Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

Fei Shi; Lvzhou Li; Qi Zhao; Xiande Zhang

Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States

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Pith's one-line read Seven-qudit AME states exist exactly in local dimension $d\ge 3$.

desk verdict Settles the last open existence question for AME(7,d) with a genuinely new construction family; the proof is explicit and checkable, and the central argument holds up. read the letter →

arxiv 2608.01011 v1 pith:P3BLC5DF submitted 2026-08-02 quant-ph

Fei Shi , Xiande Zhang , Qi Zhao , Lvzhou Li This is my paper · ORCID

classification quant-ph

keywords absolutelymaximallyentangledstatesAME(7d)sevenquditsquantumMDScodesquadratic-phaseSteanecodeFanoplaneHamming

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 settles the existence problem for seven-partite absolutely maximally entangled states: an AME(7,d) state exists if and only if $d\ge 3$. The only negative case is $d=2$, already known, and the genuinely new work supplies explicit states in every odd dimension and every dimension congruent to 2 modulo 4. The odd-dimensional family is a cyclic quadratic-phase state built from the complement of the seven-cycle, and the twice-odd family couples the Steane-code logical basis to such phase states so that every three-party reduced density is maximally mixed. A reader should care because AME states are pure quantum MDS codes and are used in quantum secret sharing, parallel teleportation, and holographic error correction; closing the seven-party case removes an outstanding entry from the standard AME table.

What carries the argument

Two explicit state families carry the proof. The odd family is the cyclic quadratic-phase state $|G_q\rangle$ with phase matrix the upper-triangular part of the complement of the seven-cycle; the AME check becomes invertibility of $3\times3$ submatrices over $\mathbb{Z}_q$. The twice-odd family is the coupled state $|\Psi_{2q}\rangle=(|0_L\rangle|G_0^{(q)}\rangle+|1_L\rangle|G_1^{(q)}\rangle)/\sqrt{2}$, where $|0_L\rangle,|1_L\rangle$ are Steane-code logical states and $|G_a^{(q)}\rangle$ are one-qudit phase-shifted copies. The central mechanism is the complementary vanishing of the binary cross block (nonzero only on the seven Fano triples $L_t=\{t,t+1,t+3\}$) and the odd cross block (vanis

What would settle it

Build $|G_q\rangle$ from Eq. (4) for an odd composite, say $q=9$, using exact arithmetic, and compute each of the 35 three-party reduced density matrices $\rho_S=\operatorname{Tr}_{S^c}|G_q\rangle\langle G_q|$; the odd-dimensional branch is false if any $\rho_S\neq I/9^3$. A cheaper check is to compute the five representative determinants in Eq. (5) modulo $q$ and confirm, via translation invariance, that every one of the 35 regenerated $\Gamma_{S,R}$ has nonzero determinant modulo $q$.

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Extended reading notes

Core claim

The paper claims the complete classification AME(7,d) exists iff $d\ge 3$. For odd $q$, it constructs $|G_q\rangle=q^{-7/2}\sum_{x\in\mathbb{Z}_q^7}\omega_q^{Q(x)}|x\rangle$ with $Q(x)=x\tilde\Gamma x^T$ and $\Gamma$ the complement of the seven-cycle; by a cited criterion, checking the state is AME reduces to showing each of 35 three-by-three submatrices $\Gamma_{S,R}$ is invertible over $\mathbb{Z}_q$, which translation symmetry reduces to five determinant computations. For $d=2q$, it forms $|\Psi_{2q}\rangle=(|0_L\rangle|G_0^{(q)}\rangle+|1_L\rangle|G_1^{(q)}\rangle)/\sqrt{2}$, identifying the qubit-qudit basis with a qudit of dimension $2q$. The binary cross term $R_{01}^S$ vanishes on al

Load-bearing premise

The load-bearing premise is the cited Theorem 3 of Ref. [8] that a quadratic-phase state over $\mathbb{Z}_q^7$ is AME whenever, for every three-subset $S$, some $3\times3$ submatrix $\Gamma_{S,R}$ is invertible over $\mathbb{Z}_q$; the paper does not prove this theorem or restate its full hypotheses, so every odd-dimension state (and every composite dimension built from it) depends on it.

Editorial extensions

If this is right

For every $d\ge 3$, the paper gives explicit normalized pure states: Eq. (4) for odd $d$, Eq. (31) for $d\equiv2\pmod4$, and product or Reed-Solomon states for the remaining powers of two.
Every seven-partite system with local dimension $d\ge3$ admits a pure state maximally entangled across every bipartition, so the seven-party AME table is complete.
The previously open cases $\mathrm{AME}(7,6)$ and $\mathrm{AME}(7,10)$ are resolved, with explicit states written out in Eq. (40).
Since AME(7,d) is equivalent to a pure quantum MDS code $[[7,0,4]]_d$, the classification proves such codes exist exactly for $d\ge3$.
The failure at $d=2$ is isolated: no other local dimension obstructs seven-party absolutely maximally entangled states.

Reading between the lines

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

One can view the construction as transferring the obstruction from the binary sector to a small exceptional set of cuts; this suggests a general recipe for AME$(n,2q)$ whenever the binary code has dual distance $\lfloor n/2\rfloor+1$ and the partner phase state kills the exceptional cuts.
The complete seven-party classification makes the qubit case $d=2$ look like the sole number-theoretic exception; it is natural to test the same binary-odd coupling on the next open case, AME(8,4).
Replacing $\mathbb{Z}_q$ in the phase state with other finite abelian groups, if the invertibility criterion carries over, would give a direct construction for even dimensions without invoking the product property, possibly simplifying the $\alpha\ge2$ branches.
Because AME(7,d) is a pure $[[7,0,4]]_d$ MDS code, the classification also closes the existence question for seven-qudit QMDS codes; the remaining hard cases shift to lengths $n\ge8$.

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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

The headline: the paper proves AME(7,d) exists iff d≥3, and it does. The genuinely new piece is the coupled binary-odd construction for d=2q (odd q≥3), which covers all previously open dimensions 6, 10, 14, and so on. The cyclic quadratic-phase construction for every odd d is a welcome uniform family, though it leans on a cited theorem rather than being fully self-contained.

What the paper does well is make the proof concrete. The odd branch reduces to five 3x3 determinants (-1, 1, 2, -1, 1), all units for odd q, so the rank condition is transparent. The twice-odd branch is worked out in real detail: binary cross blocks vanish on 28 non-Fano cuts via a clean support argument, and the seven Fano cuts are killed by explicit null-vector certificates in Table II. The case split d = 2^α m with α = 0, 1, 2, ≥3 covers every dimension, and the product property is standard. I also checked for circularity: nothing in the proof assumes the classification it is proving.

Soft spots are minor. The main external dependency is Theorem 3 of Ref. [8], which gives the sufficient condition for quadratic-phase states to be AME; the paper does not restate its full hypotheses. A referee will need to confirm the theorem applies as stated, but the specific use is only rank checks on 3x3 matrices, so verification is cheap. The d=2 nonexistence and the product property are cited standard results, which is fine. The disclosure that GPT-5.6 assisted in developing the twice-odd construction is handled honestly: the authors state they independently verified everything, and the paper supplies enough detail for an independent check. I do not see that as a red flag.

This paper is for people working on AME states, quantum MDS codes, and multipartite entanglement. It closes a known open problem with explicit constructions, and the completeness of the classification is genuinely useful for anyone querying which AME(7,d) exist. It is not a field-reshaping breakthrough, but it is a solid, complete result. The construction techniques, especially the binary-odd coupling, may carry over to other small n.

Recommendation: yes, send it to peer review. The referee should verify the hypotheses of Theorem 3 and spot-check the determinant and Table II computations, but the core argument is sound and the contribution is real.

Referee Report

0 major / 4 minor

Summary. The paper proves the complete existence classification of seven-partite absolutely maximally entangled states: AME(7,d) exists if and only if d ≥ 3. The proof is constructive. For every odd q ≥ 3, the cyclic quadratic-phase state |G_q> in Eq. (4) is shown to be AME by verifying, through the five translation representatives S = 012, 013, 014, 015, 024, that suitable 3×3 submatrices Γ_{S,R} have determinants −1, 1, 2, −1, 1, all units over Z_q for odd q. For d = 2q with q odd, the coupled state |Ψ_{2q}> in Eq. (31) combines the two logical states of the Steane code with the odd-dimensional states |G_0^{(q)}> and |G_1^{(q)}>. The binary cross-block R^S_{01} is shown to vanish on all 28 non-Fano three-subsets by a support argument using the simplex/Hamming code structure, and the odd-sector cross-block T^S_{01} is shown to vanish on the seven Fano cuts via the null-vector certificates in Table II. Together with the known nonexistence at d = 2, the known AME(7,4) state, the Reed–Solomon construction for powers of two, and the product property for AME states, the claimed classification follows.

Significance. If correct, this resolves previously open cases including AME(7,6) and AME(7,10), and completes the existence classification of AME states for all n ≤ 7. The constructions are explicit and checkable: the five submatrix determinants are computed, the Fano-cut cancellation is certified by the seven explicit null vectors in Table II, and Eq. (40) gives an explicit wavefunction for the twice-odd dimensions. The proof is largely self-contained, with only standard code-theoretic facts and one cited rank criterion used as inputs. The completeness of the case split in terms of d = 2^α m with m odd is clear. This is a clean and significant resolution of a well-known open problem.

minor comments (4)

[Odd local dimensions, Eq. (6)] The proof of Eq. (6) relies entirely on Theorem 3 of Ref. [8], whose exact hypotheses are not restated. Since the classification requires composite odd dimensions (q = 9, 15, ...), please either quote the theorem's hypotheses explicitly or include the short character-orthogonality argument showing that the stated rank condition over Z_q is sufficient for every odd q. The argument is elementary, but as written the reader cannot verify the scope of the cited theorem.
[Twice-odd local dimensions, Eq. (10)] The notation 'I23 23' in Eq. (10) is ambiguous; it should be written as I_{2^3}/2^3 (equivalently I_8/8).
[Twice-odd local dimensions, before Eq. (34)] Typo: 'Accorrding' should be 'According'.
[Twice-odd local dimensions, Eqs. (10) and (13)] The phrase 'the distance of C and (C+1) are both 4' could be misread, since C+1 contains words of weight 3. Please clarify that the minimum distance between distinct elements of the coset C+1 is 4 (as a nonlinear code), not the minimum Hamming weight of its elements.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified: constructions are explicit and the only self-citation is backed by an independent reference.

full rationale

The paper's central claim, AME(7,d) exists iff d≥3, is established by three independent construction families, each verified by direct computation rather than by assuming the conclusion. The odd-dimension branch uses Theorem 3 of Ref. [8] (Feng et al.), an external result that gives a sufficient condition for a quadratic-phase state to be AME; the paper then verifies the condition explicitly for five orbit representatives, with determinants −1,1,2,−1,1, all nonzero for odd q. This is a genuine check, not a circular use of the classification. The twice-odd construction is self-contained: the binary cross block vanishes on non-Fano cuts via the simplex/Hamming code support argument (Eq. (13)), and the odd cross block vanishes on the seven Fano cuts via explicit null-vector certificates (Table II, Eqs. (25)–(29)), with h_t c^T nonzero modulo q. The final split d=2^α m uses the external product property of AME states, cited to both Rains (Ref. [17]) and the authors' own Ref. [18]; because Rains is an independent source, the self-citation is not load-bearing. The power-of-two cases and the Reed–Solomon construction are also cited to external literature, and nonexistence at d=2 is cited to an external result. No fitted parameter is renamed as a prediction; no theorem from the present authors' prior work is invoked to forbid alternatives; no quantity is defined in terms of the target claim. The note about GPT-5.6 assistance is a transparency statement and is followed by independent verification, so it does not introduce circularity. Overall, the derivation chain is self-contained apart from standard, externally corroborated ingredients.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No unknown parameters are fitted; the only hand choice is the vector c. All other inputs are standard theorems or previously published existence results. The constructions introduce no new physical entities such as particles, forces, or dimensions.

free parameters (1)

c = (1,-1,1,0,0,0,0) = (1,-1,1,0,0,0,0)
Chosen by hand in the twice-odd construction so that h_t c^T is nonzero modulo every odd q for all seven Fano cuts; this forces T_{01}=0 on those cuts and cancels the binary cross terms.

assumptions (6)

domain assumption Theorem 3 of Ref. [8]: if for every 3-subset S some Γ_{S,R} with R⊆S^c is invertible over Z_q, then the quadratic-phase state |G_q> is AME.
Used to conclude Eq. (6) from the five determinant checks. Not proved in the paper.
standard math Tensor product of AME(7,d1) and AME(7,d2) is AME(7,d1d2) (Refs. [17,18]).
Used in the classification case split to combine known prime-power states with odd-dimension states.
domain assumption AME(7,4) exists (Refs. [19,20]).
Covers α=2 in the d=2^α m split.
domain assumption A [7,3,5] generalized Reed-Solomon code over F_{2^α}, α≥3, yields an AME(7,2^α) state (Refs. [8,19,22]).
Covers α≥3 in the case split.
domain assumption AME(7,2) does not exist (Ref. [9]).
Provides the necessity half of the iff classification.
standard math The Hamming code D=C∪(C+1) has dual distance 4, so its projection onto any 3 coordinates is uniform; the weight-3 words of C+1 have Fano-plane supports.
Used in the binary part to derive Eq. (10) and the support restriction Eq. (13); partially verified by Table I.

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Pith. "Pith review of Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States." pith.science (2026). https://pith.science/paper/P3BLC5DF

@misc{pith2026260801011,
  author       = {Pith},
  title        = {Pith review of: Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P3BLC5DF}},
  note         = {Machine review of arXiv:2608.01011}
}

abstract

We prove that an absolutely maximally entangled state of seven qudits exists if and only if the local dimension satisfies $d\geq 3$. Prior to this work, to the best of our knowledge, $\text{AME}(7,d)$ states were known to exist only when $d$ is a prime power other than $2$, or when $d$ can be expressed as a product of dimensions for which existence was already known. Since it has been proved that no $\text{AME}(7,2)$ state exists, it remains to establish existence for all $d\geq 3$. We construct cyclic quadratic-phase states for every odd local dimension and develop a coupled binary--odd-dimensional construction for every dimension congruent to $2$ modulo $4$. Together with the known power-of-two cases and the product property of AME states, these constructions cover every local dimension $d\geq 3$.

Figures

Figures reproduced from arXiv: 2608.01011 by the authors.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Symmetry-guided constructions of absolutely maximally entangled states in five open cases
quant-ph 2026-08 conditional novelty 7.0 of 10

New explicit Hermitian self-dual MDS codes yield AME(12,5), AME(18,11), AME(18,13), and their 17-party projections.

Reference graph

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Pith tools

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