Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables

Avijit Mandal; Henry D. Pfister

REVIEW 3 major objections 5 minor 29 references

Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2502.03785 v3 pith:T5P5TS45 submitted 2025-02-06 cs.IT math.ITquant-ph

Avijit Mandal , Henry D. Pfister This is my paper · ORCID

classification cs.ITmath.ITquant-ph MSC 94B0594A1781P45

keywords Reed-Mullercodesclassical-quantumchannelsHolevocapacityminimummean-squarederrorobservablecorrelationboundforquantumobservablesGelfand-Naimark-SegalinnerproductEXITareatheoremvanishingbit-errorprobability

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Who 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Appendix A.XI (proof of Theorem 14)] 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.
[Lemma 13 and Section V] 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.
[Theorem 1 (Informal) and Theorem 14] 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.

minor comments (5)

[Appendix A.II (proof of Lemma 7)] 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.
[Appendix D] 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)).
[Theorem 14, rate bound] 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.
[Abstract and Section I] The notation '2o(√ log N)' in the abstract and Theorem 1 is a typographical artifact; it should be '2^{o(\sqrt{\log N})}'.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: the core recursion and correlation bound are derived in the appendices; the questionable linear MMSE bound is an unproved gap, not a constructional identity or a fitted prediction.

full rationale

The derivation chain is not circular. The central recursive bound (Lemma 11) is proved in Appendix A.VI from the GNS orthogonal decomposition and the transitive-symmetry correlation bound (Lemma 7); it is parameter-free, does not fit any constant to data, and its proof is present in the manuscript. The EXIT-area statement used in Lemma 4 is derived in Appendix A.VII from the entropy chain rule, channel erasure structure, and transitive symmetry, so the later invocation of the EXIT area theorem in Lemma 5 is supported by the paper's own argument rather than by an unverified self-citation. The RM symmetry and nesting properties in Proposition 2 are standard facts about Reed-Muller codes and are not used to smuggle in the target result. The one genuinely weak point is Appendix A.XI, which states: 'based on the EXIT area theorem the following bound holds M(C0) = M(X0|Y∼0)ρ... ≤ 1 − (C −R(C0)) = 1 −δ.' This linear MMSE bound is not derived in the paper: Lemma 5 provides an entropy gap H(X0|Y∼0) ≤ 1 − (C−R), and Lemma 12 converts an entropy gap into at best a quadratic MMSE gap, so the combined lemmas support M(C0) ≲ 1 − (C−R)^2, not the stated linear bound. That is a soundness gap affecting the explicit constants in Theorem 14, but it is not circular: the theorem's conclusion is not equivalent by construction to this bound, no fitted parameter is renamed as a prediction, and the qualitative vanishing-error claim would survive a quadratic-gap correction. The self-citations to prior Reeves-Pfister work are used for standard RM properties and as an inspiration for the two-look recursion, but the recursion itself is re-derived for the quantum setting in the appendices. Therefore, under the requested circularity criteria, no step reduces to its own inputs by definition or by a load-bearing self-citation chain.

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

The analysis introduces no new physical entities and no fitted free parameters. The main unstated input is the validity of the classical framework of [13] and, more critically, the unproved MMSE-EXIT inequality used in Theorem 14.

assumptions (4)

ad hoc to paper The initial extrinsic MMSE satisfies M(C_0) <= 1 - (C - R(C_0)).
Asserted in Appendix A.XI without proof; Lemma 12 gives only the weaker 1 - (C - R(C_0))^2, so this is a load-bearing unproved premise.
domain assumption The MMSE observable can be chosen invariant under the code automorphism group stabilizing bit 0.
Needed to apply Lemma 7's transitive-symmetry hypothesis to M_AB. The paper does not explicitly prove that the minimizer can be symmetrized over the stabilizer.
domain assumption Reed-Muller nesting: C_|{0} union A union B equals RM(r, m-1) and the BSCQ channel output tensorizes.
This is Proposition 2c and a standard property of RM codes; it underpins the recursion between code lengths.
domain assumption Background results from reference [13] (code symmetry and nesting framework, EXIT area theorem).
The paper is explicitly built on [13], which is cited as 'To appear on arXiv, 2025'. Some framework results are used without re-derivation.

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Pith. "Pith review of Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables." pith.science (2026). https://pith.science/paper/T5P5TS45

@misc{pith2026250203785,
  author       = {Pith},
  title        = {Pith review of: Reed-Muller Codes on CQ Channels via a New Correlation Bound for Quantum Observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T5P5TS45}},
  note         = {Machine review of arXiv:2502.03785}
}

abstract

The question of whether Reed--Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels has drawn attention since it was resolved positively for the binary erasure channel by Kudekar et al.\ in 2016. In 2021, Reeves and Pfister extended this to prove the bit-error probability vanishes on BMS channels when the code rate is less than capacity. In 2023, Abbe and Sandon improved this to show the block-error probability also goes to zero. These results rely on the symmetry and nested structure of RM codes. In this work, we focus on binary-input symmetric classical-quantum (BSCQ) channels and the Holevo capacity. For a BSCQ, we consider observables that estimate the channel input in the sense of minimizing the mean-squared error (MSE). Using an orthogonal decomposition of minimum MSE (MMSE) observables under a weighted inner product, we derive a recursion for the extrinsic MMSE of a code bit. Consequently, for Reed--Muller code sequences whose rates remain below the Holevo capacity by a fixed positive amount, any prescribed set of $2^{o(\sqrt{\log N})}$ bits can be decoded sequentially with the probability of any error tending to zero.

Figures

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H. D. Pﬁster and G. Reeves, Capacity on BMS channels via code symmetry and nesting , To appear on arXiv, 2025

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On the performance of Reed-Muller codes with respect to random errors and erasures,

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