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For stabilizer codes over Pauli channels, sending above the code's own coherent information forces entanglement fidelity to decay exponentially, so constant tolerated error buys no rate.

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2026-07-30 21:48 UTC pith:2ZESKEL4

load-bearing objection Real strong converse inside the stabilizer class for product Pauli channels, with an explicit exponent and a clean win over PPT/Rains on antidegradable depolarizing noise.

arxiv 2607.23450 v1 pith:2ZESKEL4 submitted 2026-07-26 quant-ph

A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma

Marco Tomamichel This is my paper
classification quant-ph
keywords strong conversequantum capacitystabilizer codesPauli channelsblowing-up lemmacoherent informationentanglement fidelityantidegradable channels
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a strong converse for quantum communication when both the noise is a product of Pauli channels and the codes are stabilizer codes whose code space fills a full joint eigenspace. If such a code uses more logical qubits than the coherent information of its own input state, its entanglement fidelity falls exponentially in the block length, for any isometric encoder onto that space and any decoder. For memoryless channels this pins the ε-capacity of the stabilizer class to the vanishing-error stabilizer rate for every fixed error below 1; for antidegradable factors, including the depolarizing channel at error probability from 1/4 to 3/4, that capacity is zero. The argument never needs additivity or PPT relaxations: optimal decoding succeeds exactly on one event in a product error space, so the classical blowing-up lemma applies and the cheap side information it produces is charged against the coherent information. The same machinery already rules out heavy near-deterministic cores for arbitrary encoders above the coherent information, and isolates the encoder-side extraction step that would extend the strong converse to all codes.

Core claim

Within the class of full-sector stabilizer codes over product Pauli channels, any scheme whose rate exceeds the coherent information of its own input state has entanglement fidelity decaying as exp(−n E(γ)), where γ is the overshoot rate and E(γ) = (1/2)[g^{-1}(γ/2)]² > 0. Consequently the ε-quantum capacity of the class equals the vanishing-error stabilizer rate for every ε ∈ (0,1), and is zero whenever every factor is antidegradable.

What carries the argument

Exact fidelity formula for stabilizer codes: optimal entanglement fidelity equals the measure of the maximum-likelihood event A_ML in the product Pauli-error space. That identity turns decoding success into a classical set to which the blowing-up lemma applies; the resulting low-weight flag is then charged at its entropy against the coherent information via a one-shot converse on the flagged channel.

Load-bearing premise

The code space must fill an entire joint eigenspace of the stabilizer group; only then are branch fidelities binary and optimal fidelity exactly the mass of one event in the error space.

What would settle it

Exhibit a sequence of full-sector stabilizer codes over a fixed Pauli channel whose rate stays a fixed γ above the code's coherent information while entanglement fidelity remains bounded below by a positive constant independent of block length.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Among stabilizer codes, tolerating any fixed constant error does not raise the achievable quantum rate above the vanishing-error stabilizer rate.
  • For every antidegradable Pauli factor (including depolarizing noise with p ∈ [1/4, 3/4]) the stabilizer ε-capacity is zero for all ε < 1, with an explicit exponential fidelity bound.
  • Partial-transposition / Rains-type bounds cannot certify this zero-rate strong converse for depolarizing noise on p ∈ [1/4, 1/2), so the result is strictly stronger than PPT methods inside the stabilizer class.
  • The same exponential bound extends to shared-randomness mixtures, coherent superpositions of subexponentially many orthogonal stabilizer codes, and exponentially small encoder perturbations.
  • No encoder of any kind can carry a near-deterministic core of non-negligible mass once its rate exceeds its own coherent information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the open encoder-side extraction statement (Problem 7.3) holds, the strong converse would extend from stabilizer codes to all isometrically encoded schemes over memoryless Pauli channels.
  • The erasure channel is the natural next target: its classical pattern is handed to the receiver, so branch fidelities are channel-side and no-cloning may force the sharpness that stabilizer structure supplies here.
  • Small-blocklength numerical searches for coherently hedging codes that keep constant fidelity above the n-letter coherent information would directly test whether fidelity can survive without heavy cores.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 7 minor

Summary. The paper proves a strong converse for quantum communication over arbitrary products of (not necessarily identical) qubit Pauli channels, restricted to the class of stabilizer coding schemes whose code space is a full joint eigenspace of the stabilizer group, with arbitrary isometric encoder and arbitrary decoder. The central result (Theorem 6.5) is that any such (n,k) scheme with k ≥ I_c(V) + γn has entanglement fidelity F ≤ exp(−nE(γ)) with E(γ) = (1/2)[g^{−1}(γ/2)]² > 0, beyond an explicit threshold n_0(γ). The proof rests on: (i) an exact identity (Proposition 3.1) between optimal fidelity and the measure of the maximum-likelihood event A_ML in the product error space, via the sum rule Σ_ℓ f_{s,ℓ} = 1 (Eq. (26)), which in turn rests on the completeness of the logical Pauli basis on a full sector (Lemma 2.1, twirl identity (16)); (ii) a finite-blocklength blowing-up lemma (Lemma 4.1) derived from McDiarmid's inequality; (iii) a flagged-channel construction whose classical flag is charged at its entropy against coherent information (Lemmas 5.1–5.2); and (iv) a one-shot converse via twirling to an isotropic state (Lemma 5.3). Corollaries give Q_ε = Q_stab for all ε ∈ (0,1) within the class (6.7), zero capacity for antidegradable factors including depolarizing p ∈ [1/4,3/4] (6.6), and extensions to mixtures, orthogonal superpositions, and perturbations (6.9–6.11). Section 7 proves the full-sector assumption cannot be removed decoder-side (Proposition 7.2) and isolates the enc

Significance. If correct, this is the first strong converse with an explicit exponent for quantum communication over Pauli channels in any nontrivial class, and — as the paper demonstrates in Remark 8.2 — it certifies a strong converse in the antidegradable window p ∈ [1/4,1/2) where all PPT/Rains-type relaxations provably cannot, even after regularization. Particular strengths: the argument is fully explicit and finite-blocklength (thresholds n_0(γ), n_1(γ) given in closed form); it requires no additivity assumptions and no semidefinite relaxations; it covers non-memoryless product noise; the master inequality (Theorem 6.1) and Theorem 6.2 are stated for arbitrary encoders, cleanly separating what is general from what is stabilizer-specific; and Proposition 7.2 constructively proves the scope restriction is intrinsic to the method rather than an artifact, converting a potential objection into a sharp open problem (7.3) with a stated consequence (the full strong converse for memoryless Pauli channels). The qudit extension (Appendix A) is carried through with careful attention to where the F_d-linear structure is used. I verified the load-bearing chain line by line — the sum rule (26) via (16) an

minor comments (7)
  1. [§3, proof of Prop. 3.1] Typo in the proof of Proposition 3.1: "The conditional fidelity oneis therefore 1" should read "one is".
  2. [§2.3, Eq. (12) and passim] Notation for the syndrome map alternates between "σsyn" and "σ syn" (e.g., Eq. (12) vs. §2.3 text and Remark 3.2). Please uniformize.
  3. [Corollary 6.7] In Corollary 6.7 the identification Q_ε = Q_stab uses Hamada's achievability [43] for the lower bound; it would help the reader to state explicitly that Hamada's concatenated stabilizer codes satisfy the full-sector convention of Definition 2.3 (they do), so that the classes on the two sides of (61) literally coincide.
  4. [Corollary 6.10, Eq. (64)] In Eq. (64) the double-bar notation for the ℓ² norm over Kraus indices is nonstandard and the first inequality (triangle inequality applied to the vector of amplitudes ⟨Φ|(I⊗M_a)ψ_e⟩ indexed by a) deserves one clause of explanation.
  5. [References] Several references are missing terminal periods ([15], [16], [17], [34], [35], [39], [44], [46], [51], [55], [57]); a uniform pass over the bibliography is warranted.
  6. [Theorem 6.5] The constant C(γ) = e^{n_0(γ)E(γ)} in Theorem 6.5 grows very rapidly as γ ↓ 0; a brief numerical remark (e.g., for the depolarizing channel at a representative γ) on where the bound becomes nontrivial would help readers gauge its finite-blocklength content, complementing the asymptotic statement.
  7. [Remark 8.2] In Remark 8.2 the step "R(ρ_p^{⊗n}) ≥ n E_D(ρ_p)" uses superadditivity of distillable entanglement under tensoring; this is standard but currently implicit, and one line citing it would make the regularization argument self-contained.

Circularity Check

0 steps flagged

No circularity: strong converse is derived from stabilizer sector structure, classical blowing-up, and entropic flag accounting, not from fitted inputs or self-justifying definitions.

full rationale

The load-bearing chain is self-contained and non-circular. Proposition 3.1 identifies optimal fidelity with the mass of the ML event A_ML via the logical twirl and the sum rule on branch fidelities; that identity follows from the full-sector stabilizer algebra (Lemma 2.1), not from assuming the strong converse. Blowing-up (Lemma 4.1) is McDiarmid applied to Hamming distance on a product measure. The flagged channel charges side information at most ng(δ)+1 (Lemma 5.2) and the one-shot converse (Lemma 5.3) is an isotropic twirl plus data processing. Combining these yields Theorems 6.1–6.5 with an explicit exponent fixed by g^{-1}(γ/2), with no free parameters fitted to capacity data. Q_stab is imported from Hamada only as the matching vanishing-error benchmark for the ε-capacity corollary; the exponential decay above I_c(V) does not assume Hamada’s rate. Author self-citations ([15], [34], [58]) supply context or continuity tools and are not uniqueness theorems that force the claim. Scope limits (full-sector assumption, Problem 7.3) are stated openly rather than smuggled in. No self-definitional loop, fitted-as-prediction step, or renaming of a known pattern constitutes the result.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

The result is a theorem in finite-dimensional quantum information theory. It rests on standard entropy calculus, the classical blowing-up/concentration lemma, and the algebraic structure of Pauli stabilizer codes. No empirical fits. The only modeling restrictions that actually carry the claim are product Pauli noise, full-sector stabilizer code spaces, and no free two-way classical assistance.

axioms (6)
  • standard math Blowing-up / McDiarmid bounded-differences concentration for product measures on Hamming space (Lemma 4.1).
    Classical concentration imported unchanged; supplies the fattening of A_ML that creates cheap side information.
  • domain assumption Stabilizer sector structure: isotropic S gives 2^{n-k} equal-dimensional joint eigenspaces permuted by Pauli errors; logical operators form a complete orthogonal basis on a full sector (Lemma 2.1).
    Standard stabilizer formalism; full-sector filling is what makes the twirl and the sum rule Σ_ℓ f_{s,ℓ}=1 hold, hence F^*=μ(A_ML).
  • domain assumption Noise is a product of (possibly non-identical) single-qubit Pauli channels, so the error measure μ is a product measure.
    Required for blowing-up on a product probability space; stated from Section 2.2 onward.
  • standard math Coherent information one-shot converse after Haar twirl to an isotropic state (Lemma 5.3), using concavity of conditional entropy and data processing.
    Standard entropy inequalities; linear-in-error form is what tolerates constant core imperfection.
  • domain assumption No free two-way classical communication between encoder and decoder.
    Essential: Remark 8.1 notes two-way assisted capacity is positive slightly above the antidegradability threshold via recurrence+hashing, so the unassisted bound must exclude it.
  • standard math Antidegradable channels have Q^{(n)}=0 by data-processing versus complementary output (used in Corollary 6.6).
    Standard fact about antidegradability; pins I_c(V)≤0 so any positive rate overshoots.
invented entities (2)
  • θ-core (set of error patterns on which some decoder has branch fidelity ≥1-θ) no independent evidence
    purpose: Decoder-side object that blowing-up can fatten; for stabilizer codes a 0-core of mass F* exists.
    Definitional proof device (Definition 2.4), not a physical postulate; independent interest only inside the argument.
  • Flagged channel M_δ appending a minimal-weight displacement flag into the core no independent evidence
    purpose: Turns a heavy core into a high-fidelity assisted scheme whose coherent-information cost is at most ng(δ)+1.
    Constructive intermediate channel in Section 5; classical side-information accounting in the style of Ahlswede–Gács–Körner.

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Pith. "Pith review of A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma." pith.science (2026). https://pith.science/paper/2ZESKEL4

@misc{pith2026260723450,
  author       = {Pith},
  title        = {Pith review of: A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ZESKEL4}},
  note         = {Machine review of arXiv:2607.23450}
}
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We prove a strong converse for quantum communication over Pauli channels within the class of stabilizer codes. If a code whose code space is a full joint eigenspace of a stabilizer group transmits above the coherent information of its own input state, its entanglement fidelity decays exponentially in the block length; the encoder may be any isometry onto that space and the decoder any channel. For memoryless channels this determines the $\varepsilon$-quantum capacity of the class for every $\varepsilon < 1$, so that tolerating a constant error buys no rate; for antidegradable channels, such as the depolarizing channel with error probability $p \in [1/4, 3/4]$, that capacity is zero, while for $p \in [1/4,1/2)$ partial-transposition bounds provably cannot certify a strong converse. The proof uses neither additivity assumptions nor semidefinite relaxations: optimal decoding succeeds precisely on an event in a product probability space, so the blowing-up lemma of Ahlswede, G\'acs and K\"orner applies, and the side information it produces is charged against the coherent information. The argument also constrains near-deterministic decoding for codes of any kind, and we isolate the encoder-side statement that would extend it to all of them.

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This paper was first reviewed by grok-4.5 on July 30, 2026.