the magnitude is an emergent model result.

full rationale

The paper's derivation chain is self-contained. The finite-size secret key rate in Eq. (1) uses the external Z* lower bound from [15] and the simulation model/parameters from [17]; neither is authored by the present authors, so there is no load-bearing self-citation. The optimization in Eq. (20) searches over (V_A, r, p) to maximize the same key-rate expression used to define the baseline, so the optimized key rate is, by construction, at least as large as the conventional one — but this only guarantees a non-negative distance gain, not the specific 15% figure. The quantitative improvement (40.5→46.6 km for 16-APSK, etc.) is computed from the model and reported as the result of the grid search, not fitted to data and renamed a prediction. The Gram-matrix spectral method is a standard mathematical identity, not a circular argument. The paper explicitly notes it does not use the newer composable security analyses [26–29]; this is a model-validity limitation, not circularity. Likewise, the acknowledged fixed-β limitation concerns the security model's completeness, not the derivation's logical independence. The fidelity analysis is presented as a structural diagnostic and is not used to derive the key-rate result. Therefore no step in the claimed derivation reduces to its own inputs, and the central claim retains independent quantitative content.

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

The paper's positive contribution is the joint optimization of (r, p, V_A) inside an inherited model: the security bound Z* [15] and the finite-size rate model [17] are external inputs, not derived here, and the Gram matrix method is standard linear algebra. The headline improvement is the value of the model's own objective at its optimum, so the ledger is dominated by model-inheritance rather than by new free postulates. No new physical entities are introduced.

free parameters (4)
  • Modulation variance V_A = 16-APSK: 1.20 (conv.) / 1.30 (prop.); 32-APSK: 1.60 / 2.00; 64-APSK: 1.90 / 2.50 (Table 1)
    Optimized at each transmission distance along with r and p (Eq. 20); V_A = 2<n> sets the constellation scale.
  • Ring radius ratio vector r = 16-APSK: [0.41, 1]; 32-APSK: [0.25, 0.60, 1]; 64-APSK: [0.21, 0.43, 0.67, 1] (Table 1)
    The paper's core optimization variable; grid-searched over 0:0.01:1 per distance.
  • Ring probability vector p = 16-APSK: [0.71, 0.29]; 32-APSK: [0.50, 0.43, 0.07]; 64-APSK: [0.38, 0.33, 0.25, 0.04] (Table 1)
    The paper's core optimization variable; grid-searched over 0:0.01:1 per distance.
  • Rate threshold K_th for maximum transmission distance = 1e-5 bits/use
    Chosen by hand; the headline distances (40.5/46.6 km) and hence the 15% figures are defined by this threshold and would shift with a different one.
assumptions (5)
  • domain assumption Z* formula (Eq. 2) from Denys-Brown-Leverrier [15] is a valid explicit lower bound on the Alice-Bob correlation for arbitrary modulation under a Gaussian channel.
    The entire rate computation passes through Z*; the paper does not re-derive or test the tightness of this bound for the optimized constellations.
  • domain assumption Finite-size secret key rate model (Eq. 1) from [17,25] with parameters (N, m, beta, Delta_m, channel T and xi bounds) inherited unchanged from [17].
    The quantitative gains (15% etc.) are outputs of this specific model; the parameters are not listed in the paper.
  • domain assumption Gaussian thermal loss channel with transmittance T and excess noise xi, with conservative estimates T_min and xi_max.
    Standard CV-QKD channel model, used as in [17].
  • standard math Nonzero spectra of VV^dagger and V^dagger V coincide (Gram matrix method, Eqs. 10-14).
    Standard linear algebra fact; the paper uses it to reduce the eigen-decomposition dimension from N_F x N_F to M x M.
  • ad hoc to paper The Z* lower bound ranks constellations in the same order as the true secret key rate, i.e., the bound's slack does not change the comparison.
    Unflagged: the paper never checks the optimized constellations against tighter or composable security analyses [26-29]; only the fixed-beta limitation is acknowledged (Conclusion).

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Cite this review

Pith. "Pith review of Ring Optimized M-APSK Modulation for Discrete Modulated CV-QKD." pith.science (2026). https://pith.science/paper/B2TLHQVH

@misc{pith2026260801723,
  author       = {Pith},
  title        = {Pith review of: Ring Optimized M-APSK Modulation for Discrete Modulated CV-QKD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2TLHQVH}},
  note         = {Machine review of arXiv:2608.01723}
}
abstract

This paper proposes a multi ring M-APSK constellation optimization method for discrete-modulated continuous variable quantum key distribution. Unlike conventional APSK structures with fixed ring spacing and predefined ring probabilities, the proposed method optimizes the ring radius ratio and ring probability to improve the finite-size secret key rate. A method based on the Gram matrix is used to calculate the nonzero spectrum of the average state $\tau$, and fidelity is used to compare the optimized discrete average state with the Gaussian average state. The results show that the proposed structure extends the maximum transmission distance of 16-APSK by approximately 15% compared with the conventional binomial APSK structure. The optimization gain is larger for small size APSK constellations, where the average state has a larger structural gap from Gaussian modulation.

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