{"id":"c1322171-cefb-48f7-ad93-ddf03b0f4a51","arxiv_id":"2608.11122","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A finite-energy GKP Bell test with position-momentum binning and photon-number-mod-four measurements yields CHSH violation and Bell-pair extractability above about 4.2 dB squeezing.","lead":"This paper computes the CHSH score a finite-energy GKP-entangled oscillator pair would produce under four explicit measurements, including a photon-number-mod-four readout. It finds Bell violation and a quantitative self-testing guarantee above about 4.2 to 5 dB of per-peak squeezing, with a calibrated displacement improving the score.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The paper's central claim has two components: (i) the model calculation of CHSH scores for a number-filtered GKP Bell state with the specified full-oscillator measurements, and (ii) the implication from those scores via Kaniewski's bound to Bell-pair extractability. Component (ii) is a corollary of known results extended to separable Hilbert spaces, and the provided proof in Appendix B is sound: the finite-rank approximation preserves the CHSH expectation asymptotically, and the local channels are constructed as normal CPTP maps. Component (i) is a numerical evaluation of finite-dimensional expressions derived exactly in Section 5; the formulas (Eq. 56) are exact, the observables are genuine reflections, and the convergence tests in Tables 3 and 4 demonstrate stability of all quoted digits and threshold crossings. The reader's stated weakest assumption - that the physical source matches the idealized number-filtered state and lossless measurements - is explicitly declared by the authors as a scope boundary (Sec. 7.2), and the abstract and Section 7.3 clearly separate honest-model predictions from device-independent certification based on an observed score. Therefore I do not find a load-bearing flaw. The remaining risk is purely numerical reproducibility, which the ancillary archive addresses; since I did not execute the code, a re-implementation check is the appropriate verification.","tokens_in":18462,"tokens_out":27285,"duration_ms":224342,"concrete_test":"Independently re-implement Eq. (56) with a different numerical method (e.g., exact Mehler-kernel quadrature versus Fock-basis exponentiation) using the ancillary archive, and confirm the fixed and calibrated CHSH scores in Table 1 and the threshold values in Eqs. (69)-(72) to the stated six-digit precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central argument. The model calculation is internally consistent: the observables in Eqs. (28), (34), (37), and (49) are genuine binary reflections on the full oscillator Hilbert space, the state in Eq. (22) is a normalized bipartite state, and Eq. (56) correctly evaluates the physical correlations while retaining finite-energy effects. The infinite-dimensional extension of Kaniewski's bound in Corollary 2.2 is proved by a valid finite-rank approximation, and the numerical convergence tests in Tables 3-4 are extensive. The idealized source and lossless measurements are explicitly declared scope boundaries, not hidden assumptions: the self-testing statement depends only on the observed score, and the model thresholds are clearly labeled as honest-model predictions. The only residual risk is that the numerical code or ancillary data contain an undetected bug, which the provided reproducibility package is designed to rule out; I did not independently execute the code.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a CHSH test on a two-mode finite-energy GKP state, with Alice's tilted measurements realized by a fixed binary coarse-graining of photon number modulo four and its displaced conjugate, and Bob's measurements by periodic position and momentum binning. The physical correlations are evaluated exactly within the number-filtered family using a Gram-matrix formula, and the resulting CHSH score is fed into Kaniewski's analytic extraction bound, which the authors extend to separable Hilbert spaces by finite-rank approximation. For the canonical displacement d=sqrt(pi), the model predicts Bell violation above 4.56 dB and nontrivial extractability above 5.02 dB; calibrating only the displacement over a fixed interval lowers these thresholds to 4.21 dB and 4.58 dB. The authors carefully distinguish honest-model predictions from loss, detection-efficiency, and finite-sample thresholds, and they explain that a device-independent guarantee follows from any valid confidence lower bound on the observed CHSH score.","tokens_in":18561,"tokens_out":15718,"duration_ms":136080,"significance":"If the calculations are correct, this is a valuable step for continuous-variable device-independent protocols: it provides a concrete, full-Hilbert-space realization of the tilted settings that bypasses the known homodyne-only obstruction, with quantitative squeezing thresholds and an explicit extractability statement. The analytical core is clean and mostly self-contained: Eq. (56) is an exact evaluation of physical correlations, Proposition 3.2 is proved directly, and Corollary 2.2 extends Kaniewski's bound to separable spaces by a valid finite-rank argument. The numerical work is unusually careful, with one-parameter and coupled-cutoff convergence tests in Tables 3 and 4 and an ancillary reproducibility package; the paper also explicitly flags that the reported roots are convergence estimates rather than certified interval enclosures. The idealized lossless source and measurement model is declared as a scope boundary in Sec. 7.2, and the device-independent conclusion is correctly stated to depend only on the observed score, not on the GKP interpretation. These strengths make the paper a solid contribution to the literature on bosonic nonlocality and self-testing.","major_comments":[],"minor_comments":[{"comment":"The abstract and conclusion state the thresholds as \"above 4.56 dB\" and \"above 5.02 dB\" without qualification, but Table 4 explicitly says the six-digit roots are not certified interval enclosures. Please add a short qualifier such as \"model estimate\" in the abstract and conclusion so the numerical nature of these thresholds is not misread as a rigorous bound.","section":"Abstract and Sec. 6.3/Table 4"},{"comment":"The calibrated score is defined as the maximum of S(beta,d) over the a priori interval I, so the reported improvement over the fixed prescription is partly by construction; although this is disclosed in the text, a one-sentence reminder in the abstract or conclusion would help prevent readers from interpreting the calibrated gain as a parameter-free prediction.","section":"Sec. 4.5, Eq. (53)"},{"comment":"Because the large calibrated gain is attributed specifically to the deterministic phase-bit rule, the comparison with the one-bit POVM would be easier to evaluate if the independently calibrated one-bit curve were shown in Figure 1 or if Table 5 included one-bit calibrated values at additional representative squeezing levels beyond the 5-dB entry.","section":"Sec. 6.4, Table 5"},{"comment":"The statement that a deviation of at most eta in each of the four product correlators gives an observed-score lower bound of S(beta,d) - 4 eta would benefit from an explicit definition of eta as the absolute deviation of each correlator; as written, the bound is clear but the type of deviation could be stated more precisely.","section":"Sec. 7.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is within the journal's scope and the authors are exemplary in declaring their model assumptions and the numerical nature of their thresholds. The main editorial risk is that the abstract's unqualified threshold numbers will be quoted without the numerical-estimate caveat; asking for that qualification in the abstract would strengthen the manuscript. I found no load-bearing technical flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a careful, honest model calculation, not an experimental claim. The new content is the full-oscillator finite-energy evaluation of a CHSH self-test with GKP states where the tilted settings come from photon-number-mod-four and its displaced conjugate. The individual ingredients are known (quarter-rotation identity, modular-number readout, diagonal homodyne as Y), but the combination, including explicit squeezing thresholds and a one-parameter calibration, is genuinely new. If the numerics are right, it gives a continuous-variable path around the homodyne-only no-go theorem.\n\nStrengths: The math is clean. The Gram-matrix formula is exact, Proposition 3.2 is correct, and Corollary 2.2 extending Kaniewski's bound to separable Hilbert spaces is proved by a valid finite-rank approximation. The numerical convergence tests are extensive, including coupled cutoffs. The paper goes out of its way to separate model predictions from device-independent guarantees: the self-testing statement depends only on the observed CHSH score, not on GKP assumptions, and the authors say so. The calibration is disclosed as an optimization over d, with a fixed interval and no hidden fine-tuning of the odd-sector rule. The comparison with the one-bit POVM is informative.\n\nSoft spots: The reported thresholds are numerical roots, not certified interval enclosures; the authors acknowledge this. The source state is the number-filtered Bell state, and no loss, inefficiency, or finite-sample confidence bounds are provided; those are explicitly scoped out. The calibrated improvement is partly by construction, since it optimizes d over an interval, but that is disclosed and the canonical fixed-d results are also reported. In an actual experiment, the predicted thresholds are model predictions, not experimental loss budgets. The paper would be stronger if it included a simple finite-sample confidence analysis, but the logical separation makes that a missing piece rather than a flaw.\n\nWho it's for: researchers working on CV device-independent protocols, GKP error correction, and modular-variable measurement. It's a solid construction that should receive serious peer review. My own verdict is positive; I would cite it as a reference for finite-energy CHSH self-testing.","headline":"A clean and honest full-oscillator calculation of a finite-energy GKP CHSH test, with well-separated model predictions and device-independent claims; the soft spots are declared scoping boundaries rather than hidden flaws.","tokens_in":19125,"tokens_out":1796,"would_cite":true,"duration_ms":16270,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"CHSH score from a GKP state certifies Bell pairs above 4.56 dB","keywords":["GKP code","CHSH self-testing","continuous-variable Bell test","photon-number-mod-four measurement","finite-energy GKP states","Kaniewski extractability bound","Bell-pair extraction","device-independent certification"],"falsifier":"Within the honest model, recompute the CHSH score for $|\\Psi_\\beta\\rangle$ at 5 dB with $M_4$ and $D(\\sqrt{\\pi})$: the paper predicts $S_{\\rm fix}=2.10052$ and, after calibrating the displacement, $S_{\\rm cal}=2.21292$; a materially different result in an independent exact calculation would falsify the numerical claim. A physical implementation whose odd-sector probability departs from the predicted $p_{\\rm odd}$ values of 0.3285 (canonical) and 0.3633 (calibrated) at 5 dB would indicate that the realized source differs from the assumed filter model.","tokens_in":18210,"feed_emoji":"⚛️","tokens_out":14225,"duration_ms":107641,"temperature":0.7,"pith_summary":"This paper asks whether a finite-energy Gottesman–Kitaev–Preskill (GKP) oscillator state can power a practical Bell-pair self-test, and answers yes with explicit full-oscillator measurements. The authors construct four binary measurements — periodic binnings of position and momentum, plus a fixed binary coarse-graining of photon number modulo four and its displaced conjugate — and evaluate their correlations on a number-filtered GKP Bell state without postselection or logical post-correction. For the canonical displacement $d=\\sqrt{\\pi}$, the model predicts CHSH violation above 4.56 dB of per-peak squeezing and a nontrivial Bell-pair extractability bound above 5.02 dB. Calibrating only the displacement amplitude lowers these thresholds to 4.21 dB and 4.58 dB, and at 12 dB raises the score from 2.69486 to 2.78858 and the target-state overlap bound from 0.90758 to 0.97243. These are honest-model predictions: any valid confidence lower bound on an observed score plugs into the self-testing theorem to give a device-independent, dimension-independent extractability guarantee.","feed_headline":"CHSH score from a GKP state certifies Bell pairs above 4.56 dB","feed_subtitle":"Calibrating the displacement lowers the Bell threshold to 4.21 dB and raises the self-tested overlap to 0.97 at 12 dB.","key_machinery":"The central objects are the number-filtered GKP columns $W_\\beta = e^{-\\beta \\hat n} W_{\\rm ideal}$ and the modular-four phase-bit reflection $M_4 = \\Pi_0+\\Pi_1-\\Pi_2-\\Pi_3$, together with its displaced conjugate $D(d)^\\dagger M_4 D(d)$. The load-bearing identity is the phase-fixed quarter-rotation covariance $F W_\\beta = W_\\beta H$ with $F=e^{i\\pi \\hat n/2}$, which makes $M_4$ an exact finite-energy Hadamard on the filtered code; the displaced setting has no such simplification because the displacement fails to preserve parity and the code manifold. Physical correlations are computed by the Gram-matrix formula $\\langle M\\otimes N\\rangle_\\beta = \\langle\\psi_B|\\Gamma_\\beta(M)\\otimes \\Gamma_\\beta(N)|\\psi_B\\rangle / \\langle\\psi_B|S_\\beta\\otimes S_\\beta|\\psi_B\\rangle$ with $\\Gamma_\\beta(M)=W_\\beta^\\dagger M W_\\beta$, retaining the finite codeword overlap and the filter-induced logical distortion rather than compressing measurements to their logical action. The Kaniewski affine bound $f_K(S)=\\max(1/2, (4+5\\sqrt{2})S/16 - (1+2\\sqrt{2})/4)$ converts the resulting CHSH value into a Bell-pair extractability guarantee.","core_discovery":"On its own terms, the paper establishes that the two-mode number-filtered GKP Bell state $|\\Psi_\\beta\\rangle = (W_\\beta\\otimes W_\\beta)|\\psi_B\\rangle/\\sqrt{N_\\beta}$ can produce a CHSH score in the self-testing regime using only homodyne binning and photon-number-modulo-four readouts. The first tilted setting is the phase-bit measurement $M_4 = \\Pi_0+\\Pi_1-\\Pi_2-\\Pi_3$, an exact logical Hadamard on the filtered code, and the second is its displaced conjugate $D(d)^\\dagger M_4 D(d)$, evaluated directly in the full oscillator Hilbert space without postselection. Because the displacement does not preserve the centered code manifold, it activates the odd photon-number sectors, whose fixed binary assignments materially affect the score. With $d=\\sqrt{\\pi}$, the predicted CHSH value exceeds the local bound above 4.56 dB per-peak squeezing and the Kaniewski extractability bound $f_K(S)$ becomes nontrivial above 5.02 dB; calibrating only $d$ in a fixed interval lowers these crossings to 4.21 dB and 4.58 dB. The same score-to-extractability relation, applied to a confidence lower bound on an observed black-box score, yields a dimension-independent guarantee of Bell-pair extraction via Corollary 2.2.","pith_inferences":["An implication the authors leave implicit is that including photon loss would shift the thresholds upward, because a single annihilation event moves an even residue sector into an odd sector with the opposite binary assignment, but the score-to-extractability statement would survive for whichever score is observed.","Beyond the paper, the predicted role of the odd sectors could be tested directly by measuring the odd-sector probability $p_{\\rm odd}$ as a function of squeezing and checking that the deterministic phase-bit assignment, not merely the displacement, drives the score gain.","A natural extension would be to carry the same modular-four construction into finite-sample statistical analysis, treating calibration and Bell-test data as independent; the authors explicitly leave that construction outside this paper's scope."],"forward_implications":["With these scores, a GKP-based bipartite Bell test can reach the self-testing regime at moderate squeezing without a logical T gate, because modular photon-number readout supplies the tilted bases that the homodyne-only Pauli palette cannot provide.","Any valid confidence lower bound $S_L$ on an observed CHSH score yields the device-independent guarantee $\\Xi(\\rho\\to\\psi_B)\\ge f_K(S_L)$, independent of the oscillator dimension and without assuming the GKP source model.","The calibrated prescription fixes the displacement before Bell-test data are collected; if the finite-energy parameter is misestimated, only the observed score changes, and the same bound still applies to it.","The deterministic phase-bit coarse-graining is essential to the large calibrated gain: independently calibrating the one-bit POVM with randomized odd-sector outcomes gives only a small improvement, so the gain is not an artifact of optimizing residue-class labels.","The additional X/Z checks approach their ideal values faster than the displaced tilted setting and connect the construction to the six-setting verification protocols used for graph states."],"supporting_citations":[{"why":"supplies the GKP code, the phase-fixed quarter-rotation identity, and the modular-photon-number interpretation of the tilted readout.","marker":"[7]"},{"why":"provides the analytic extractability bound that converts a CHSH score into a Bell-pair overlap guarantee.","marker":"[3]"},{"why":"proves the homodyne-only Pauli palette cannot violate CHSH on an encoded Bell pair, making the modular-four tilted settings necessary.","marker":"[9]"},{"why":"gives the earlier GKP CHSH proposal whose logical-T basis change the present route replaces.","marker":"[8]"},{"why":"supplies the exact conversion between the number-filter parameter and the squeezing/comb parameter used to report the dB thresholds.","marker":"[12]"},{"why":"defines the CHSH operator and the local bound the predicted scores must exceed.","marker":"[14]"},{"why":"supplies the circuit-QED proposal for implementing the controlled quarter rotations underlying the modular-four readout.","marker":"[11]"},{"why":"motivates the additional X/Z checks and their use in verification protocols.","marker":"[13]"}],"fun_headline_variants":["Calibration lowers GKP Bell-pair threshold to 4.21 dB","GKP CHSH test certifies Bell pairs with robust self-testing","Finite-energy GKP self-testing beats 4.56 dB squeezing","Displacement-activated odd sectors boost GKP self-testing","From 0.91 to 0.97: calibration raises GKP Bell-pair overlap"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the physical source and measurements are exactly the idealized number-filtered GKP Bell state and the lossless projective modular-four readout with the fixed coarse-graining; if real loss or detector inefficiency enters, the predicted thresholds no longer apply to the experiment, although the score-to-extractability bound would still hold.","fun_headline_variants_meta":{"raw":{"variants":["Calibration lowers GKP Bell-pair threshold to 4.21 dB","GKP CHSH test certifies Bell pairs with robust self-testing","Finite-energy GKP self-testing beats 4.56 dB squeezing","Displacement-activated odd sectors boost GKP self-testing","From 0.91 to 0.97: calibration raises GKP Bell-pair overlap"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000842,"raw_usage":{"total_tokens":3784,"prompt_tokens":1176,"completion_tokens":2608,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":792,"completion_tokens_details":{"reasoning_tokens":2508}},"tokens_in":792,"tokens_out":2608,"duration_ms":14821,"temperature":1.0,"reasoning_tokens":2508,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:49:00.185661+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Within the honest model, recompute the CHSH score for $|\\Psi_\\beta\\rangle$ at 5 dB with $M_4$ and $D(\\sqrt{\\pi})$: the paper predicts $S_{\\rm fix}=2.10052$ and, after calibrating the displacement, $S_{\\rm cal}=2.21292$; a materially different result in an independent exact calculation would falsify the numerical claim. A physical implementation whose odd-sector probability departs from the predicted $p_{\\rm odd}$ values of 0.3285 (canonical) and 0.3633 (calibrated) at 5 dB would indicate that the realized source differs from the assumed filter model.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"proves the homodyne-only Pauli palette cannot violate CHSH on an encoded Bell pair, making the modular-four tilted settings necessary."},{"cited_title":"Marshall and C","cited_arxiv_id":null,"evidence_quote":"gives the earlier GKP CHSH proposal whose logical-T basis change the present route replaces."},{"cited_title":"Hayashi and M","cited_arxiv_id":null,"evidence_quote":"motivates the additional X/Z checks and their use in verification protocols."}],"review_version":1}