Unclassified existence parameters for homogeneous AME states

Problem

For which of the parameter pairs specified below does an absolutely maximally entangled state exist? A normalized vector \(|\psi\rangle\in(\mathbb C^d)^{\otimes n}\) is an \(\operatorname{AME}(n,d)\) state when

\begin{equation} \operatorname{Tr}_{S^c}\!\left(|\psi\rangle\!\langle\psi|\right) =\frac{I_{d^{|S|}}}{d^{|S|}} \quad\text{for every }S\subseteq\{1,\ldots,n\} \text{ with }|S|\leq\left\lfloor\frac n2\right\rfloor. \tag{1} \end{equation}

Determine whether a state satisfying Eq. (1) exists for every pair in

\begin{equation} \mathcal U:=\bigl\{(8,6),(8,10),(9,6),(9,10),(10,6),(10,10), (11,3),(11,6),(11,10),(12,6),(12,10)\bigr\}. \tag{2} \end{equation}

Thus the task is to classify every pair in Eq. (2) by existence or nonexistence, without restricting to stabilizer or minimal-support states.

Source

The unresolved parameter list is drawn from the AME existence table and open tasks compiled by Rajchel-Mieldzioć et al., with cases removed when later constructions settled them [RBR+26].

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Existence under Eq. (1) is equivalent to a pure one-dimensional quantum error-correcting code with parameters \(((n,1,\lfloor n/2\rfloor+1))_d\). Quantum-code bounds and shadow inequalities therefore provide nonexistence tests, but they do not settle the pairs in Eq. (2) [Sco04], [HG20].
General constructions yield \(\operatorname{AME}(5,d)\) and \(\operatorname{AME}(6,d)\) for every \(d\geq2\), while the qubit existence problem is completely classified: an \(\operatorname{AME}(n,2)\) state exists exactly for \(n\in\{2,3,5,6\}\). These results delimit, but do not decide, Eq. (2) [RBR+26].
Two 2026 advances remove stale benchmark cases: the seven-party classification proves \(\operatorname{AME}(7,d)\) exists exactly when \(d\geq3\), and an exact Hermitian self-dual MDS code constructs \(\operatorname{AME}(12,5)\). Neither result decides any pair retained in Eq. (2) [SZZL26], [BB26].

Comment

The set in Eq. (2) is the nonredundant remainder of the audited DeepMind benchmark list: the solved pairs \((7,6)\), \((7,10)\), and \((12,5)\) are excluded, while the unresolved \((8,4)\) case is covered by the eight-ququart perfect-tensor problem. The related problems on real AME states, minimal-support AME states, and finite nontrivial local-unitary moduli of AME states respectively impose a real-coefficient constraint, ask for minimal support, and classify local-unitary orbits; none duplicates the unrestricted existence predicates collected here.

References

[Sco04]: A. J. Scott, “Multipartite Entanglement, Quantum-Error-Correcting Codes, and Entangling Power of Quantum Evolutions,” Physical Review A 69, 052330 (2004).DOI arXiv
[HG20]: F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces,” Quantum 4, 284 (2020).DOI arXiv
[RBR+26]: G. Rajchel-Mieldzioć, R. Bistroń, A. Rico, A. Lakshminarayan, and K. Życzkowski, “Absolutely Maximally Entangled Pure States of Multipartite Quantum Systems,” Reports on Progress in Physics 89, 057601 (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
[BB26]: S. Bevins and Y. Bidav, “Symmetry-Guided Constructions of Absolutely Maximally Entangled States in Five Open Cases,” arXiv:2608.05781v2 (2026).DOI arXiv

Unclassified existence parameters for homogeneous AME states

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