{"id":"1f232997-4ea0-46e1-b1b8-aad18bd9a5a3","arxiv_id":"2502.03785","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Reed-Muller codes achieve vanishing bit-error probability below Holevo capacity on binary-input symmetric classical-quantum channels.","lead":"Reed-Muller codes are shown to have vanishing bit-error probability when used over a class of quantum channels called binary-input symmetric classical-quantum channels, as long as the code rate stays below the Holevo capacity. The proof introduces a new correlation bound for quantum observables and a recursive analysis of minimum mean-squared error estimators.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 14's stated (7/8)^{k - ceil(3/delta)} rate relies on an unproved linear MMSE bound; only a quadratic bound follows from the paper's Lemmas 5 and 12.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: the linear bound on the initial MMSE in the proof of Theorem 14 is asserted without derivation, and the paper's own lemmas supply only a quadratic bound. I checked the surrounding argument to see whether this gap is fatal to the central qualitative claim. The recursive Lemma 11 and the correlation bound Lemma 7 appear internally consistent; the issues with kappa_k not being constant in Lemma 13 are easily patched because kappa_k remains below 1/2, making the kappa0 = 1/2 choice conservative. The sequential-decoding claim via the quantum union bound is plausible from the per-bit bound alone. The only serious defect is the initial linear MMSE bound: replacing it with the quadratic bound changes the decay rate from (7/8)^(k - O(1/delta)) to (7/8)^(k - O(1/delta^2)), but still yields exponential decay in k. Consequently, the informal capacity-approaching result survives, while the formal Theorem 14 requires correction. The verdict CONDITIONAL is therefore appropriate, and my read does not change it.","tokens_in":19438,"tokens_out":23232,"duration_ms":209391,"concrete_test":"Re-derive the initial MMSE bound from the stated lemmas: set delta = C - R(C0). Lemma 5 gives H(X0|Y~0) <= 1 - delta. Applying Lemma 12 with delta' = delta ln2 yields M(C0) <= 1 - (delta ln2)^2. Propagate this quadratic gap through the proof of Lemma 13 (with kappa0 = 1/2) and the induction in Theorem 14. Check whether the resulting bound is P_b(C_k) <= (1/4)(7/8)^(k - O(1/delta^2)). If yes, the theorem's stated (7/8)^(k - ceil(3/delta)) is false and must be corrected; if a direct proof of the linear bound M(C0) <= 1 - delta can be supplied, the original statement survives.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix A.XI asserts M(C0) <= 1 - (C - R(C0)) 'based on the EXIT area theorem'. The paper's own Lemma 5 only supplies H(X0|Y~0) <= 1 - (C - R(C0)). Lemma 12 converts an entropy gap u into M <= 1 - u^2 when u = delta/ln2. Combining these gives at best M(C0) <= 1 - ((C-R(C0)) ln2)^2, not the linear bound used in Theorem 14. Since Lemma 13's recursion multiplies the gap by (1-kappa0)/2 each level, the threshold k0 = ceil(3/delta) in Theorem 14 is an order of magnitude too small (delta vs delta^2). Thus the formal error bound P_b(C_k) <= (1/4)(7/8)^(k - ceil(3/delta)) does not follow from the stated lemmas. The qualitative claim - vanishing bit-error probability below capacity - is likely recoverable with the quadratic gap, because the recursion still yields exponential decay in k, and choosing k = O(log L) = o(sqrt(m)) still supports the 2^{o(sqrt(log N))} sequential-decoding statement. However, Theorem 14's explicit constants and rate are not proven. A secondary, non-fatal issue: Lemma 13 assumes kappa_k = kappa_0, but in the nested RM sequence kappa_k = (2^{m+k-2}-1)/(2^{m+k-1}-1) < 1/2 and increases toward 1/2; this is conservative when using kappa0 = 1/2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that for binary-input symmetric classical-quantum (BSCQ) channels, sequences of Reed-Muller codes whose rates stay a fixed positive amount below the Holevo capacity have vanishing bit-error probability under sequential decoding of a prescribed set of 2^{o(sqrt(log N))} bits. The technical core is a new correlation bound for quantum observables with transitive symmetry, obtained via the GNS inner product (Lemma 7), and an MMSE recursion for the extrinsic estimate of a single code bit (Lemma 11). These lead to Lemma 13, which gives exponential decay of the extrinsic MMSE along the nested sequence C_k = RM(r, m+k), and to Theorem 14, which states the explicit bit-error bound P_b(C_k) <= (1/4)(7/8)^{k - ceil(3/delta)} whenever R(C_0) <= C - delta. The main formal result is thus Theorem 14, with the abstract's sequential-decoding statement following by a union bound over the prescribed bits.","tokens_in":19758,"tokens_out":11022,"duration_ms":103733,"significance":"If the main theorem is correct, the paper would extend the classical Reed-Muller vanishing-bit-error result of Reeves and Pfister to quantum channels, using only the Holevo capacity rather than a capacity definition tied to classical output processing. The proposed MMSE observable framework for binary hypothesis testing and the correlation bound Lemma 7 are new and potentially useful beyond RM codes. The paper is refreshingly parameter-free: no constants are fit to data, and the main recursion is derived rather than assumed. However, the proof of Theorem 14 contains a load-bearing unsupported inequality, and the rate of the nested sequence is not sufficient for the informal capacity-approaching statement in Theorem 1. The qualitative claim that bit error vanishes for rates below capacity is plausible and likely recoverable, but the explicit bounds as stated are not established by the supplied lemmas.","major_comments":[{"comment":"The proof asserts 'based on the EXIT area theorem' that M(C_0) <= 1 - (C - R(C_0)) = 1 - delta. This linear bound is not derived anywhere and is inconsistent with the paper's own Lemmas 5 and 12. Lemma 5 gives H(X_0|Y_{\\sim 0}) <= 1 - delta (with delta = C - R(C_0)), while Lemma 12's final implication is: if the conditional entropy is at most 1 - delta/ln 2, then the MMSE is at most 1 - delta^2. Applying Lemma 5 to Lemma 12 with delta' = delta ln 2 yields only M(C_0) <= 1 - (C - R(C_0))^2 (ln 2)^2, not the linear bound used in A.XI. Appendix B also concludes a quadratic bound, M <= 1 - (1 - H)^2. Since Lemma 13's contraction rate multiplies the gap by (1 - kappa_0)/2 at each level, the threshold k_0 = ceil(3/delta) and the decay rate (7/8)^{k - ceil(3/delta)} in Theorem 14 do not follow from the stated lemmas; a quadratic gap would change k_0 to order 1/delta^2. The qualitative vanishing bit-error statement may survive this repair, but Theorem 14 as written is not proven.","section":"Appendix A.XI (proof of Theorem 14)"},{"comment":"Lemma 13 assumes kappa_k = kappa_0 for all k, but for the nested sequence C_k = RM(r, m+k) the parameter kappa_k = |A_k|/(|A_k| + |B_k|) is not constant: writing the partition as in Proposition 2(c), kappa_k = (2^{m+k-2} - 1)/(2^{m+k-1} - 1), which increases toward 1/2 as k grows. The proof as written therefore does not directly apply to the stated sequence. The lemma is repairable because kappa_k <= 1/2 and the recursion of Lemma 11 is monotone increasing in kappa for M(C_k) < 1, so using kappa_0 = 1/2 is conservative; however, the current statement and proof should be revised to state and use this monotonicity explicitly.","section":"Lemma 13 and Section V"},{"comment":"The informal Theorem 1 claims a sequence RM(r_m, m) with rate converging to C - eta and bit-error probability at most e^{-c eta sqrt(m)}. Theorem 14, however, concerns the fixed-order nested sequence C_k = RM(r, m+k) with r fixed. For fixed r, the rate R(C_k) tends to zero as k grows (roughly like 2^{-k} times a polynomial in m+k), so this sequence does not have rates converging to a positive constant C - eta. To obtain a capacity-approaching sequence one would need r to grow with the blocklength, which is not covered by Theorem 14. The paper does not explain how Theorem 14 implies the informal theorem's rate statement, nor how the exponent sqrt(m) in the informal bound is obtained from the (7/8)^k decay with k = O(sqrt(m)) while preserving the stipulated rate gap.","section":"Theorem 1 (Informal) and Theorem 14"}],"minor_comments":[{"comment":"In the proof, the shorthand 'S_AC := S_{A \\cup B}' should read 'S_AC := S_{A \\cup C}', since the supports of F and G are A \\cup B and A \\cup C respectively.","section":"Appendix A.II (proof of Lemma 7)"},{"comment":"The displayed inequalities read 'M(C) \\le 4P_b(C)(1 - 4P_b(C))' and similarly for M(C'); by Lemma 10 the second factor should be (1 - P_b(C)), not (1 - 4P_b(C)).","section":"Appendix D"},{"comment":"The assertion R(C_k) >= R(C_0) - k/(2 sqrt(m)) is used without proof or reference. It is plausible from the single-step drop R(r,m+1) - R(r,m) = -(1/2) binom(m,r)/2^m plus a bound on the central binomial coefficient, but the derivation should be included.","section":"Theorem 14, rate bound"},{"comment":"The notation '2o(√ log N)' in the abstract and Theorem 1 is a typographical artifact; it should be '2^{o(\\sqrt{\\log N})}'.","section":"Abstract and Section I"},{"comment":"Some subscript/grouping ambiguities occur, e.g., '\\rho_0' versus '\\rho^0' and '\\rho_{Y_{\\sim 0}}' versus '\\rho_{Y_{\\sim 0}}^{0}'. A consistent convention for the all-zero codeword background state would improve readability.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cs.IT and the core idea is promising. The main obstruction is the unsupported linear MMSE bound in the proof of Theorem 14; this is a local but load-bearing gap. I believe the authors can repair it by replacing the linear bound with the quadratic bound and adjusting k_0 and the decay-rate statement, while preserving the qualitative result. The rate-to-capacity gap in the informal Theorem 1 is more than a wording issue and should be addressed either by proving a corresponding statement with growing r or by explicitly restricting the claims to fixed-order nested sequences whose rates stay below capacity. I would be willing to look at a revision that fixes these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth engaging: it takes the Reeves–Pfister framework for Reed–Muller codes and pushes it into binary-input symmetric classical-quantum channels, and it introduces a genuinely new tool along the way. Lemma 7, the correlation bound for observables under the GNS inner product, is the real contribution. The proof is transparent, the symmetry argument is clean, and it appears correct. The recursive MMSE relation (Lemma 11) and the entropy-to-MMSE conversion (Lemma 12) are also carefully derived. The paper builds explicitly on [13], and the citations are appropriate; there is no parameter fitting and no circularity that I can see.\n\nThe soft spot is real, and it is exactly where the stress-test points. Theorem 14's error bound uses the claim, stated in Appendix A.XI, that M(C0) <= 1 - (C - R(C0)). That linear inequality is not proven in the paper. Lemma 5 gives H(X0|Y~0) <= 1 - (C-R), and Lemma 12 converts an entropy gap of delta/ln 2 into M <= 1 - delta^2. Combining these gives only M(C0) <= 1 - ((C-R) ln2)^2, a quadratic gap. The threshold k0 = ceil(3/delta) in Theorem 14 is therefore too small by an order of magnitude, and the stated (7/8)^{k - ceil(3/delta)} decay does not follow from the preceding lemmas. This is a load-bearing gap in the formal statement.\n\nThe good news is that the qualitative result is probably salvageable. Lemma 13's recursion multiplies the gap by (1-kappa0)/2 each level, so a quadratic gap still gives exponential decay in k, just with delta^2 instead of delta. Since the proof only needs k = O(log L) = o(sqrt(m)), the 2^{o(sqrt(log N))} sequential-decoding statement should survive a correction. There is also a minor mismatch between the informal Theorem 1, which says the rate converges to C - eta, and the fixed-degree RM sequence in Theorem 14, where the rate drifts downward as k grows; the authors should state a proper sequence of codes or clarify the sense in which capacity is approached. Lemma 13's assumption kappa_k = kappa_0 is not exactly true but is conservative, so that is a minor issue.\n\nWho should read this: quantum coding theorists, and anyone following the RM-capacity line. It deserves a serious referee, but not acceptance as-is. I would ask the authors to either prove the linear MMSE bound or reformulate Theorem 14 with the quadratic gap; the core technique is solid enough that the effort is worthwhile.","headline":"A genuinely new correlation bound plus a plausible RM-on-CQ result, but Theorem 14's explicit decay relies on an unproved linear MMSE inequality; the qualitative vanishing-error claim likely survives with a quadratic correction.","tokens_in":20295,"tokens_out":3511,"would_cite":true,"duration_ms":31727,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94B05","94A17","81P45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Reed-Muller code sequences are shown to have vanishing bit-error probability on any binary-input symmetric classical-quantum channel whose rate is below the Holevo capacity.","keywords":["Reed-Muller codes","classical-quantum channels","Holevo capacity","minimum mean-squared error observable","correlation bound for quantum observables","Gelfand-Naimark-Segal inner product","EXIT area theorem","vanishing bit-error probability"],"falsifier":"Choose a concrete BSCQ channel, e.g. uniform input with $\\rho_0=|0\\rangle\\langle 0|$ and $\\rho_1=|+\\rangle\\langle +|$, and for small $m$ directly compute the MMSE of bit 0 of $\\mathrm{RM}(r,m)$ from the extrinsic output. If that computed value exceeds $1-(C-R)$, the exponential rate claimed in Theorem 14 is false as stated; a quadratic gap would still allow vanishing error but with a slower rate.","tokens_in":19180,"feed_emoji":"⚛️","tokens_out":12002,"duration_ms":105477,"temperature":0.7,"pith_summary":"Reed–Muller (RM) codes are known to make bit-error probability vanish on classical binary memoryless symmetric channels below capacity; this paper extends that statement to binary-input symmetric classical–quantum (BSCQ) channels, where the relevant limit is the Holevo capacity—the maximum rate for reliable classical communication over the quantum channel. The main result is that for an RM code sequence whose rate stays below the Holevo capacity by a fixed positive gap, the bit-error probability of decoding any single bit with the optimal quantum measurement decays exponentially along the nested code sequence, and therefore any prescribed set of $2^{o(\\sqrt{\\log N})}$ bits can be decoded sequentially with total error tending to zero. The proof works by assigning a minimum mean-squared error (MMSE) observable to each code bit and deriving a recursive inequality that relates the MMSE of $\\mathrm{RM}(r,m)$ to the MMSE of two half-length projections $\\mathrm{RM}(r,m-1)$. A correlation bound for quantum observables with transitive symmetry carries the recursion, and the EXIT area theorem supplies the starting gap from capacity.","feed_headline":"Reed-Muller codes reach vanishing bit error on quantum channels","feed_subtitle":"Below the Holevo capacity, all but a vanishing set of codeword bits can be decoded reliably.","key_machinery":"The load-bearing object is the MMSE observable $M(C)$ for a code bit—the Hermitian operator on the extrinsic quantum output that minimizes mean-squared error in estimating the transmitted input bit—together with its orthogonal decomposition under the Gelfand–Naimark–Segal inner product $\\langle F,G\\rangle_\\rho = \\mathrm{Tr}(G^\\dagger \\rho F)$. The correlation bound (Lemma 7) states that if two observables are supported on overlapping blocks $A\\cup B$ and $A\\cup C$, and a coordinate permutation fixing $A$ carries one to the other while $F$'s symmetry group is transitive on $B$, then $\\langle F,G\\rangle_{\\rho^{\\otimes n}} \\le \\kappa \\|F\\|^2_{\\rho^{\\otimes n}} + (1-\\kappa)\\langle F,I\\rangle^2_{\\rho^{\\otimes n}}$ with $\\kappa = |A|/(|A|+|B|)$. Applied to the MMSE observables of the two half-size projections of $\\mathrm{RM}(r,m)$, this gives the recursion of Lemma 11, and the nested, doubly transitive structure of RM codes supplies the symmetry needed to run the recursion down the code sequence.","core_discovery":"On the paper's own terms, the central claim is Theorem 14: for a BSCQ channel $W$ with Holevo capacity $C$, the nested RM code sequence $C_k = \\mathrm{RM}(r, m+k)$ has rate at least $R(C_0) - k/(2\\sqrt{m})$, and if $R(C_0) \\le C - \\delta$ then the extrinsic bit-error probability satisfies $P_b(C_k) \\le \\frac14 (\\frac78)^{k - \\lceil 3/\\delta \\rceil}$. Since $R(C_k)$ can be kept below $C$ for a window of length proportional to $\\sqrt{m}$, the bit-error probability vanishes as $m$ grows, and the quantum union bound converts this single-bit statement into sequential decoding of any prescribed set of $2^{o(\\sqrt{\\log N})}$ positions. The route is: Lemma 10 bounds the MMSE of a bit above in terms of the Helstrom error probability; Lemma 11 gives the two-look recursion $M(C) \\le \\frac{1+\\kappa}{2} M(C') + \\frac{1-\\kappa}{2} M(C')^2$ for the two half-size projections $C' = \\mathrm{RM}(r,m-1)$; and the EXIT area lemma bounds the initial MMSE by $1-\\delta$. The paper's contribution is to make each of these steps work for quantum observables, with the correlation inequality as the new ingredient.","pith_inferences":["If the linear initial bound $M(C_0)\\le 1-(C-R(C_0))$ used at the base of Theorem 14 is weakened to the quadratic bound that Lemma 12 actually establishes, the recursion still gives vanishing error with a slower decay rate; the qualitative conclusion appears robust to this gap.","The correlation bound is stated for observables with transitive symmetry, so it should extend to other code families with the same nesting and double-transitivity, such as generalized Reed–Muller codes; the paper does not state this as a theorem.","Because the proof reduces the problem to a one-dimensional MMSE recursion, a direct computation of the base MMSE for small $m$ on a given BSCQ channel would translate Theorem 14 into finite-length error estimates, a testable extension the paper does not carry out.","The KMS inner-product variant noted in the paper suggests that the recursion is not an artifact of the GNS choice of inner product, so the technique may transfer to other noncommutative settings."],"forward_implications":["For any BSCQ channel, RM code sequences with rate a fixed amount below the Holevo capacity have single-bit error probability decaying like $(7/8)^{k}$ along the nested sequence, so the bit-error rate vanishes as the block length grows.","Any prescribed set of $2^{o(\\sqrt{\\log N})}$ positions can be decoded sequentially with total error probability tending to zero, because the quantum union bound makes the error accumulate additively over a sub-exponentially small set.","This extends the classical vanishing-bit-error result for RM codes on binary memoryless symmetric channels to quantum outputs, giving a quantum counterpart for symmetric classical-quantum channels.","The MMSE-to-Helstrom comparison means that reliable bitwise decoding on such channels can be certified by bounding the MMSE, a scalar quantity, rather than by constructing full block decoders.","Block-error probability is not settled: the theorem controls prescribed sets of bits, not the entire codeword simultaneously."],"supporting_citations":[{"why":"Establishes the vanishing bit-error result for RM codes on BMS channels whose symmetry and nesting properties the present proof reuses.","marker":"[11]"},{"why":"Supplies the two-look recursion and symmetry/nesting framework for BMS channels that this paper adapts to quantum observables.","marker":"[13]"},{"why":"Provides the Gelfand-Naimark-Segal inner product used for the orthogonal decomposition of observables.","marker":"[22]"},{"why":"Gives the quantum union bound that turns the single-bit error bound into sequential decoding of many bits.","marker":"[25]"},{"why":"Defines the Helstrom error probability and the optimal binary measurement that the MMSE is compared with.","marker":"[27]"},{"why":"Supplies the relation between Helstrom error probability and conditional min-entropy used in Lemma 12.","marker":"[28]"},{"why":"Fixes the mutual-information convention used in the EXIT-area integration lemma.","marker":"[21]"}],"fun_headline_variants":["RM codes achieve vanishing bit error on quantum channels below capacity","Reed-Muller codes beat quantum channels with a correlation bound","Sub-capacity Reed-Muller codes push error probability to zero","New correlation inequality unlocks RM decoding on quantum channels","RM codes hit zero bit error on quantum channels using a correlation bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit decay rate rests on an unproved initial bound saying that the minimum mean-squared error of decoding a single bit starts at most one minus the gap between capacity and code rate; only a weaker quadratic form of that bound is actually derived.","fun_headline_variants_meta":{"raw":{"variants":["RM codes achieve vanishing bit error on quantum channels below capacity","Reed-Muller codes beat quantum channels with a correlation bound","Sub-capacity Reed-Muller codes push error probability to zero","New correlation inequality unlocks RM decoding on quantum channels","RM codes hit zero bit error on quantum channels using a correlation bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000819,"raw_usage":{"total_tokens":3658,"prompt_tokens":1087,"completion_tokens":2571,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":703,"completion_tokens_details":{"reasoning_tokens":2502}},"tokens_in":703,"tokens_out":2571,"duration_ms":17140,"temperature":1.0,"reasoning_tokens":2502,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T00:47:58.556043+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a concrete BSCQ channel, e.g. uniform input with $\\rho_0=|0\\rangle\\langle 0|$ and $\\rho_1=|+\\rangle\\langle +|$, and for small $m$ directly compute the MMSE of bit 0 of $\\mathrm{RM}(r,m)$ from the extrinsic output. If that computed value exceeds $1-(C-R)$, the exponential rate claimed in Theorem 14 is false as stated; a quadratic gap would still allow vanishing error but with a slower rate.","supporting_citations":[{"cited_title":"Reed–Muller codes on BMS channels achieve vanishing bit-error probability for all rates below capacity,","cited_arxiv_id":null,"evidence_quote":"Establishes the vanishing bit-error result for RM codes on BMS channels whose symmetry and nesting properties the present proof reuses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-look recursion and symmetry/nesting framework for BMS channels that this paper adapts to quantum observables."},{"cited_title":"Arveson, An invitation to C*-algebras","cited_arxiv_id":null,"evidence_quote":"Provides the Gelfand-Naimark-Segal inner product used for the orthogonal decomposition of observables."},{"cited_title":"Quantum union bounds for sequential projec- tive measurements,","cited_arxiv_id":null,"evidence_quote":"Gives the quantum union bound that turns the single-bit error bound into sequential decoding of many bits."},{"cited_title":"Quantum detection and estimation theory,","cited_arxiv_id":null,"evidence_quote":"Defines the Helstrom error probability and the optimal binary measurement that the MMSE is compared with."},{"cited_title":"Tomamichel, Quantum information processing with finite resources: mathematical foundations","cited_arxiv_id":null,"evidence_quote":"Supplies the relation between Helstrom error probability and conditional min-entropy used in Lemma 12."}],"review_version":1}