An absolutely maximally entangled state of eight ququarts

An absolutely maximally entangled (AME) state of $n$ parties is as entangled as possible across every split into two halves: whichever half you look at, it is in a completely random state. Such states are known for many combinations of party number and local dimension, and since a 2026 preprint reported settling the seven-party case, the smallest open case is eight parties with four levels each.

Problem

A pure state $\lvert\psi\rangle\in(\mathbb{C}^d)^{\otimes n}$ is $\mathrm{AME}(n,d)$ if every reduced state on $\lfloor n/2\rfloor$ parties is maximally mixed:

$$\operatorname{Tr}_{S^c}\lvert\psi\rangle\langle\psi\rvert=\frac{I}{d^{\,|S|}}\qquad\text{for every } S\subseteq\{1,\dots,n\},\ |S|=\lfloor n/2\rfloor .$$

Question. Does an $\mathrm{AME}(8,4)$ state exist? That is, is there a unit vector in $(\mathbb{C}^4)^{\otimes 8}$ all of whose four-party marginals equal $I/256$?

By a result of Huber and Grassl, this is equivalent to the existence of a quantum code $((7,4,4))_4$: a four-dimensional subspace of seven ququarts with minimum distance four, which meets the quantum Singleton bound [1].

Background

AME states are pure quantum codes of maximal distance, the quantum relatives of classical MDS codes [1]. They are used for quantum secret sharing, multi-user teleportation and error correction, and as the perfect tensors of holographic codes [5]. The existence question for general $n$ and $d$ is No. 35 in the Open Quantum Problems collection of IQOQI Vienna (contact F. Huber, 2017), which already named eight ququarts among the smallest unresolved cases [6]. A 2026 review lists it first among its open questions [5].

What is known

• The general length bound for quantum MDS codes of Huber and Grassl, $n\le D^2+d-2$ for local dimension $D$ and distance $d\ge3$, allows these parameters [1].
• No $[[8,0,5]]_4$ stabilizer code exists, so no AME(8,4) state is a stabilizer state over $\mathbb{F}_4$; Ball, Moreno and Simoens proved this through the geometry of stabilizer codes over fields of even order [2].
• AME states of $4n$ parties with even local dimension cannot be canonical graph states over $\mathbb{Z}_d$, which excludes $\mathbb{Z}_4$ graph states here [3].
• In 2026 Shi, Zhang, Zhao and Li reported a proof that $\mathrm{AME}(7,d)$ exists exactly when $d\ge3$, completing the classification for up to seven parties (unrefereed preprint). They name AME(8,4) as the first remaining open case [4].

So a solution, if it exists, lies outside the standard stabilizer and graph-state constructions, and a nonexistence proof has to handle arbitrary, possibly non-additive, codes.

What would count as progress

• An explicit AME(8,4) state: $4^8=65{,}536$ amplitudes whose 35 four-versus-four marginals are all $I/256$. This can be checked numerically in seconds and, with exact algebraic entries, rigorously. An explicit $((7,4,4))_4$ code is equivalent.
• A proof of nonexistence, for example a semidefinite-programming or shadow-inequality certificate in exact arithmetic that rules out $((7,4,4))_4$ codes.
• Partial results that narrow the search, such as excluding further structured classes (states with a given symmetry group, or non-additive codes built from stabilizer pieces).
• The question is stated in Lean as ame_8_4_open in OpenQuantumProblems/35.lean of formal-conjectures.

References

1. F. Huber and M. Grassl, "Quantum codes of maximal distance and highly entangled subspaces", Quantum 4, 284 (2020). doi:10.22331/q-2020-06-18-284
2. S. Ball, E. Moreno and R. Simoens, "Stabilizer codes over fields of even order", IEEE Transactions on Information Theory 71, 3707–3718 (2025). doi:10.1109/TIT.2024.3454480
3. J. Wójcik, O. Makuta, W. Bruzda and R. Augusiak, "On non-existence of absolutely maximally entangled canonical graph states in even local dimensions", preprint (2026). arXiv:2603.18193
4. F. Shi, X. Zhang, Q. Zhao and L. Li, "Complete existence classification of seven-partite absolutely maximally entangled states", preprint (2026). arXiv:2608.01011
5. 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:10.1088/1361-6633/ae672a
6. F. Huber (contact), "Existence of absolutely maximally entangled pure states", Open Quantum Problems, IQOQI Vienna, problem 35 (posted 2017). Archived page

Source

Imported from QIQCOP Zoo op_b08ad9d4371ed0cb, curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.