Existence of an eight-ququart perfect tensor

Unsolved ID op_b08ad9d4371ed0cb Last edited 4 September 2026

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Fields: Quantum Resource Theory
Quantum Error Correction
Topics: Absolutely maximally entangled states
Quantum coding theory

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].

Source

Shi, Zhang, Zhao, and Li complete the seven-party AME classification and 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 \(N\leq 6\). Thus any solution of (1) must have strictly more than \(4^{4}\) nonzero coefficients in every local product basis [Ber19].
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. This excludes canonical \(\mathbb{Z}_{4}\) graph-state constructions for (1), but their theorem does not exclude stabilizer constructions over \(\mathbb{F}_{4}\) or general non-stabilizer states [WMB+26].
Shi, Zhang, Zhao, and Li proved that \(\operatorname{AME}(7,d)\) exists exactly for \(d\geq 3\), completing the homogeneous existence classification through seven parties. They identify \(\operatorname{AME}(8,4)\) as the first remaining open case [SZZL26].

Comment

The unresolved question concerns arbitrary complex states satisfying (1). Existing no-go results cover minimal-support states and canonical \(\mathbb{Z}_{4}\) graph states, but do not rule out non-minimal-support \(\mathbb{F}_{4}\) stabilizer states or non-stabilizer constructions.

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
[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

Existence of an eight-ququart perfect tensor

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