Binary code rate bounds via classical--quantum channels

Omar Alrabiah; Venkatesan Guruswami

Binary code rate bounds via classical--quantum channels

T0 review · 0 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves a single "pretty good criterion" that recovers all four classical binary code rate bounds and yields strict improvements over both MRRW bounds.

desk verdict A genuine breakthrough: one quantum criterion recovers all four classic binary rate bounds and strictly improves both MRRW bounds, and the core proof holds up on inspection. read the letter →

arxiv 2608.09347 v1 pith:ITCGC6IS submitted 2026-08-10 cs.IT math.COmath.ITquant-ph

Omar Alrabiah , Venkatesan Guruswami This is my paper · ORCID

classification cs.ITmath.COmath.ITquant-ph MSC 94B6581P4594A17

keywords binarycodesrate-distanceboundsprettygoodmeasurementclassical-quantumchannelsMRRWHolevoinformationposteriorsamplingchannelmasking

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

The paper proposes a single information-theoretic criterion for upper-bounding the rate of binary error-correcting codes. It says that if a binary-input output-symmetric classical-quantum channel has pretty-good-measurement bit error rate below $\delta$, then every length-$n$ binary code of relative distance $\delta$ has rate at most the channel's Holevo capacity, up to an $O(n^{-1/2})$ correction. Asymptotically this gives $R_2(\delta)\le \chi(\sigma_0,\sigma_1)$. Choosing four familiar channels rediscovers the Plotkin, Elias-Bassalygo, and both MRRW bounds, while choosing two new mixed-state channels gives an upper bound strictly smaller than each MRRW bound for every $\delta\in(0,1/2)$. If correct, these are the first asymptotic improvements over the fifty-year-old MRRW benchmark, and the proof turns a hard combinatorial problem into a channel-design optimization.

What carries the argument

The pretty good measurement, the square-root quantum analog of posterior sampling, is the object that carries the argument. Its key structural lemma is PGM data processing: applying any quantum channel to the outputs cannot decrease the PGM bit error rate, a fact proved in the paper with a quantum recovery map. Together with closure under coarse-graining and the fact that the uniform prior maximizes PGM bit error, this yields the expected-Hamming-distance bound $\mathbb{E}[d(\hat{c},c)]\le np_e(\sigma_0,\sigma_1)$; the cq channel-coding strong converse then converts the distance-driven constant success probability into the rate bound. The new channels are the mixed-qubit channel MQC, obtained from pure-state outputs by an X-Pauli bit-flip, and its masked version 2MQC, obtained by channel masking.

What would settle it

Evaluate the variational formulas for $R_{\mathrm{MQC}}(\delta)$ and $R_{\mathrm{2MQC}}(\delta)$ at a fixed distance such as $\delta=1/4$; if either exceeds the corresponding MRRW value, Theorems 2 and 3 are false, and more directly, any binary code of relative distance $\delta$ whose rate exceeds $\chi(\sigma_0,\sigma_1)$ for a channel with $p_e(\sigma_0,\sigma_1)<\delta$ would refute the pretty good criterion itself.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1, the pretty good criterion: fix a binary-input output-symmetric classical-quantum channel with output states $\sigma_0,\sigma_1$ and uniform-prior PGM bit error rate $p_e(\sigma_0,\sigma_1)<\delta$; then every binary code $\mathcal{C}\subseteq\{0,1\}^n$ with minimum distance at least $\delta n$ satisfies $R(\mathcal{C})\le \chi(\sigma_0,\sigma_1)+O(n^{-1/2})$, and asymptotically $R_2(\delta)\le\chi(\sigma_0,\sigma_1)$. The derivation goes through the expected Hamming distance between the transmitted codeword and the block PGM output, which is at most $n p_e(\sigma_0,\sigma_1)$; the distance assumption turns that bound into constant block-decoding success, and the cq strong converse converts constant success into the capacity upper bound. Instantiating the criterion with the BEC, BSC, pure-state channel, and masked pure-state channel recovers Plotkin, Elias-Bassalygo, and the two MRRW bounds, while the mixed-qubit channel MQC and masked mixed-qubit channel 2MQC strictly improve the first and second MRRW bounds respectively throughout $(0,1/2)$.

Load-bearing premise

The load-bearing premise is a data-processing inequality: throwing away quantum information via any quantum channel cannot lower the pretty-good-measurement bit error rate, and the expected-Hamming-distance bound together with every rate bound in the paper relies on it.

Editorial extensions

If this is right

For every $\delta\in(0,1/2)$, $R_2(\delta)\le R_{\mathrm{MQC}}(\delta)<R_{\mathrm{MRRW}}(\delta)$: the first MRRW bound is strictly improved at every relative distance.
For every $\delta\in(0,1/2)$, $R_2(\delta)\le R_{\mathrm{2MQC}}(\delta)<R^{(2)}_{\mathrm{MRRW}}(\delta)$: the second MRRW bound is also strictly improved at every relative distance.
All four classical bounds become corollaries of one theorem, so improving binary code bounds reduces to choosing a channel with smaller capacity under a fixed PGM bit-error constraint.
The criterion extends to $q$-ary output-symmetric cq channels, yielding $q$-ary analogues of Plotkin, Elias-Bassalygo, and the first linear-programming bound together with a $q$-ary mixed-channel family.
For binary linear codes whose duals are generated by weight-3 parity checks, a one-check PSC decoder gives a rate bound below the first MRRW bound everywhere and below the existing sparse-dual benchmark for $\delta\in[1/6,1/2)$.

Reading between the lines

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

The strictness proofs use a logarithmic small-argument expansion at one endpoint of the optimization; this suggests a reusable perturbation mechanism for any channel family with a pure-state endpoint, which is my reading rather than a claim the paper makes.
Since the paper notes that MQC covers all output-symmetric qubit channels up to symmetry, the only remaining room for improvement inside this framework must come from four-dimensional or higher outputs; that programmatic conclusion is my inference from their Remark 5.2.
The same criterion is applied to structured code families and to $q$-ary alphabets, so one could try further structured channels for other code families; the paper leaves that search open.

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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

This paper is the real thing. The pretty good criterion (Theorem 1) is a single, clean statement: if the PGM bit error rate of a binary-input output-symmetric cq channel is below δ, then every binary code of relative distance δ has rate at most the channel capacity, up to O(n^{-1/2}). From that one theorem they recover Plotkin, Elias–Bassalygo, and both MRRW bounds, and then go further: the MQC and 2MQC channels give strict asymptotic improvements over the first and second MRRW bounds for every δ in (0,1/2). That is the first improvement over MRRW in roughly five decades, if it holds.

What the paper does well: the proof is coherent and unusually self-contained. The expected Hamming distance bound (Theorem 17) is the load-bearing step, and it rests on the PGM data-processing inequality (Lemma 10). The appendix proof via the Petz map is sound; I checked the weighted-inner-product contraction argument and found no gap. The strict improvement arguments are genuinely analytic, not just numerical: local perturbations around endpoints give inequalities valid for every δ, not sampled points. The rederivations of the four classical bounds are elegant and useful in themselves.

Soft spots, in proportion: there is no shipped code or machine-checked proof, and the numerical tables are floating-point estimates, not certified optima. That is a verification gap, not an identified error. The formal expansion in Section 6.2 is explicitly heuristic and does not carry the theorems. The concurrent-work note about OpenAI and the AI-use statement are unusual but transparent; the mathematics is proved in the paper, so I do not treat those as flaws. The citation pattern is fair; self-citations appear mainly in narrative sections and are not load-bearing.

The main risk is that long, intricate proofs can hide a sign error or a missing case, and the payoff is large enough that independent verification is warranted. But on reading, the central claim is well-supported and the weakest structural step is proved, not assumed.

Who this is for: coding theorists, quantum information theorists, and anyone tracking the rate-distance problem. It deserves a serious referee. My recommendation is to accept subject to careful verification of Lemma 10 and the perturbation analyses, and to send it to a referee comfortable with both quantum information and classical coding bounds.

Referee Report

0 major / 6 minor

Summary. The paper proposes a general sufficient condition, called the "pretty good criterion" (Theorem 1), for upper-bounding the asymptotic rate of binary codes with relative distance δ. The criterion states that if a binary-input output-symmetric classical-quantum channel has PGM bit error rate below δ, then every length-n code of relative distance δ has rate at most the channel's Holevo information plus O(n^{-1/2}). The authors prove this by deriving an expected-Hamming-distance bound from bitwise PGM properties, converting it into a constant block-decoding success probability, and then invoking a cq-channel strong converse. They show that the BEC, BSC, pure-state channel, and masked pure-state channel rederive the Plotkin, Elias-Bassalygo, first MRRW, and second MRRW bounds, respectively. They then introduce two new channel families, the mixed-qubit channel (MQC) and the masked mixed-qubit channel (2MQC), and prove strict asymptotic improvements over the first and second MRRW bounds for every δ in (0,1/2). The paper also sketches q-ary extensions and gives an LDPC-code refinement.

Significance. If correct, this is a substantial contribution: it provides the first asymptotic improvements over the long-standing MRRW bounds, and it unifies four classical rate-distance bounds as consequences of one channel-coding criterion. The main proof is structurally sound: Theorem 1 rests on the established cq strong converse and on self-contained appendix proofs for PGM coarse-graining, the PGM data-processing inequality via the Petz map, and the uniform-prior maximization lemma. The strict-improvement arguments in Propositions 25 and 26 are analytic and cover every δ in (0,1/2), not just numerically verified points. The paper is also transparent about what is rigorous and what is heuristic: the low-rate expansion in Section 6.2 is explicitly labeled as numerical/heuristic, and Table 2 is described as floating-point estimates rather than certified optima. The numerical tables are therefore not load-bearing for the main claims. The q-ary and LDPC sections broaden the framework and provide a promising research direction, although they are less developed than the binary results.

minor comments (6)

[Appendix A.4] In the proof of Theorem 7, the equality -Tr(σ0 log \bar σ)=S(\bar σ) is asserted without explanation; it follows from the output-symmetry relations Uσ0U†=σ1 and U\bar σU†=\bar σ, and this step should be spelled out for the reader.
[Section 7] The q-MC row of Table 3 and the associated spectral formulas are stated without derivations; since Section 7 is explicitly a blueprint, please mark these entries clearly as unproven sketches so that they are not mistaken for theorems.
[Section 6.2] The low-rate expansion is presented in the main text but is heuristic; consider moving it to an appendix or adding a stronger disclaimer that it is not part of the proof of Proposition 25.
[Sections 1.7 and 1.9] The references to specific GPT models and to OpenAI's concurrent work appear in the mathematical narrative; these statements might be better placed in the acknowledgments or a separate remarks section to keep the technical exposition neutral.
[Eq. (34)] The factorization in Eq. (34) is correct, but writing it as p_e = 1/2[1-(1-2η)^2(1-2r)] would match the form of Eq. (19) and remove a small notational hurdle for the reader.
[Section 1.1, footnote 2] The footnote correctly observes that the output-symmetry lift preserves p_e and χ, but it should explicitly note that the O(n^{-1/2}) constant in Theorem 1 may change under the lift while the asymptotic rate bound is unchanged.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained, with external citations only for standard strong-converse and bitwise-optimality facts, and no fitted parameter is renamed as a prediction.

full rationale

The paper's central claim is the pretty good criterion (Theorem 1): if the uniform-prior PGM bit error rate satisfies p_e(σ0,σ1) < δ, then every binary code of relative distance δ has rate at most χ(σ0,σ1) + O(n^{-1/2}). This is not circular: the bit-to-block step E[d(ĉ,c)] ≤ n p_e is proved internally in Theorem 17 using Lemmas 9, 10, and 11, whose proofs are supplied in Appendices A.5, A.2, and A.6. In particular, Lemma 10 (PGM error monotonicity under quantum channels) is proved via the Petz recovery map and the Petz–Gram contraction (Fact 32), not assumed or imported through a self-citation. Corollary 18 then converts p_e < δ into constant block-decoding success, and the cq strong converse is cited to Winter and Ogawa–Nagaoka, with a full proof also included in Appendix A.4. The rederivations of the four known bounds are exact computations for explicit channels: BEC, BSC, PSC, and masked PSC. No channel parameter is fitted to the target function R2(δ); the channels are fixed families, and the old bounds emerge as special cases. The new MQC and 2MQC bounds are variational upper bounds, and the strict improvements over the first and second MRRW bounds are proven by explicit local perturbations (Propositions 25 and 26); the numerical tables and figures are illustrative and not load-bearing. The paper's self-citations appear only in narrative and AI-use contexts and none supports a theorem. The only externally cited facts used at a proof step, Fact 28 and Theorem 7, come from independent works and are either reproduced or standard. No step in the derivation reduces by definition to its own input, and no uniqueness claim from the authors' prior work is invoked to force a choice. The absence of machine-checked verification is a reproducibility gap, not a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim is essentially self-contained: the main lemmas are proven, and the only external inputs are standard quantum information theorems and one external decodability fact. The channel parameters in the MQC and 2MQC bounds are optimization variables of a defined problem, not fitted constants, so they are not counted as free parameters.

assumptions (3)

standard math Cq channel coding strong converse (Theorem 7)
The rate upper bound hinges on it; the paper also gives a self-contained proof in Appendix A.4.
domain assumption Bitwise PGM optimality for binary linear codes over the PSC (Fact 28)
External result from PR25 used in the LDPC refinement (Theorem 4) and in the intuitive PSC argument; not needed for the general pretty good criterion.
standard math Petz recovery map and its contraction property (Fact 32)
Used to prove the PGM data-processing lemma; the paper proves these properties in Appendix A.1, but they are standard quantum information results.

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Pith. "Pith review of Binary code rate bounds via classical--quantum channels." pith.science (2026). https://pith.science/paper/ITCGC6IS

@misc{pith2026260809347,
  author       = {Pith},
  title        = {Pith review of: Binary code rate bounds via classical--quantum channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ITCGC6IS}},
  note         = {Machine review of arXiv:2608.09347}
}

abstract

We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.'' If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior sampling---of a binary-input output-symmetric classical--quantum (cq) channel lies below $\delta$, then every length-$n$ binary code, linear or nonlinear, of relative distance $\delta$ has rate at most the channel's capacity, up to an $O(n^{-1/2})$ correction. Rate--distance bounds thereby reduce to a channel design problem, wherein the task is to minimize channel capacity subject to the posterior bit error rate constraint. Via the pretty good criterion, the binary erasure channel (BEC) yields Plotkin, the binary symmetric channel (BSC) yields Elias--Bassalygo, the pure-state channel (PSC) yields the first MRRW bound, and a masked PSC yields the second MRRW bound exactly. This framework is then instantiated with new channels to improve upon the MRRW bounds. Specifically, the mixed-qubit channel (MQC), a mixed-state version of PSC, strictly improves the first MRRW bound at every $0 < \delta < \frac{1}{2}$, while the masked mixed-qubit channel (2MQC) strictly improves the second MRRW bound throughout the same interval.

Figures

Figures reproduced from arXiv: 2608.09347 by the authors.

**Figure 1.** Five benchmark rate curves, shown for 0 ⩽ 𝛿 ⩽ 1 2 . The Plotkin [Plo60], Elias– Bassalygo [Eli55, Bas65], and first MRRW [MRRW77] curves are closed form, while the second MRRW [MRRW77] and MQC curves are sampled one-dimensional minima. The two-variable 2MQC infimum from (46) is not plotted; analytically, it lies strictly below the second MRRW throughout 0 < 𝛿 < 1 2 by Proposition 26. The inset magnifies the marked w… view at source ↗

**Figure 2.** A plot of the MQC Holevo information function [PITH_FULL_IMAGE:figures/full_fig_p041_2.png] view at source ↗

**Figure 3.** Numerically minimized values of Δ(𝛿) = 𝑅MRRW(𝛿) − 𝑅MQC(𝛿) for 0.2 ⩽ 𝛿 ⩽ 0.499, shown on a logarithmic vertical scale. The strict-improvement proposition (Proposition 25) proves Δ(𝛿) > 0 for every 𝛿 ∈ (0, 1 2 ). The formal calculation below suggests that Δ( 1 2 − 𝜀) = 𝜀 4 log e + 𝑂(𝜀 6 log(1/𝜀)). Formal expansion. We now heuristically examine the scale visible in [PITH_FULL_IMAGE:figures/full_fig_p042_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Rate upper bounds for binary linear codes whose duals are generated by checks of weight [PITH_FULL_IMAGE:figures/full_fig_p055_4.png]

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