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Error correction, authentication, and false acceptance, probabilities for communication over noisy quantum channels: converse upper bounds on the bit transmission rate

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims a strict converse upper bound on noisy-quantum bit transmission rates, with decoding error and false acceptance made arbitrarily small.

desk verdict Despite a reasonable question and a few true textbook inequalities, the central bound is conditional on unproven pruned alphabets and the proof has unsupported steps; this paper did not clear the bar for review. read the letter →

arxiv 2507.03035 v1 pith:BFGTHQJA submitted 2025-07-03 quant-ph cs.ITmath.ITmath.PR

Pete Rigas This is my paper · ORCID
classification quant-phcs.ITmath.ITmath.PR MSC 81P0281Q02
keywords bittransmissionratequantumchannelserrorcorrectionauthenticationfalseacceptanceconverseboundprunedalphabetsmutualinformation
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

This paper attempts to establish a converse upper bound on the bit transmission rate for sending classical bit codewords through noisy quantum channels, under the deliberately paradoxical condition that the Alice–Bob channel is noisier than the Bob–Eve channel. The claimed bound is a strict inequality: the usual expression $\sup_{P_X} \min(I(X;Y), \min_z H_Q(Y|Z=z) - H_P(Y|X))$ is strictly bounded above by a piecewise log-log function of the sizes of pruned alphabets $X^*, Y^*, Z^*$. If true, this would mean Alice and Bob can keep decoding error and false acceptance simultaneously small even when Eve has a cleaner channel, and it would give a quantitative target for noise-resilient error-correcting codes and authentication protocols. The proof route is an entropy-cardinality estimate combined with a pruning procedure that removes letters from the players' alphabets, so the practical force of the result depends on whether such prunings can always be found.

What carries the argument

The carrying mechanism is the pruning of alphabets: sub-alphabets $X^*, Y^*, Z^*$ are chosen so that false-acceptance and decoding-error probabilities vanish, and the mutual informations and conditional entropies are then bounded by logarithms of alphabet cardinalities, producing the four-case log-log upper bound (*). The overlap function $O(X,Y,Z)$ records whether Eve's letters intersect Alice's or Bob's alphabets and controls whether the noise ordering implies an advantage. Two supporting lemmas carry the stochastic claims: inverse monotonicity of Hamming-ball radii with respect to channel noise (Lemma 1), and the iff correspondence between high error-correction probability and low false-acceptance probability (Lemma 3).

What would settle it

Take a binary-symmetric quantum channel with finite alphabets $X,Y,Z$ and noise levels $N_{A\leftrightarrow B} > N_{B\leftrightarrow E}$, and enumerate all candidate prunings $X^* \subseteq X$, $Y^* \subseteq Y$, $Z^* \subseteq Z$. If for every pruning either $P_{FA} > 0$, $P_{DE} > 0$, or the cardinality relation required by one of the four cases of (*) fails, then Theorem 1's upper bound does not hold as stated for that channel; a numerical search over small alphabets, say $|X|,|Y|,|Z| \le 4$, would settle the existence assumption directly.

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Extended reading notes

Core claim

The paper's central claim is that the lower-bound expression for the bit transmission rate $r$ studied in prior work admits a converse: under the noise ordering $N_{A\leftrightarrow B} > N_{B\leftrightarrow E}$, the supremum over input distributions of $\min(I(X;Y), \min_z H_Q(Y|Z=z) - H_P(Y|X))$ is strictly less than the four-case piecewise function of $\log\log$ ratios of alphabet sizes shown in (*), where $X^*, Y^*, Z^*$ are pruned sub-alphabets of Alice, Bob, and Eve obtained by deleting letters that obstruct error correction and authentication. With this strict upper bound in place, the paper asserts that Alice and Bob can make decoding error and false acceptance occur with arbitrarily small probability. The paper further asserts stochastic domination of the error-correction and false-acceptance probabilities across the two quantum channels, and the existence of protocols that map $n$-bit codewords into the authenticated space for rates $r > h(q) - h(p)$, extending the earlier existence result to the converse rate regime.

Load-bearing premise

The load-bearing premise is that for every channel in the claimed regime there exist pruned alphabets $X^*, Y^*, Z^*$ such that deleting letters makes false-acceptance and decoding-error probabilities vanish and places the alphabet sizes in one of the four orderings required by the bound (*); the paper asserts this pruning procedure but does not construct the pruned alphabets or prove they exist.

Editorial extensions

If this is right

  • If Theorem 1 holds, any attempted transmission rate above the piecewise log-log bound is impossible when the Alice–Bob channel is noisier than the Bob–Eve channel, so noise-resilient coding must operate below that threshold.
  • The result is a converse in the precise sense that the lower-bound expression previously identified for $r$ cannot be pushed above the log-log cap, confining the 'paradoxical' noisier-channel advantage to a specific rate regime.
  • Under the bound, decoding error and false acceptance can be driven to zero together, so message authentication and error correction can coexist even though Eve's channel is cleaner.
  • Theorem 2 makes the advantage quantitative: error correction over Alice–Bob succeeds with higher probability than over Bob–Eve, and the ratio $p_{EC,A\leftrightarrow B}/p_{FA,A\leftrightarrow B}$ is bounded below.
  • Theorem 3 guarantees, for every $r > h(q) - h(p)$, a protocol that maps $n$-bit codewords into the authenticated space, extending the earlier existence construction to the converse regime.

Reading between the lines

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

  • A direct test of the paper's claim is to search, for small alphabets, for channels with $N_{A\leftrightarrow B} > N_{B\leftrightarrow E}$ where no pruning can make $P_{FA}$ and $P_{DE}$ vanish while respecting the cardinality orderings in (*); a single such channel would show the asserted upper bound is vacuous where it is meant to apply.
  • The log-log form means the rate cap grows extremely slowly with alphabet size, so if the converse is correct even exponentially large alphabets buy only modest rate; the pruning procedure, not the entropy estimate, would then be the operative constraint for applications.
  • The stochastic domination and the lower bound on $p_{EC,A\leftrightarrow B}/p_{FA,A\leftrightarrow B}$ could be read as a resource inequality between the two channels, which may connect to composable-security statements for authentication without a shared secret key, though the paper does not develop that connection.
  • A concrete extension would be to formulate pruning as an optimization problem over sub-alphabets and compute, for fixed channel families, the largest pruned alphabets for which the vanishing conditions hold; that would turn the existence assumption into a computable quantity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript claims to establish a converse upper bound on the bit transmission rate r for classical communication over noisy quantum channels with authentication. Theorem 1 states that sup over input distributions of min(I(X;Y), min_z H_Q(Y|Z=z) - H_P(Y|X)) is strictly bounded by a piecewise log-log expression involving the sizes of pruned alphabets X*, Y*, Z*. Theorem 2 asserts a stochastic domination relation between error-correction and false-acceptance probabilities when the Alice-Bob channel is noisier than the Bob-Eve channel. Theorem 3 asserts that for r > h(q)-h(p) there exist protocols mapping bit codewords into the authenticated space with high probability. The proofs, in Section 3, are sequences of formal manipulations; the central derivation in Section 3.1.2 does not justify the passage to the log-log expression, and the pruned alphabets are introduced in Section 2.5.2 without an existence proof.

Significance. A genuine converse bound for the bit transmission rate in this setting would be a meaningful contribution: the cited work [38] established lower bounds, and an upper bound depending on alphabet sizes would be a surprising and useful complement. The manuscript also has the merit of being explicit about its dependence on [38], including quoting the relevant theorem and Proposition 4. However, the central claims are not supported by the proofs as written. The log-log bound in Theorem 1 depends on pruned alphabets whose existence is merely stipulated, and the proof contains invalid algebraic steps. The result in Theorem 3 reverses the rate condition of the cited Theorem 3 of [38] without supplying the required argument. These issues are foundational, not cosmetic.

major comments (5)
  1. [Section 2.5.2] The pruned alphabets X*, Y*, Z* are defined by requiring the false-acceptance and decoding-error probabilities to vanish after pruning, as formalized in conditions (X*,1), (Y*,1), and (Z*,1). No construction or existence proof is given, and the four cases of the bound (*) require specific cardinality orderings such as |X| > |Y*| > |Z| that are never shown to be compatible with the vanishing-error conditions. Since the statement of Theorem 1 then asserts that Alice and Bob can guarantee arbitrarily small decoding-error and false-acceptance probabilities, the bound assumes the very property it is supposed to establish. If such pruned alphabets cannot be realized, the bound in (*) is vacuous or undefined.
  2. [Section 3.1.2] The proof of Theorem 1 contains an unjustified passage to the log-log expression. Starting from the display labeled (∗∗), the proof replaces ratios of logarithms of alphabet sizes with ratios of a logarithm and an alphabet cardinality, and it moves suprema and minima inside logarithms without any monotonicity or continuity argument. The final four-case formula in (*) is simply asserted after these manipulations. This step is load-bearing because it produces the claimed upper bound; without a rigorous derivation, Theorem 1 is not proven.
  3. [Section 3.1.2 (first display)] The proof uses the equality sup_PX min(log I, min_z log(H_Q - H_P)) ≡ sup_PX min(log I, min_z log H_Q) - sup_PX min(log I, log H_P). This identity is not valid in general: a supremum of a minimum of differences cannot be decomposed into a difference of two suprema of minima. The same type of splitting is used repeatedly in the chain of inequalities leading to (*). Since this decomposition is essential to the derivation, the argument fails at this step.
  4. [Theorem 3 / Section 3.3.2] Theorem 3 states the existence of suitable protocols for every r > h(q)-h(p), while the cited Theorem 3 of [38], reproduced in Section 2.5.2, gives existence for r < h(q)-h(p). The proof in Section 3.3.2 simply says that Proposition 4 of [38] can be applied; that proposition only bounds the simulator distance by max(pde, pfa) and says nothing about the sign of r - (h(q)-h(p)). No argument is provided for why reversing the rate inequality is legitimate, and the statement contradicts the range of the cited theorem. This is a load-bearing gap in the manuscript's third main result.
  5. [Section 3.2.2] The proof of Theorem 2 relies on a large number of unspecified constants C*, C**, C***, C****, C******, and C_final, and it assumes that several limits as |Y| and |Z| tend to infinity exist, are finite, and can be interchanged with suprema and sums. For example, the proof asserts that the limits involving P(alphabets y,z) and p_fa,B<->E are finite and strictly positive without derivation, and it uses an equivalence C* > 1 ⇔ c*_{y,z}<1 that is not justified. The final product of constants is asserted to be positive rather than proved. The stochastic domination claim is therefore not established by the presented argument.
minor comments (5)
  1. [Throughout] There are numerous typographical errors and inconsistencies in notation, including 'evesdrop' in the abstract, 'referree' in the introduction, and inconsistent use of sup/inf in the definitions of p_EC and p_FA in Section 1.3.
  2. [Equation (*)] The log-log expressions in (*) require specification of their domain: if the arguments are ratios smaller than 1, the iterated logarithm is not real-valued, and the manuscript does not address which branch or convention is intended.
  3. [Section 2.1] The text states p_F A,B<->E > p_F A,A<->B, but Theorem 2 as stated in Section 1.4 does not include this comparison; either the theorem statement should be expanded or the assertion in Section 2.1 should be removed.
  4. [Section 2.2] The reference for typical sequences is cited as '[]' on page 24; the citation is incomplete.
  5. [Section 1.3] The paper introduces many objects, such as the operators T_XOR and T_FFL and the Schmidt basis, that are not used in the proofs of the main results; this makes the exposition hard to follow.

Circularity Check

2 steps flagged · score 8.0 of 10

Theorem 1's converse bound and Theorem 2's stochastic domination are wired into the definitions of pruned alphabets and the overlap threshold c*; without independent existence proofs, the central results reduce to their assumptions.

  1. self definitional [Section 2.5.2, definitions of X*, Y*, Z* and conditions (X*,1)-(Z*,1); Theorem 1 statement in Section 1.4; proof in Section 3.1.2]
    "Introduce the pruned alphabets, X∗ ≡ ⋃_{x∗∈X\(X)∗} {subalphabets (X)∗ ⊊ X : PF A (X, Y, Z), PDE (X, Y, Z) > 0}, ... The goal of being able to maintain Quantum advantage of being able to simultaneously achieve authentication, and error correction, with high probability when NA← →B > NB← →E is illustrated with the conditions, PF A (X∗, Y, Z), PDE (X∗, Y, Z) ≡ 0, ( X∗,1)"

    The pruned alphabets are defined as subalphabets on which the false-acceptance and decoding-error probabilities vanish (conditions (X*,1)-(Z*,1)). Theorem 1's claimed upper bound (*) is then expressed entirely through these pruned alphabets, and the theorem concludes that, under this bound, Alice and Bob can guarantee arbitrarily small decoding error and false acceptance. No construction or existence proof for such alphabets is given, so the theorem's guarantee is assumed by definition; the bound is undefined or vacuous unless alphabets with the very vanishing-error property already exist. This is exactly the property the theorem claims to establish.

  2. self definitional [Section 2.5.2, 'Quantifying the relationship...' and 'Putting it all together'; proof of Theorem 2 in Section 3.2.2]
    "PF A, PDE ≡ 0 ⇐ ⇒O (X, Y, Z) ≡ c∗ ⇐ ⇒NA− →B > NB− →E, PF A, PDE > 0 ⇐ ⇒O (X, Y, Z) ≡ C ∗ ⇐ ⇒NA− →B < NB− →E."

    This block postulates, as a definitional equivalence, that vanishing/nonvanishing of false-acceptance and decoding-error probabilities is equivalent to a threshold c* of the alphabet-overlap function, and that this threshold is equivalent to the noise ordering NA→B > NB→E. The proof of Theorem 2 then manipulates the overlap function and recovers the stochastic domination pEC,A→B > pEC,B→E and pFA,B→E > pFA,A→B as a consequence. The theorem's content is therefore imported through the stipulated equivalence, rather than derived from independent channel properties.

full rationale

The central claim of the paper, Theorem 1, is a strict upper bound in terms of pruned alphabets X*, Y*, Z*, but those alphabets are introduced in Section 2.5.2 precisely as subalphabets making the false-acceptance and decoding-error probabilities vanish. The theorem then uses those alphabets to assert that, under the bound, decoding error and false acceptance occur with arbitrarily small probability. Since no independent construction or existence argument is provided, the theorem's conclusion is an assumption embedded in the definitions; the four cardinality cases of (*) are likewise stipulated rather than shown compatible with the vanishing-error conditions. Theorem 2 is similarly self-definitional: the stochastic domination is encoded in the stipulated equivalences among PF_A, PDE, the overlap threshold c*, and the noise ordering. Separately, the proof of Theorem 3 applies [38]'s Proposition 4 while reversing the inequality from r < h(q)-h(p) to r > h(q)-h(p) without bridging the reversal, and Section 3.1.1 contains an unjustified loglog manipulation; these are additional correctness concerns but not needed for the circularity verdict. Overall, the main results reduce by definition to the objects they purport to derive, so the circularity score is high.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central claim rests on an unproven pruning procedure, on standard entropy-alphabet inequalities, and on arbitrary constants introduced in the proofs. This ledger shows that the paper's contribution is mostly definitional and does not provide independent support for the stated results.

free parameters (3)
  • Pruning choice for alphabets X*, Y*, Z* = not specified
    The upper bound in Theorem 1 depends on pruned alphabets introduced in Section 2.5.2; no algorithm or existence proof is given, so the bound depends on a hand-chosen pruning.
  • Threshold c* for alphabet overlap = c* > 0, unspecified
    Introduced ad hoc in Section 1.2 to distinguish regimes N_A > N_B and N_A < N_B; no derivation of existence is provided.
  • Constants C*, C**, C***, and C_final in the proof of Theorem 2 = unspecified positive constants
    The stochastic domination proof in Section 3.2.2 introduces arbitrary constants, some said to tend to infinity and others claimed finite, with no justification.
assumptions (4)
  • standard math Conditional entropy decreases when the alphabet is restricted: H(Y|X) > H(Y*|X) for Y* subset of Y.
    Used in inequality (H) in the proof of Theorem 1; the statement is true for properly defined subalphabets, but its application here is not clean because Y* is not rigorously specified.
  • ad hoc to paper Pruned alphabets X*, Y*, Z* exist and satisfy the cardinality assumptions in the four cases of (*).
    Stated in Section 2.5.2 and used without proof in Theorem 1; no construction or existence proof is given.
  • domain assumption The binary symmetric channel model with 0 <= p < q <= 1/2 and noise thresholds N_A > N_B characterizes the quantum channels.
    The paper works entirely within this simplified BSC model inherited from [38]; no physical derivation of the noise model is provided.
  • ad hoc to paper The limits appearing in the proof of Theorem 2 are finite and can be interchanged with sums and suprema.
    Section 3.2.2 assumes finiteness of several limits and then splits products of limits without proof; this is a load-bearing unverified assumption.
invented entities (2)
  • Pruned alphabets X*, Y*, Z*
    purpose: To shrink the alphabets so that the false acceptance and decoding error probabilities vanish, making the claimed upper bound in Theorem 1 hold.
    No independent construction is provided; their existence is assumed ad hoc in Section 2.5.2, and they are defined in terms of the very probabilities the theorem aims to bound.
  • Overlap function O(X,Y,Z)
    purpose: To quantify the overlap between Alice, Bob, and Eve's alphabets and to connect alphabet overlap with the relative noise levels of the channels.
    Defined in Section 1.2 but never given a concrete probabilistic meaning or independent justification; it is used to state conditions rather than to derive them.

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Cite this review

Pith. "Pith review of Error correction, authentication, and false acceptance, probabilities for communication over noisy quantum channels: converse upper bounds on the bit transmission rate." pith.science (2026). https://pith.science/paper/BFGTHQJA

@misc{pith2026250703035,
  author       = {Pith},
  title        = {Pith review of: Error correction, authentication, and false acceptance, probabilities for communication over noisy quantum channels: converse upper bounds on the bit transmission rate},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFGTHQJA}},
  note         = {Machine review of arXiv:2507.03035}
}
read the original abstract

We obtain strict upper bounds on the bit transmission rate for communication of Classical bit codewords over Quantum channels. Albeit previous arguments in arXiv: 1804.01797 which have demonstrated that lower bounds can be shown to hold for the bit transmission rate without the presence of significant noise over the channel shared by Alice and Bob for the purposes of encoding, decoding, transmission and authentication, the author suggests that upper bounding the bit transmission rate could be of use towards classifying paradoxical aspects of communication protocols, as well as constructing error correcting codes which are resilient to noise. The upper bound that is obtained in this work for the bit transmission rate, as a converse result, is dependent upon the natural logarithm of the size of each player's alphabet, as well as smaller alphabets, which can be leveraged for simultaneously realizing Quantum advantage for maximizing error correction and minimizing false acceptance. Crucially, the upper bound to the bit transmission rate is dependent upon a pruning procedure, which seeks to determine whether letters from player's alphabets can be removed so that prospective Quantum advantage, in order for Alice and Bob to implement error correction protocols with high probability, despite the fact that there is more noise over the channel between Alice and Bob in comparison to that between Bob and Eve.

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