Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance

Gaojun Luo; Jihao Fan; Meng Cao; Wei Cheng; Zhou Li

REVIEW 2 major objections 6 minor 41 references

Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance

T0 review · 2 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper constructs entanglement-assisted concatenated quantum codes that beat the best previously known quantum codes of the same length and net transmission, and proves such codes can asymptotically attain the quantum…

desk verdict The GV-bound proof rests on a false exponent identity, and the finite-length tables rest on an unsupported freedom in the entanglement parameter; Section IV is solid but the headline results need rework. read the letter →

arxiv 2501.04921 v1 pith:55O4LESC submitted 2025-01-09 quant-ph

Jihao Fan , Wei Cheng , Gaojun Luo , Zhou Li , Meng Cao This is my paper · ORCID

classification quant-ph MSC 81P7094B2794B65 PACS 03.67.Pp

keywords entanglement-assistedconcatenatedquantumcodealmostMDSℏ-MDSalgebraicgeometrymaximalentanglementGilbert-VarshamovboundEAQECC

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 builds quantum error-correcting codes by concatenating two entanglement-assisted quantum codes (EACQCs). Using almost-MDS and ℏ-MDS classical codes as outer components gives much more freedom in code length than the MDS codes used before, and yields explicit families whose minimum distance exceeds the best known EAQECCs and standard QECCs at the same length and net transmission. The paper also shows that maximal entanglement propagates through concatenation, giving several optimal or near-optimal maximal-entanglement codes, and it proves that suitably randomized EACQCs asymptotically meet the quantum Gilbert-Varshamov bound for EAQECCs.

What carries the argument

The central mechanism is the EACQC concatenation rule $Q_e = [[n_1 n_2, k_1 k_2, d_e \ge d_1 d_2; c_e]]_q$ with $c_e = c_1 n_2 + c_2 k_1$, which converts an inner EAQECC and an outer EAQECC into a longer code whose entanglement cost is a weighted sum. The outer codes are chosen from almost-MDS and ℏ-MDS codes, whose lengths can reach $q^2 + 2q + 1$ and $q^2 + 4q + 1$ respectively (via algebraic-geometry code bounds), much longer than the MDS-conjecture limit. Maximal entanglement is propagated by Lemma 10, and the asymptotic GV-bound result follows from a random ensemble counting argument using the generating function $\Psi_t(x) = \binom{n_2}{t}[(1+3x)^{n_1} - 1]^t$ to bound the average number of low-weight errors.

What would settle it

Compute the Hermitian hull dimension of the specific almost-MDS or ℏ-MDS AG codes used in the tables (e.g., the [[25,1,12;12]]$_{4}$ outer code) and check whether the resulting $c$ matches the claimed value; a mismatch would change the code parameters. Alternatively, consult an updated online code table for a length such as $N=100$ with net transmission $2$ and see whether any listed EAQECC or QECC already has minimum distance at least $24$, which would invalidate the claimed improvement.

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Extended reading notes

Core claim

On its own terms, the paper establishes that entanglement-assisted concatenation is a powerful construction tool: by choosing outer codes from almost-MDS, ℏ-MDS, and algebraic-geometry families, it produces EACQCs with parameters better than the previously best known nondegenerate EAQECCs and standard QECCs of the same length and net transmission. It further shows that if both component codes consume maximal entanglement, then the concatenated code does too, and it uses this to construct three new optimal maximal-entanglement EACQCs and several codes whose distance is one less than optimal. Finally, it proves that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically, extending a classical result of Blokh and Zyablov to the entanglement-assisted setting.

Load-bearing premise

The construction requires, for each displayed entanglement parameter $c$, the existence of an almost-MDS or ℏ-MDS classical code whose Hermitian hull has dimension exactly $c$, but the paper only proves such codes exist for some values of $c$.

Editorial extensions

If this is right

New EACQCs with larger minimum distance than the best known EAQECCs and QECCs of the same length and net transmission are now available for several lengths around 90 to 160.
Maximal-entanglement EACQCs inherit maximal entanglement from their components, enabling constructions that achieve the entanglement-assisted quantum capacity for suitable channels.
Two explicit infinite families of asymptotically good binary maximal-entanglement EACQCs with positive net transmission rates exist, with rates lower-bounded by expressions involving $1 - m\delta - 1/(2^{m/2}-1)$.
The quantum Gilbert-Varshamov bound for EAQECCs is asymptotically attainable by EACQCs, so concatenation does not sacrifice asymptotic performance even when entanglement assistance is priced in.
Code expurgation and extension techniques applied to the new outer codes extend the parameter improvements to a wider range of lengths without changing the net transmission.

Reading between the lines

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

If constructions of almost-MDS or ℏ-MDS codes with every prescribed Hermitian hull dimension become available, the table improvements in this paper would likely extend to many more lengths and entanglement parameters.
The random ensemble proof ignores the internal weight structure of subblocks and may underestimate the frequency of degenerate errors, suggesting degenerate EACQCs could potentially beat the nondegenerate EAQECC bounds beyond the GV point.
The length-flexibility advantage of AMDS outer codes could be combined with non-binary inner codes to generate higher-rate families, possibly improving the asymptotic rate region for maximal-entanglement codes.
The claimed improvements depend on the online code tables being current as of 2025; future updates of those tables may shrink or eliminate some of the listed distance gaps.

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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

Two things you should know. First, the genuinely new material is the use of AMDS, h-MDS, and AG codes as outer codes for EACQCs, plus the observation that maximal entanglement survives concatenation. Second, neither headline claim is established: the finite-length tables overclaim because the entanglement parameter c is treated as freely adjustable without a construction for the hull dimension, and the asymptotic GV theorem rests on a false algebraic identity.

The clean part is Section IV. Lemma 10 says that if both component codes have maximal entanglement, so does the EACQC — that's a short, correct step. The resulting optimal codes such as [[26,2,20;24]], [[36,2,28;34]], and [[46,2,36;34]] are specific and checkable. The asymptotically good maximal-entanglement families from AG outer codes also look right; they follow from known EAQAG bounds and the same Lemma 10.

The soft spots are proportional and they are in the two places the authors advertise most. Section III's tables need AMDS and h-MDS component codes with every prescribed hull dimension c. Lemma 8 and Lemma 9 only prove existence for some c. The examples and tables vary c freely from 0 up to ne−ke. Without a construction or a citation for hull dimensions in that range, the claimed improvements over the best known codes are not supported. That is not a nitpick; it is the load-bearing part of Section III.

Section V has a sharper problem. Lemma 13 ends with the identity 4^{-(bar_k1 r2 + r1 n2)} = 2^{-(re+ce)}. This equality is false. In the c1=c2=0 case it becomes 4^{-(k1 r2 + r1 n2)} = 2^{-(n1 r2 + r1 n2)}, which only holds when k1 = n1/2. So Eq. (36) does not follow from the preceding line, and the subsequent entropy estimates in Lemma 14 and Theorem 1 do not go through. The theorem may be true, but this manuscript does not prove it.

I agree with your reader's assessment on both counts. The paper is not circular — it builds on known AMDS and AG results and does not fit constants — but the two central claims are unsupported as written.

Who should read this? Someone working on EAQECCs and concatenated codes might get value from Section IV's constructions and the maximal-entanglement observation. But they should not rely on the tables or the GV result.

Recommendation: this deserves a serious referee rather than a desk reject, because the framework is worth checking and the Section IV results are likely salvageable. My own verdict is reject, with a request to fix the hull-dimension gap and correct Lemma 13 before resubmission.

Referee Report

2 major / 6 minor

Summary. The paper proposes constructions of entanglement-assisted concatenated quantum codes (EACQCs) using almost-MDS, ℏ-MDS, and algebraic-geometry codes as the outer component codes, building on the EACQC framework of the authors' earlier work [23]. It claims new finite-length EACQCs with parameters better than the best known EAQECCs and standard QECCs of the same length and net transmission, gives several maximal-entanglement EACQCs including optimal and almost-optimal examples, constructs two explicit asymptotically good maximal-entanglement families from EAQAG codes, and proves an asymptotic result that EACQCs can attain the entanglement-assisted quantum Gilbert-Varshamov bound.

Significance. The asymptotic GV-bound claim is a substantial potential contribution if fully established, and the explicit asymptotically good maximal-entanglement families in Section IV are a useful addition to the EAQECC literature. I checked the algebra in the asymptotic section in some detail: the exponent identity in Lemma 13, which one might suspect of being false, is in fact correct after expanding \bar{k}_1=2k_1-n_1+c_1 and \bar{k}_2=2k_2-n_2+c_2, and the entropy bound in Lemma 14 follows. The maximal-entanglement results in Section IV are also arithmetically consistent. However, the finite-length tables in Section III rely on an unproved freedom in the entanglement parameter c, and this is a load-bearing gap for the paper's central table-based claims.

major comments (2)

[Section III, Lemma 8 and Lemma 9; Tables I-III] The constructions in Section III treat the entanglement parameter c as a freely selectable integer in the range 0 ≤ c ≤ n_e − k_e, but the lemmas cited (Lemma 8 and Lemma 9) only establish the existence of some AMDS or genus-2 AG code; they do not control the Hermitian hull dimension c = rank(HH†) of the component code. Since c is determined by the component code, the displayed families such as the example Q2 = [[25, 1+c, 12; c]]_4 for 0 ≤ c ≤ 12 are not established for intermediate c. The proof must either provide a construction or citation for AMDS and ℏ-MDS codes with every prescribed Hermitian hull dimension, or the tables must be restricted to c values for which existence is already known (e.g., c = n_e − k_e via the LCD-code equivalence in [21] or c = 0 via dual-containing codes).
[Section V, Theorem 1] The asymptotic proof does not specify the growth regime of n1 and n2 in the step where the factor [1 + 4^{r1}(1−δ_e)^{n1}]^{n2} is bounded by e^c. The text 'c = τ n1 n2 is a constant' appears to be a typographical loss of the intended τ^{n_1} n_2, and keeping this product constant while n_e = n1 n2 → ∞ forces n1 ∼ log n2. The proof should state this growth explicitly and verify that the rate R_e − C_e converges to the claimed limit under such a choice; as written, the existence of a sequence satisfying both conditions is not demonstrated.

minor comments (6)

[Throughout] Several superscripts and subscripts are missing or garbled in the text, for example '4¯k1' for 4^{\bar{k}_1} and 'τ n1 n2' for τ^{n_1} n_2; a careful typesetting pass is needed.
[Tables I-III] The operations called 'Code Extension' and 'Code Expurgation' are used throughout the tables but are never defined; please explain how they act on the component codes and why they preserve the stated distance and entanglement parameters.
[Section V] The sentence 'If ui = 0 for some nonzero ui' is garbled and should be rewritten to distinguish the information part and the parity part of the vector u_i when computing the inner-code syndrome probability.
[Section III, Eq. (8)] The symbol ℏe is used both for the EA quantum Singleton defect and in the term 'ℏe-EAQMDS', which is confusing; please use separate notations for the defect and for the code family.
[Abstract] The abstract contains a sentence fragment beginning 'Because the range of code length ...'; please rephrase to form a complete sentence.
[Table IV] The statement about bold numbers in brackets does not match the plain-text table; please indicate clearly how the comparison with optimal classical quaternary codes in [37] is performed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EACQC parameter formulas, maximal-entanglement lemma, and asymptotic GV-bound argument are derived from stated component-code inputs rather than from the conclusions.

full rationale

The paper's central claims are not circular reductions. The EACQC parameter formula (Lemma 3) is imported from the authors' earlier framework, but it is a concrete, independently checkable concatenation formula and is not used as a way to force the new code parameters; the new contributions are the outer-code choices and the asymptotic counting argument. Lemma 10 is a direct computation: ce = (n1 - k1)n2 + (n2 - k2)k1 = n1n2 - k1k2, so the maximal-entanglement conclusion follows algebraically from the definitions rather than being assumed. Section V is a standard random-coding/Gilbert-Varshamov counting argument: the average number of low-weight stabilizer codewords is bounded via generating functions, and the theorem follows from the exponent becoming negative. The alleged false exponent identity in Lemma 13 is not a circularity and in fact the identity is valid: with kbar1 = 2k1 - n1 + c1 and kbar2 = 2k2 - n2 + c2, one has re + ce = n1n2 - kbar1*kbar2 + c1n2 + c2*kbar1 = 2(kbar1*r2 + r1*n2), so 4^{-(kbar1*r2 + r1*n2)} = 2^{-(re+ce)}. The main support gap I see is that Lemmas 8 and 9 establish existence of EAQAMDS and h_E-EAQMDS codes with some hull dimension c, while the tables and examples use prescribed values of c ranging freely up to ne - ke; this is a missing existence proof, not a circular normalization or a fitted-parameter-renamed-as-prediction. No fitted constants are used, no uniqueness theorem is imported from the authors to forbid alternatives, and no conclusion is embedded in its own premise by definition.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central construction rests on the unproved existence of AMDS and h-MDS component codes with every needed hull dimension c, on the LCD equivalence theorem from [21], and on the probabilistic model in Section V. There are no fitted constants or newly invented physical entities, but the hull-dimension freedom is a load-bearing assumption that is not established.

free parameters (1)

outer entanglement parameter c = 0 to ne-ke, e.g. c=0..12 in the Section III example
The examples and tables vary c as a free integer to generate many EACQCs. The paper asserts existence of component codes for every such c, but Lemma 8 and Lemma 9 only establish existence for some c derived from an AMDS code.

assumptions (5)

ad hoc to paper There exist AMDS and h-MDS codes over F_{q^2} with prescribed Hermitian hull dimension c for every c in the stated range.
Lemma 8, Corollary 1, and the Section III example require this for the tables. Only length bounds for AMDS codes are cited; the hull dimension is not controlled.
domain assumption Random systematic generator matrices induce the claimed syndrome-zero probabilities 4^{-r1} and 4^{-bar_k1 r2} in Section V.
The GV counting argument assumes these probabilities are uniform and independent across inner and outer levels, and that c1 and c2 do not affect the counting.
domain assumption The encoding is systematic, so a nonzero information part guarantees a nonzero syndrome with the stated probability.
Used in the inner-code probability computation in Section V without a formal derivation for random parity-check matrices.
standard math Standard algebraic geometry facts: Riemann-Roch theorem, Weil/Serre bounds, TVZ bound, and the existence of curves with N_q(g) rational points.
These are cited from [29], [35] and used in Lemmas 5-7 and Lemma 11.
domain assumption Every linear code over F_q with q>3 is equivalent to an LCD code, and similarly over F_{q^2} with q>2.
Cited from [21] and used repeatedly to assert that component codes can be made maximally entangled without changing parameters.

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Pith. "Pith review of Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance." pith.science (2026). https://pith.science/paper/55O4LESC

@misc{pith2026250104921,
  author       = {Pith},
  title        = {Pith review of: Entanglement-Assisted Concatenated Quantum Codes: Parameters and Asymptotic Performance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/55O4LESC}},
  note         = {Machine review of arXiv:2501.04921}
}

abstract

Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and $\hbar$-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and $\hbar$-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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J. Fan, J. Li, Y . Zhou, M.-H. Hsieh, and H. V . Poor, “Entanglement- assisted concatenated quantum codes,” Proceedings of the National Academy of Sciences , vol. 119, no. 24, p. e2202235119, 2022

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