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Symmetry-guided constructions of absolutely maximally entangled states in five open cases

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Pith's one-line read Explicit Hermitian self-dual MDS codes prove five previously open AME states exist.

desk verdict Resolving five open AME cases with explicit codes; credible result with one reproducibility gap in the unprinted minor enumeration. read the letter →

arxiv 2608.05781 v2 pith:PP4NBG4R submitted 2026-08-06 quant-ph cs.ITmath.IT

Samuel Bevins , Yunus Bidav This is my paper · ORCID
classification quant-phcs.ITmath.IT
keywords absolutelymaximallyentangledstatesHermitianself-dualMDScodesquantumstabilizerconstructiongroup-circulantmatricesfinite-Fouriertransformsquare-minorenumerationZ_3^2symmetry
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 proves that five absolutely maximally entangled (AME) states exist, for party numbers and local dimensions (12,5), (17,11), (18,11), (17,13), and (18,13); two of these parameters were previously listed as open. The strategy is constructive and exact. The authors exhibit three generator matrices and verify that each row space is a Hermitian self-dual maximum distance separable (MDS) code, meaning the code is its own Hermitian dual and attains the Singleton distance bound. The standard stabilizer construction converts each such code into a pure quantum MDS code, which is exactly an AME state, and projecting one party out of the even-party cases produces the two odd-party cases. Because AME states act as quantum error-correcting codes and as ingredients for quantum secret sharing and multiunitary matrices, the settled parameters become available resources.

What carries the argument

The load-bearing mechanism is the systematic generator matrix $G=[I_k|A]$ over $\mathbb{F}_{q^2}$, together with Criterion 1: the row space of $G$ is a Hermitian self-dual $[2k,k,k+1]_{q^2}$ MDS code exactly when $A A^{\top} = -I_k$ and every nonempty square minor of $A$ is nonzero. The first equation is the self-duality check, and the square-minor condition is systematic superregularity, which forces the MDS distance. For the length-eighteen cases, a regular $\mathbb{Z}_3^2$ coordinate orbit forces $A_{x,y}=a(y-x)$ for $x,y\in \mathbb{Z}_3\times\mathbb{Z}_3$, so the $9\times9$ block is determined by the nine-entry kernel $a$; self-duality becomes the convolution equation $\sum_{h} a(h+g)\overline{a(h)}=-\delta_{g,0}$, which the finite-Fourier transform separates into independent character blocks. The final certificate is purely arithmetic: exact products and complete enumeration of square minors, 923 for the $6\times6$ matrix and 48,619 for each $9\times9$ matrix.

What would settle it

Recompute, in exact arithmetic, the Hermitian products $A_5 A_5^{\top}$, $A_{11} A_{11}^{\top}$, and $A_{13} A_{13}^{\top}$ from the printed matrices, reconstructing $A_{11}$ and $A_{13}$ from the kernels in Eqs. (11) and (13) via Eq. (5), and enumerate every nonempty square minor of each $A$ block. A single zero square minor in any of the three matrices would invalidate the MDS claim and hence the corresponding AME existence statement; a single failed Hermitian product would invalidate the self-duality claim.

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Extended reading notes

Core claim

The paper's central claim is Theorem 2: the matrices in Eqs. (8), (11), and (13) define Hermitian self-dual MDS codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$, and consequently the AME states in Eq. (1) exist. The bridge is exact: an $\mathbb{F}_{q^2}$-linear Hermitian self-dual MDS code of length $2k$ and dimension $k$ maps through the stabilizer construction to a pure quantum MDS code $[[2k,0,k+1]]_q$, which is an $\mathrm{AME}(2k,q)$ stabilizer state; the reduction $\mathrm{AME}(2k,q) \Rightarrow \mathrm{AME}(2k-1,q)$ by projecting any one party yields the two odd-party states. The length-eighteen matrices were not found by brute force over all $9\times 9$ blocks: imposing a regular $\mathbb{Z}_3^2$ coordinate orbit forces a group-circulant form determined by a nine-entry kernel, and a finite-Fourier transform separates the self-duality equations before the MDS minor checks are run.

Load-bearing premise

The load-bearing premise is that the complete enumeration of all 48,619 square minors of each length-eighteen matrix was performed correctly; the paper asserts the computation but provides no code, log, or independent certificate.

Editorial extensions

If this is right

  • The five AME parameters in Eq. (1) are removed from the open cases, so protocols needing an AME(12,5), AME(17,11), AME(18,11), AME(17,13), or AME(18,13) state have an explicit stabilizer construction to use.
  • For $q=11$ and $q=13$ there now exist quantum MDS codes $[[18,0,10]]_q$ and, by projection, $[[17,0,9]]_q$, reaching the maximum distance allowed by the quantum Singleton bound.
  • The $q=5$ length-twelve code is not monomially equivalent to a generalized Reed–Solomon code, so the existence proof genuinely goes outside that classical family.
  • The certificate does not classify the states, so local-unitary and stochastic local operations with classical communication (SLOCC) equivalence classes among these constructions remain open questions.

Reading between the lines

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

  • If the length-eighteen certificates hold, the same $\mathbb{Z}_3^2$ group-circulant reduction could be tried for other groups and finite fields, making Hermitian self-dual MDS searches feasible at parameters where full $9\times9$ brute force is not.
  • The paper separates the search from the certificate, so a skeptical reader can verify the theorem directly from the printed matrices without reproducing the automorphism computation.
  • A natural follow-up is to test the two length-eighteen codes for monomial equivalence against each other and against other distance-ten constructions, parallel to the paper's non-GRS check for the $q=5$ code.
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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

1 major / 5 minor

Summary. The paper constructs explicit generator matrices for Hermitian self-dual MDS codes with parameters [12,6,7]_25, [18,9,10]_121, and [18,9,10]_169. Using the standard stabilizer-to-AME correspondence, it proves the existence of AME(12,5), AME(18,11), and AME(18,13), and via one-party projection also AME(17,11) and AME(17,13). The construction is based on group-circulant kernels with Z_3^2 symmetry, discovered from a direct search at length 12. The proof relies on two exact conditions: A A^T = -I (Hermitian self-duality) and superregularity (all square minors nonzero). The self-duality identities are stated explicitly; the superregularity is asserted from a computer enumeration without a published certificate.

Significance. If the square-minor enumerations are correct, the paper resolves five previously open questions in the AME existence table, including two cases at q=11 and two at q=13. The group-circulant symmetry reduction is a valuable technique for future AME constructions. The paper is careful to distinguish the search from the certificate, and the printed matrices provide a concrete starting point for independent verification. The main weakness is the lack of a reproducible verification artifact for the 97,238 square-minor checks, which is essential for confidence in the theorem.

major comments (1)
  1. [Section IV, proof of Theorem 2] The proof's two computational claims are not independently verifiable from the manuscript. The identities in Eq. (15) are asserted as 'exact arithmetic' without showing the products, and the superregularity check is described only by the sentence 'Gaussian elimination over the same fields shows that every nonempty square minor is nonzero.' No code, log, certificate, or checksum is provided for the 923 (A_5) and 48,619 (each A_q) minor computations, nor for the finite-field arithmetic in Eq. (15). Since Theorem 2's five AME existence statements depend on these exact matrices being Hermitian self-dual and superregular, this is a load-bearing gap. In particular, a single zero minor among the 97,238 checked would invalidate AME(18,11), AME(18,13), AME(17,11), and AME(17,13). The abstract's phrase 'certified by ... complete square-minor enumeration' overstates what the paper actually supplies. Please provide a reproducible verification artifact (e.g., a short script in Sage or Python that reconstructs the matrices from Eqs. (8), (11), (13) and verifies both conditions, or a machine-checked certificate of all minor determinants). Such an artifact would allow referees and readers to confirm the theorem rather than take the assertion on faith.
minor comments (5)
  1. [Section II, Eq. (2)] The displayed identity 'AA T = −I k' should be 'A A^T = -I_k' with proper superscripts, and 'detA[R,C]̸= 0' should include the missing parentheses around the submatrix.
  2. [Section III, Eq. (6)] The convolution sum in Eq. (6) is over the group H, but the field in which the arithmetic takes place is not explicitly stated; please specify that all operations in the sum are in F_{q^2}.
  3. [Section V] The sentence 'related Euclidean permutation-group analysis appears in Ref. [24]' is vague; a brief indication of the specific result or connection would help the reader.
  4. [References] Several references are arXiv preprints dated 2026 (e.g., Refs. [10], [29], [39]); please confirm their accessibility and ensure the citations point to the correct versions.
  5. [General] Providing the three matrices in a machine-readable supplementary file would greatly facilitate independent verification, even in addition to the requested verification script.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof is an explicit construction certified by direct arithmetic checks, independent of the search that found the matrices.

full rationale

The paper's central claim is an existence proof: the three printed matrices define Hermitian self-dual MDS codes, verified by the two checks in Eq. (2) and the displayed identities in Eq. (15). Criterion 1 is a standard equivalence theorem, and the matrices themselves are given explicitly, so the self-duality and superregularity conditions are finite, checkable statements rather than fitted parameters or predictions. The symmetry-reduced search is explicitly declared secondary to the certificate; Section V states 'the three printed matrices and the two checks in Eq. (2) suffice to prove the theorem.' No parameter is fitted to a subset of data and then renamed as a prediction. There are no self-citations by the authors carrying any load-bearing premise: the cited theorems on stabilizer codes, AME projection, and GRS bounds are external results with independent mathematical content. The only notable weakness is a reproducibility gap: the proof asserts that Gaussian elimination verifies all 48,619 square minors for each length-18 matrix without printing code, logs, or certificates. That is a verification artifact issue, not circularity, because the asserted checks are not logically equivalent to the theorem's conclusion by construction and an independent reader could in principle perform them. The open status of the parameters in the Huber-Wyderka table is contextual background, not an input to the derivation. Thus no circular step is present; the construction is self-contained modulo the unprinted exhaustive minor check, whose absence affects verifiability rather than circularity.

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

The certificate is self-contained except for standard results and the correctness of the huge exact computation. No numbers are fit to data; the field polynomials are standard irreducible choices, and no new physical entities are introduced.

assumptions (4)
  • standard math Criterion 1: G=[I_k|A] over F_{q^2} is Hermitian self-dual MDS iff A\bar{A}^T=-I_k and all square minors of A are nonzero.
    Quoted from Refs [13,14]; used as the core certificate in Theorem 2.
  • domain assumption Stabilizer correspondence: a Hermitian self-dual MDS [2k,k,k+1]_{q^2} code yields a pure quantum MDS [[2k,0,k+1]]_q state and hence an AME(2k,q) state.
    Standard result cited to Refs [15,16]; enables the move from classical codes to AME states.
  • domain assumption Projection: AME(2k,q) implies AME(2k-1,q) by projecting one party to a basis state.
    Standard theorem cited to Ref [2]; used to get the two length-17 states.
  • domain assumption The printed field embeddings and conjugation rules in Eqs. (7), (10), and (12) are correct.
    Exact arithmetic in these finite fields is the basis of the self-duality product checks; the formulas are explicit and checkable.

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Cite this review

Pith. "Pith review of Symmetry-guided constructions of absolutely maximally entangled states in five open cases." pith.science (2026). https://pith.science/paper/PP4NBG4R

@misc{pith2026260805781,
  author       = {Pith},
  title        = {Pith review of: Symmetry-guided constructions of absolutely maximally entangled states in five open cases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PP4NBG4R}},
  note         = {Machine review of arXiv:2608.05781}
}
abstract

We give explicit Hermitian self-dual maximum distance separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. The stabilizer construction proves the existence of ${\rm AME}(12,5)$, ${\rm AME}(18,11)$, and ${\rm AME}(18,13)$ states; one-party projection also gives ${\rm AME}(17,11)$ and ${\rm AME}(17,13)$. The first code was found by a direct search. A regular $\mathbb{Z}_3^2$ coordinate orbit of its automorphism group suggested a group-circulant form that reduces each length-eighteen construction to a nine-entry kernel. The printed matrices are certified by exact Hermitian products and complete square-minor enumeration.

Figures

Figures reproduced from arXiv: 2608.05781 by the authors.

Figure 1
Figure 1. FIG. 1. Two-stage search and certification pipeline. Steps 1–4: the length-twelve search without prescribed coordinate symmetry, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The geometry behind the symmetry transfer (steps 4 and 5 of Fig. 1). (a) What the automorphism computation [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Candidate counts at the stages of the two eighteen-party searches, grouped by stage on a shared logarithmic axis: blue [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: The local-Clifford reduction at q = 11 preserves the prescribed symmetry: the resulting matrix consists of 3 × 3 circulant blocks, verified for all 36 blocks. The q = 13 reduction chose pivots that break this pattern. C. All single-vertex deletions For each eighteen-ve…
Figure 4
Figure 4. Figure 4: ; all three matrices are supplied as both NumPy ar￾rays and plain text, and Appendix C describes the checks tying the packaged copies to the printed matrices. The weighted graphs defined by the three matrices are drawn in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The eighteen-vertex graph matrices, with diagonal phase terms set to zero as described in the text; color encodes [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The weighted graphs of the three even-party graph-state representatives, with vertices in coordinate order on a circle. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Entries of the archived Huber–Wyderka table snapshot of August 1, 2026[ [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The archived generators [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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