An absolutely maximally entangled state of eight ququarts

Find a pure state of eight four-level systems whose every four-party marginal is maximally mixed, or prove that none exists.

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	1	+# An absolutely maximally entangled state of eight ququarts
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+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:
	8	+
	9	+$$\operatorname{Tr}_{S^c}\lvert\psi\rangle\langle\psi\rvert=\frac{I}{d^{\,\|S\|}}\qquad\text{for every } S\subseteq\{1,\dots,n\},\ \|S\|=\lfloor n/2\rfloor .$$
	10	+
	11	+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 $I/256$?
	12	+
	13	+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].
	14	+
	15	+## Background
	16	+
	17	+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].
	18	+
	19	+## What is known
	20	+
	21	+- 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].
	22	+- 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].
	23	+- 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].
	24	+- 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].
	25	+
	26	+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.
	27	+
	28	+## What would count as progress
	29	+
	30	+- 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.
	31	+- A proof of nonexistence, for example a semidefinite-programming or shadow-inequality certificate in exact arithmetic that rules out $((7,4,4))_4$ codes.
	32	+- 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).
	33	+- The question is stated in Lean as `ame_8_4_open` in [OpenQuantumProblems/35.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OpenQuantumProblems/35.lean) of formal-conjectures.
	34	+
	35	+## References
	36	+
	37	+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](https://doi.org/10.22331/q-2020-06-18-284)
	38	+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](https://doi.org/10.1109/TIT.2024.3454480)
	39	+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](https://arxiv.org/abs/2603.18193)
	40	+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](https://arxiv.org/abs/2608.01011)
	41	+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](https://doi.org/10.1088/1361-6633/ae672a)
	42	+6. F. Huber (contact), "Existence of absolutely maximally entangled pure states", Open Quantum Problems, IQOQI Vienna, problem 35 (posted 2017). [Archived page](https://web.archive.org/web/20231029013508/https://oqp.iqoqi.oeaw.ac.at/existence-of-absolutely-maximally-entangled-pure-states)
	43	+
	44	+## Source
	45	+
	46	+Imported from [QIQCOP Zoo op_b08ad9d4371ed0cb](https://qiqc-op.com/problem/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.