{"id":"61efa739-22ac-4aed-8955-67caf028ffb0","arxiv_id":"2502.09514","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Continuous-variable quantum MacWilliams identities yield quantum Cohn-Elkies and Levenshtein bounds, and conditional optimality of E8/Leech GKP codes.","lead":"This paper introduces a continuous-variable version of the quantum MacWilliams identities, which relate two ways of weighting the errors that strike a quantum code, and uses them to prove new upper bounds on the size and distance of quantum error-correcting codes against displacement noise.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline optimality of E8/Leech GKP codes rests on unproved Assumptions 1 and 2; if either sup-at-0 quotient fails for some d≤d_max, Theorem 3 and the abstract's optimality claim lose their justification.","rationale":"The reader's weakest_assumption is exactly the one I would flag. I agree with CONDITIONAL: the trace-class core (Theorems 1, 2, and 4) is derived cleanly, and the proof of Theorem 1 from Lemma 1 and the MacWilliams identity is internally consistent; the Levenshtein bound has a real proof in Appendix D. The paper is honest about the conditional status, but the abstract and conclusion present optimality as 'conditional on Assumptions 1 and 2', so the unproved numerical assumptions are the load-bearing point. My recommendation is UNCHANGED because the reader already made the verdict conditional on proving these assumptions. I would not move to reject: the assumptions are plausible, and the paper does not hide them. The GKP normalization issue is real but secondary: it affects the ideal-GKP weight-distribution example and the claimed formal MacWilliams identities for GKP codes, not the finite-K upper bound of Theorem 3. The proposed interval-arithmetic check is sufficient to settle whether the assumptions land.","tokens_in":25059,"tokens_out":18490,"duration_ms":164335,"concrete_test":"Certify Assumptions 1 and 2 with interval arithmetic: implement the explicit Viazovska and Cohn–Kumar–Miller–Radchenko–Viazovska formulas for f8, hat(f8), f24, hat(f24), and use a validated ODE or Taylor-model bound to enclose the quotients on [0,1] for d=d_max^(8) and d=d_max^(24). If the certified upper bound equals the value at x=0, the assumptions hold; if the bound exceeds that value, the optimality claim is not established. An independent 100-digit evaluation at a dense set of points could falsify a visible interior maximum, but only a rigorous enclosure settles the supremum.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central advertised claim—that no non-lattice code can beat E8/Leech GKP distances—is Theorem 3, whose proof is literally conditional on Assumption 1 and Assumption 2 (Section 6.2, Eqs. 73–74). The assumptions assert that on x∈[0,1] the quotients f8(√2 x)/hat(f8)(d^2 x/√2) and f24(2x)/hat(f24)(d^2 x/2) have their supremum at x=0, equal to (2π)^4 and (2π)^12, for all d≤d_max. The paper supplies only Fig. 3 as evidence and explicitly states that no proofs of the equalities are provided. A failure at any d in the allowed range—for example, an interior maximum exceeding the x=0 value—would make the bound in Theorem 3 weaker than the E8/Leech code point, so the optimality argument would no longer go through. This is load-bearing because Theorem 1 alone is a conditional upper bound; its tightness for the magic functions is exactly what the assumptions supply. A secondary gap is the formal GKP weight-distribution derivation in Section 3.2.4, where the Poisson-summation coefficient and the relation K^2=|det(L/√(2π))| appear underived and normalization-sensitive; however Theorem 3's upper bound does not depend on that section, so the principal risk remains Assumptions 1–2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces continuous-variable analogues of quantum weight distributions and a corresponding MacWilliams identity (Theorem 4) for pairs of trace-class operators. It uses these identities to derive a continuous-variable quantum Cohn-Elkies bound (Theorem 1), a quantum Levenshtein sphere-packing bound (Theorem 2), and, conditional on two numerical assumptions, bounds that match the parameters of ideal E8- and Leech-lattice GKP codes (Theorem 3). The paper also defines an approximate quantum error detection code (QEDC) with parameter epsilon, derives a relation between QEDC quality and the weight distributions (Lemma 1), gives an occupation-number bound, and presents examples including coherent, Fock, cat, and ideal GKP codes.","tokens_in":25369,"tokens_out":11788,"duration_ms":98490,"significance":"The core derivations of Theorems 1 and 2 are clean and parameter-free: they follow from the new MacWilliams identity, the QEDC ratio condition, and known properties of the Levenshtein auxiliary functions. If Theorem 3's assumptions hold, the result would be notable because it extends optimality of E8 and Leech sphere packings to a statement about all continuous-variable quantum codes, not just lattice-based GKP codes. The approximate-QEDC definition is operational and the paper is honest about the conditional status of the main optimality claim, though the conclusion section overstates it. The significance is therefore contingent on the validity of Assumptions 1 and 2 and on the correctness of the GKP normalization in Section 3.2.4.","major_comments":[{"comment":"The paper's central advertised result, the optimality of E8 and Leech GKP codes, is proved only under Assumptions 1 and 2, which the manuscript explicitly states it does not prove and for which only Fig. 3 is supplied. These assumptions assert that the quotients in Eqs. (73) and (74) attain their suprema at x = 0 for all d up to d_max; if an interior maximum ever exceeds the value at x = 0, the Theorem 3 bound becomes weaker than the E8/Leech code point and the optimality conclusion does not follow. Section 7 nonetheless states that 'we have shown that the distances achieved by ideal GKP codes based on the E8 and Leech lattices cannot be exceeded by any physical construction,' which is stronger than what the conditional proof establishes. Please provide a proof of the assumptions, supply certified numerical verification (e.g., interval arithmetic) over the full d-ranges, or reclassify Theorem 3 as a conjecture and adjust the abstract and conclusion accordingly.","section":"Section 6.2, Assumptions 1-2 and Theorem 3"},{"comment":"The formal derivation of the MacWilliams identity for ideal GKP codes contains normalization problems. With the Fourier convention of Eq. (5), Poisson summation for the symplectic dual lattice L^\\perp gives a coefficient involving det L in the stated form, not |det L/sqrt(2pi)|^{1/2} as printed; the printed quantity is dimensionally inconsistent. In addition, the relation K^2 = |det L/sqrt(2pi)| does not match the standard GKP parameter relation: for the square GKP qubit L = 2 sqrt(pi) Z^2, one has det L = 4 pi and K = 2, so K = det L/(2 pi)^N, whereas K^2 = |det L/sqrt(2pi)| would give an incorrect value. As a result, the identity (21) does not follow for A and B as defined in Eqs. (37)-(38) without a correction of the normalization. Since Section 3.2.4 is presented as deriving the MacWilliams identities for ideal GKP codes, this needs to be fixed or the section explicitly restricted; the main upper-bound theorems do not depend on this section, but the claimed scope of the framework does.","section":"Section 3.2.4, Eqs. (37)-(40)"}],"minor_comments":[{"comment":"The word 'emply' should be 'employ', and the sentence 'the quantity in parenthesis must be equal to B' should clarify that the equality holds in the sense of distributions after testing against arbitrary rapidly decaying functions.","section":"Section 3.2.4, Eq. (40)"},{"comment":"The plots do not label which d values correspond to which curves, nor do they indicate how d_max is determined; the statement that the quotient 'visibly achieves' a maximum is not quantitative. Please add a legend and, ideally, a table of computed suprema or certified numerical bounds.","section":"Figure 3"},{"comment":"The phrase 'considered by Van Lint one of the most fundamental results' is grammatically awkward, and reference [11] in the introduction should be attributed to Cohn et al. rather than to Cohn alone.","section":"Introduction"},{"comment":"The abstract says the paper 'argue[s]' optimality, while Section 7 says 'we have shown'; these two statements should be made consistent given that Theorem 3 is conditional on Assumptions 1 and 2.","section":"Abstract and Section 7"},{"comment":"The first inequality in the proof chain requires the assumed non-negativity of \\hat f; this is stated in Theorem 1, but it would be helpful to restate it in the proof for readability.","section":"Eq. (65)"}],"recommendation":"major_revision","confidential_remarks":"This is a competently written manuscript with a clean core derivation of the MacWilliams identity and the resulting bounds. The main issue is that Theorem 3, the paper's headline optimality claim, is conditional on unproved numerical assumptions, and the GKP weight-distribution section has a normalization inconsistency that should be checked against Ref. [20]. If the authors can provide a proof or certified numerics for Assumptions 1-2 and correct the GKP normalization, the paper would be a strong contribution. My recommendation of major_revision reflects that the assumptions are load-bearing rather than cosmetic; the paper may be acceptable as a conditional result if the claims are carefully reworded, but in its current form the conclusion overstates what is established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the quant-ph paper by Burchards on CV quantum MacWilliams identities. Here's my take. The continuous-variable quantum MacWilliams identity (Theorem 4) and the resulting quantum Cohn-Elkies bound (Theorem 1) are genuinely new and well derived. The identity relating A and B through the Bessel kernel is clean, and the proof via Fourier decomposition of displacement operators is solid. The Levenshtein-type bound (Theorem 2) is a useful concrete application, with a real proof of the needed sup-at-zero lemma (Appendix D). The examples—coherent, Fock, cat, and GKP codes—are illustrative and check out.\n\nThe paper is also honest about its main limitation. The headline optimality of E8 and Leech GKP codes (Theorem 3) is conditional on Assumptions 1 and 2, which assert that certain quotients of the magic functions achieve their suprema at x=0 for all d up to d_max. These are only verified numerically in Fig. 3, no proofs. If either quotient develops an interior maximum above the x=0 value at some d in range, the bound weakens and the optimality conclusion disappears. The stress-test note correctly identifies this as load-bearing. I don't think it's a fatal flaw—the assumptions are explicit, the numerics look convincing, and the claim is advertised as conditional—but a referee should push for either a proof (or at least rigorous interval arithmetic) or a downgrade of the claim to 'suggests'.\n\nA secondary gap is in Section 3.2.4, where the GKP weight distributions (Eqs. 37-38) rely on a Poisson summation and the relation K^2=|det L/√(2π)| that isn't derived. It's normalization-sensitive, but Theorem 3 doesn't depend on this section, so it's a clarity issue rather than a load-bearing one.\n\nOverall, the core machinery is likely correct and will be useful for CV quantum coding. The citation pattern is appropriate—the relevant classical Cohn-Elkies, Viazovska, and lattice-code literature is all there. I'd send this to peer review with a request to address the assumptions. The paper deserves a serious referee.","headline":"New CV MacWilliams identities and a clean quantum Cohn-Elkies bound, but the E8/Leech optimality claim rests on two unproved numerical assumptions.","tokens_in":25876,"tokens_out":1881,"would_cite":true,"duration_ms":17391,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P70","94B65","11H31"],"pacs":["03.67.Pp"],"model":"deepseek-v4-flash","headline":"New continuous-variable MacWilliams identities bound all quantum codes against displacement noise and put E8 and Leech GKP codes at the optimal distance.","keywords":["continuous-variable quantum error correction","MacWilliams identities","weight enumerators","displacement noise channel","Cohn-Elkies bound","Levenshtein bound","GKP codes","sphere packing"],"falsifier":"Evaluate $f_8(\\sqrt2 x)/\\hat f_8(d^2 x/\\sqrt2)$ and $f_{24}(2x)/\\hat f_{24}(d^2 x/2)$ on $[0,1]$ with rigorous interval arithmetic for $d\\le d^{(8)}_{\\max}$ and $d\\le d^{(24)}_{\\max}$; a single point where either quotient exceeds $(2\\pi)^4$ or $(2\\pi)^{12}$, respectively, disproves Assumption 1 or 2 and collapses Theorem 3. Separately, computing both sides of the GKP MacWilliams identity (37)--(38) for the square GKP code with the stated normalization $K^2=|\\det L/\\sqrt{2\\pi}|$ would test the Poisson-summation step.","tokens_in":24834,"feed_emoji":"⚛️","tokens_out":8007,"duration_ms":66092,"temperature":0.7,"pith_summary":"This paper develops a continuous-variable analogue of the quantum MacWilliams identities and uses it to bound the parameters of any quantum error-correcting code that protects against displacement noise. For any two trace-class operators on $N$ modes it defines a primary weight distribution $A(r)$ and a dual weight distribution $B(r)$, and proves they are related by an invertible Bessel integral transform. Applied to a code projector, the identities turn the quantum error-detection conditions into the statement that $A(r)/(K B(r))$ stays near $1$ for all displacements shorter than the code distance. From this the paper derives a quantum version of the Cohn-Elkies sphere-packing bound, a quantum Levenshtein bound, and, conditional on two numerical assumptions, the conclusion that ideal $E_8$ and Leech-lattice GKP codes attain the largest possible distance among all codes, not just lattice codes. If correct, this would mean no physical bosonic code construction can beat the best known lattice GKP codes against displacement noise.","feed_headline":"New bound: no ideal code beats E8 and Leech GKP distances","feed_subtitle":"New continuous-variable MacWilliams identities cap displacement-noise code sizes, matching the best lattice constructions.","key_machinery":"The load-bearing object is the continuous-variable weight-distribution pair $(A,B)$, together with the Bessel integral kernel $r^N J_{N-1}(rx)/x^{N-1}$ that implements the radial Fourier transform between them. For projectors onto code subspaces, the quantum error-correction conditions are exactly the statement that $A(r)=K B(r)$ for all $r$ below the code distance, so the MacWilliams identity lets one translate constraints on $A$ into constraints on $B$. The bounds then arise by inserting admissible radial test functions $f$ whose Fourier transforms change sign at the distance $d$, exactly as in the classical sphere-packing linear program.","core_discovery":"The central result is Theorem 1, a continuous-variable quantum Cohn-Elkies bound: any $[[N,K,d,\\epsilon]]$ approximate error-detecting code satisfies $K \\le \\frac{1}{1-\\epsilon}\\sup_{x\\in[0,d]} f(x)/\\hat f(x)$ for every bounded non-negative radial function whose Fourier transform is non-negative below $d$ and non-positive above $d$. The bound follows from the paper's main technical tool, Theorem 4: for any trace-class operators $\\hat O_1,\\hat O_2$, the dual weight distribution is the Bessel-kernel integral transform $B(r)=r^N\\int_0^\\infty dx\\, J_{N-1}(rx) x^{1-N} A(x)$ of the primary distribution, an involutive relation that plays the role of the Krawtchouk transform in classical coding theory. Specializing to the magic functions that solve the 8- and 24-dimensional sphere-packing problems gives the paper's headline application: conditional on two unproved supremum assumptions, any $[[4,K,d,\\epsilon]]$-QEDC obeys $K d^8\\le(4\\pi)^4/(1-\\epsilon)$ for $d\\le 3.4286$ and any $[[12,K,d,\\epsilon]]$-QEDC obeys $K d^{24}\\le(8\\pi)^{12}/(1-\\epsilon)$ for $d\\le 4.9193$, so ideal $E_8$- and Leech-based GKP codes are optimal in distance among all codes.","pith_inferences":["A rigorous proof of Assumptions 1 and 2, for example through interval arithmetic or analytic bounds on the magic functions, would upgrade the optimality of $E_8$ and Leech GKP codes from conditional to unconditional; the paper leaves this as the main open step.","The same Bessel-kernel MacWilliams identity applies to arbitrary trace-class operators, so it may provide a new invariant for continuous-variable states beyond codes, such as a quantitative description of how much of a state's characteristic function sits at each displacement length.","Because the bounds are not tight for approximate $\\epsilon>0$ codes, the paper leaves open the possibility of finite-energy constructions that beat the naive envelope-based GKP codes while approaching the ideal-distance limit.","Testing the normalization $K^2=|\\det L/\\sqrt{2\\pi}|$ in the Poisson summation step against a direct computation for square or hexagonal GKP codes would confirm whether the formal identity extends beyond the stabilizer-code intuition."],"forward_implications":["Every $[[N,K,d,\\epsilon]]$-QEDC automatically satisfies the quantum Cohn-Elkies inequality, giving a universal upper bound on logical dimension versus distance and noise quality.","The quantum Levenshtein bound shows that for a fixed logical dimension $K$, the attainable distance grows at most as $O(\\sqrt N)$ in the number of modes.","Under Assumptions 1 and 2, no ideal code on 4 or 12 modes can exceed the distances of the $E_8$ and Leech GKP codes; any $K\\ge2$ code with larger distance must be approximate with $\\epsilon>0$.","Approximate finite-energy GKP codes inherit a distance of roughly half the ideal code's minimum dual-lattice length, with error quality $\\epsilon$ exponentially small in $1/\\Delta^2$; as $\\epsilon\\to0$ the distance does not approach the ideal value continuously.","The weight distributions of any code projector below its distance saturate the inequality $A\\le K B$, so the MacWilliams identity gives a necessary and sufficient condition for approximate error detection in terms of weight distributions."],"supporting_citations":[{"why":"Supplies the discrete-variable quantum weight distributions and MacWilliams identities that this paper generalizes to continuous variables.","marker":"[4]"},{"why":"Supplies the classical Cohn-Elkies linear-programming bound, the admissible-function conditions, and the Levenshtein functions used in Theorem 1 and Theorem 2.","marker":"[8]"},{"why":"Supplies the $E_8$ magic function $f_8$ used in Theorem 3 and Assumption 1.","marker":"[10]"},{"why":"Supplies the Leech-lattice magic function $f_{24}$ used in Theorem 3 and Assumption 2.","marker":"[37]"},{"why":"Provides the lattice formulation of GKP codes and the relation $K^2=|\\det L/\\sqrt{2\\pi}|$ used in the formal Poisson summation for GKP weight distributions.","marker":"[20]"},{"why":"Establishes the quantum error-correction conditions that connect code distance to the weight distributions.","marker":"[32]"},{"why":"States the classical Levenshtein sphere-packing bound whose quantum version is derived as Theorem 2.","marker":"[35]"},{"why":"Provides the Poisson summation and theta-function background used to derive the MacWilliams identity for GKP codes.","marker":"[29]"},{"why":"Defines ideal GKP codes, the objects whose optimality Theorem 3 addresses.","marker":"[26]"}],"fun_headline_variants":["Quantum Cohn–Elkies bound: E8 and Leech GKP optimal","No displacement-noise code beats E8 or Leech GKP distances","Continuous-variable MacWilliams identities cap code sizes","GKP codes on E8 and Leech achieve optimal distances"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the quotient suprema in Assumptions 1 and 2 really occur at $x=0$ throughout the stated distance ranges, a fact the paper checks only by plotting, together with a formally applied Poisson summation whose normalization constant $K^2=|\\det L/\\sqrt{2\\pi}|$ is not derived.","fun_headline_variants_meta":{"raw":{"variants":["Quantum Cohn–Elkies bound: E8 and Leech GKP optimal","No displacement-noise code beats E8 or Leech GKP distances","Continuous-variable MacWilliams identities cap code sizes","GKP codes on E8 and Leech achieve optimal distances"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000565,"raw_usage":{"total_tokens":2708,"prompt_tokens":1001,"completion_tokens":1707,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":617,"completion_tokens_details":{"reasoning_tokens":1633}},"tokens_in":617,"tokens_out":1707,"duration_ms":16005,"temperature":1.0,"reasoning_tokens":1633,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:13:25.132339+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $f_8(\\sqrt2 x)/\\hat f_8(d^2 x/\\sqrt2)$ and $f_{24}(2x)/\\hat f_{24}(d^2 x/2)$ on $[0,1]$ with rigorous interval arithmetic for $d\\le d^{(8)}_{\\max}$ and $d\\le d^{(24)}_{\\max}$; a single point where either quotient exceeds $(2\\pi)^4$ or $(2\\pi)^{12}$, respectively, disproves Assumption 1 or 2 and collapses Theorem 3. Separately, computing both sides of the GKP MacWilliams identity (37)--(38) for the square GKP code with the stated normalization $K^2=|\\det L/\\sqrt{2\\pi}|$ would test the Poisson-summation step.","supporting_citations":[{"cited_title":"Quantum Analog of the MacWilliams Identities for Classical Coding Theory","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-variable quantum weight distributions and MacWilliams identities that this paper generalizes to continuous variables."},{"cited_title":"New Upper Bounds on Sphere Packings I","cited_arxiv_id":null,"evidence_quote":"Supplies the classical Cohn-Elkies linear-programming bound, the admissible-function conditions, and the Levenshtein functions used in Theorem 1 and Theorem 2."},{"cited_title":"The Sphere Packing Problem in Dimension 8","cited_arxiv_id":null,"evidence_quote":"Supplies the $E_8$ magic function $f_8$ used in Theorem 3 and Assumption 1."},{"cited_title":"The Sphere Packing Problem in Dimension 24","cited_arxiv_id":null,"evidence_quote":"Supplies the Leech-lattice magic function $f_{24}$ used in Theorem 3 and Assumption 2."},{"cited_title":"Gottesman-Kitaev-Preskill Codes: A Lattice Perspective","cited_arxiv_id":null,"evidence_quote":"Provides the lattice formulation of GKP codes and the relation $K^2=|\\det L/\\sqrt{2\\pi}|$ used in the formal Poisson summation for GKP weight distributions."},{"cited_title":"A Theory of Quantum Error-Correcting Codes","cited_arxiv_id":null,"evidence_quote":"Establishes the quantum error-correction conditions that connect code distance to the weight distributions."},{"cited_title":"On Bounds for Packings in N-Dimensional Euclidean Space","cited_arxiv_id":null,"evidence_quote":"States the classical Levenshtein sphere-packing bound whose quantum version is derived as Theorem 2."},{"cited_title":"Theta Functions and Weighted Theta Functions of Euclidean Lattices, with Some Applications","cited_arxiv_id":null,"evidence_quote":"Provides the Poisson summation and theta-function background used to derive the MacWilliams identity for GKP codes."},{"cited_title":"Encoding a Qubit in an Oscillator","cited_arxiv_id":null,"evidence_quote":"Defines ideal GKP codes, the objects whose optimality Theorem 3 addresses."}],"review_version":1}