Eight-ququart AME state and the ((7,4,4)) ququart MDS code

Problem

Does an absolutely maximally entangled state of eight ququarts exist? More precisely, with \([8]=\{1,\ldots,8\}\), determine whether there is a unit vector \(|\psi\rangle\in(\mathbb{C}^{4})^{\otimes 8}\) such that

\begin{equation} \operatorname{Tr}_{S^{c}}\!\left(|\psi\rangle\!\langle\psi|\right) =\frac{I_{4^{4}}}{4^{4}} \qquad\text{for every }S\subseteq[8]\text{ with }|S|=4. \tag{1} \end{equation}

Condition (1) is equivalently the existence of a rank-eight perfect tensor of bond dimension \(4\): every flattening across a \(4|4\) partition is proportional to a unitary [PYHP15]. Under the pure-code correspondence, it is also equivalent to a pure quantum MDS code with parameters \(\bigl[\!\bigl[8,0,5\bigr]\!\bigr]_{4}\) [SZZL26]. Equivalently, the question asks whether there is a quantum maximum-distance-separable code with parameters \(((7,4,4))_{4}\): a four-dimensional subspace \(\mathcal{Q}\subseteq(\mathbb{C}^{4})^{\otimes 7}\) whose orthogonal projector \(P\) satisfies

\begin{equation} PEP=c(E)\,P \quad\text{for every operator }E\text{ acting nontrivially on at most three ququarts}, \qquad \dim\mathcal{Q}=4=4^{\,7-2(4-1)}, \tag{2} \end{equation}

where \(c(E)\in\mathbb{C}\) and the last equality is saturation of the quantum Singleton bound [HG20].

Source

Shi, Zhang, Zhao, and Li complete the seven-party AME classification and, in their Conclusion and Discussion, explicitly identify \(\operatorname{AME}(8,4)\) as the first remaining homogeneous existence case [SZZL26].

Progress

Bernal proved that a minimally supported \(\operatorname{AME}(N,4)\) state exists only for \(N\leq 6\) (Theorem 6). Thus any solution of (1) must have strictly more than \(4^{4}\) nonzero coefficients in every local product basis [Ber19].
Huber and Grassl proved that every quantum MDS code is pure (Theorem 5) and that a pure \(((n,1,n/2+1))_{D}\) code, that is, an \(\operatorname{AME}(n,D)\) state, exists exactly when an \(((n-1,D,n/2))_{D}\) code exists (Observation 1 and Proposition 7). For \(n=8\) and \(D=4\) this makes (1) and (2) equivalent. Given an orthonormal basis \(\{|a_L\rangle\}_{a=0}^{3}\) of a code satisfying (2), the state \(|\psi\rangle=\tfrac12\sum_{a=0}^{3}|a\rangle_R|a_L\rangle\), with \(R\) a reference ququart, satisfies (1); conversely, a Schmidt decomposition of such a state across one ququart versus the other seven yields the code. Their shadow-inequality bounds hold for general, not necessarily stabilizer, quantum MDS codes but do not exclude \(\bigl[\!\bigl[8,0,5\bigr]\!\bigr]_{4}\): in their table for local dimension \(4\), the family with \(n+k=8\) has upper bound \(\bigl[\!\bigl[8,0,5\bigr]\!\bigr]_{4}\), while the highest distance then attained by a known code in that family is that of \(\bigl[\!\bigl[6,2,3\bigr]\!\bigr]_{4}\) (Corollary 11 and Table 3) [HG20].
Ball, Moreno, and Simoens proved, with a computer-assisted classification, that no \(\bigl[\!\bigl[7,1,4\bigr]\!\bigr]_{4}\) stabilizer code exists, and deduced by projection that no \(\bigl[\!\bigl[8,0,5\bigr]\!\bigr]_{4}\) stabilizer code exists (Section 5.2 and Lemma 49 of arXiv version 2). Their stabilizers are abelian groups of Pauli operators labelled by \(\mathbb{F}_{4}\); they correspond to additive codes in \(\mathbb{F}_{4}^{2n}\) that are self-orthogonal under the trace-symplectic form and need not be \(\mathbb{F}_{4}\)-linear (Theorem 17), equivalently to binary stabilizer codes on \(2n\) qubits grouped in pairs (Theorem 21). Hence no state locally unitarily equivalent to an \(\mathbb{F}_{4}\) stabilizer state satisfies (1), and every code satisfying (2) lies outside the \(\mathbb{F}_{4}\) stabilizer formalism. At the start of Section 5 they also remark that stabilizer codes with these parameters over the Pauli group labelled by \(\mathbb{Z}/4\mathbb{Z}\) would yield binary \(\bigl[\!\bigl[7,1,4\bigr]\!\bigr]_{2}\) and \(\bigl[\!\bigl[8,0,5\bigr]\!\bigr]_{2}\) codes, which do not exist. These arguments do not address non-stabilizer states [BMS25].
Wójcik, Makuta, Bruzda, and Augusiak proved that no canonical \(\mathbb{Z}_{d}\) graph state can be \(\operatorname{AME}(N,d)\) when \(4\) divides \(N\) and \(d\) is even (Theorem 1). This excludes canonical \(\mathbb{Z}_{4}\) graph-state constructions for (1). Their argument uses the ring \(\mathbb{Z}_{d}\) and does not apply to stabilizer states over \(\mathbb{F}_{4}\): the authors point out that an \(\mathbb{F}_{4}\) stabilizer \(\operatorname{AME}(4,4)\) state exists, although their theorem excludes a canonical \(\mathbb{Z}_{4}\) graph state with those parameters [WMB+26].
Shi, Zhang, Zhao, and Li proved that \(\operatorname{AME}(7,d)\) exists exactly for \(d\geq 3\) (Theorem 1), settling the previously open cases \(\operatorname{AME}(7,6)\) and \(\operatorname{AME}(7,10)\) and, with earlier results, completing the homogeneous existence classification through seven parties. They identify \(\operatorname{AME}(8,4)\) as the first remaining open case (Conclusion and Discussion) [SZZL26].

Comment

The unresolved question concerns arbitrary complex states satisfying (1), equivalently arbitrary, possibly non-additive, codes satisfying (2). Existing no-go results exclude minimal-support states, all stabilizer states over \(\mathbb{F}_{4}\) (including those given by additive codes that are not \(\mathbb{F}_{4}\)-linear), and canonical \(\mathbb{Z}_{4}\) graph states; Ball, Moreno, and Simoens also remark that stabilizer codes over \(\mathbb{Z}/4\mathbb{Z}\) with these parameters do not exist. A construction must therefore have more than \(4^{4}\) nonzero coefficients in every local product basis and lie, up to local unitaries, outside both stabilizer formalisms, whereas a nonexistence proof must treat general quantum codes, for which the shadow-inequality bounds of Huber and Grassl do not suffice. A literature check through 15 September 2026 found no such construction or proof.

References

[PYHP15]: F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, “Holographic Quantum Error-Correcting Codes: Toy Models for the Bulk/Boundary Correspondence,” Journal of High Energy Physics 2015(6), 149 (2015).DOI arXiv
[Ber19]: A. Bernal, “On the Existence of Absolutely Maximally Entangled States of Minimal Support II,” Quantum Physics Letters 8, 1–4 (2019).DOI arXiv
[HG20]: F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces,” Quantum 4, 284 (2020).DOI arXiv
[BMS25]: S. Ball, E. Moreno, and R. Simoens, “Stabilizer Codes Over Fields of Even Order,” IEEE Transactions on Information Theory 71(5), 3707–3718 (2025); arXiv title “Stabiliser codes over fields of even order.”DOI arXiv
[WMB+26]: J. Wójcik, O. Makuta, W. Bruzda, and R. Augusiak, “On Non-Existence of Stabilizer Absolutely Maximally Entangled States in Even Local Dimensions,” arXiv:2603.18193 (2026).DOI arXiv
[SZZL26]: F. Shi, X. Zhang, Q. Zhao, and L. Li, “Complete Existence Classification of Seven-Partite Absolutely Maximally Entangled States,” arXiv:2608.01011 (2026).DOI arXiv

Eight-ququart AME state and the ((7,4,4)) ququart MDS code

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