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arXiv:2608.05781

Symmetry-guided constructions of absolutely maximally entangled states in five open cases

Samuel Bevins, Yunus Bidav

quant-phcs.ITSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations• Revised Aug 10, 2026 (v2)
v1 · Aug 6, 2026v2 · Aug 10, 2026
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Abstract

We give explicit Hermitian self-dual maximum distance separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. The stabilizer construction proves the existence of ${\rm AME}(12,5)$, ${\rm AME}(18,11)$, and ${\rm AME}(18,13)$ states; one-party projection also gives ${\rm AME}(17,11)$ and ${\rm AME}(17,13)$. The first code was found by a direct search. A regular $\mathbb{Z}_3^2$ coordinate orbit of its automorphism group suggested a group-circulant form that reduces each length-eighteen construction to a nine-entry kernel. The printed matrices are certified by exact Hermitian products and complete square-minor enumeration.

Comments
4 pages. Results were found and written up with the help of Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI)

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Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of nnn parties with local dimension qqq a power of a prime for all q≥n−1q \geq n-1q≥n−1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q≥n−1q \geq n-1q≥n−1 prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ ⁣[n,1,n/2] ⁣]q[\![n,1,n/2]\!]_q[[n,1,n/2]]q​ for nnn even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to n=8n = 8n=8.
Abstract Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q ⩾ n − 1 . Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q ⩾ n − 1 prime, we show how to construct stabilizer QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [ [ n , 1 , n / 2 ] ] q for n even with the highest distance allowed by the quantum Singleton bound.
We address the relation between absolutely maximally entangled (AME) states and quantum- maximum-distance-separable (QMDS) codes by constructing whole families of QMDS codes from AME states that have stabilizer representations. We introduce a generic reduction-friendly form for the generator set of the stabilizer representation of an AME state, from which the stabilizer form for children codes, all QMDS, can be obtained. Our method would work for relevant high-dimensional codes as well as qubit-based codes. We then relate this to optimal codes for one-way quantum repeaters, by minimizing the short-term infrastructure cost as well as the long-term running cost of such quantum repeaters. This would allow us to obtain the optimal QMDS code, derived from an AME parent state, that can be used in such quantum repeaters.
The group of transversal gates and the group of local symmetries are important features of quantum error correcting codes and pure quantum states, respectively; the former provides fault-tolerant operations on a code while the latter tells us about a state's reachability via stochastic local operations with classical communication. We prove that there exists a bijection between local unitary (LU) orbits of absolutely maximally entangled (AME) states in (CD)⊗n(\mathbb{C}^D)^{\otimes n}(CD)⊗n where nnn is even, also known as perfect tensors, and LU orbits of ((n−1,D,n/2))D((n-1,D,n/2))_D((n−1,D,n/2))D​ quantum error correcting codes. Furthermore, there is a close connection between the local symmetries of an AME state and the transversal gates of its corresponding quantum error correcting code. We explore in detail the 4-qutrit AME state ∣Φ⟩|Φ\rangle∣Φ⟩ and its corresponding ((3,3,2))3((3,3,2))_3((3,3,2))3​ qutrit code C\mathcal{C}C. We show that ∣Φ⟩|Φ\rangle∣Φ⟩ and C\mathcal{C}C are both unique up to the action of the LU group. We find generators of the local symmetry group of ∣Φ⟩|Φ\rangle∣Φ⟩ and the group of transversal gates on C\mathcal{C}C. Our proofs rely on prior results by Huber and Grassl (2020), Hebenstreit et al. (2016), and Rather et al. (2023). We use Vinberg's theory of Lie algebras to study the special case of ∣Φ⟩|Φ\rangle∣Φ⟩ and C\mathcal{C}C.
Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols and the construction of quantum error correcting codes. To our knowledge, the only known systematic construction of these states is based on a class of classical error correction codes known as maximum distance separable. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from maximum distance separable codes. Furthermore, we used our method to construct several examples of absolutely maximally entangled states whose existence was open so far.