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This study presents the exact construction and certification of new absolutely maximally entangled states using symmetry-guided methods in nonbinary quantum codes, advancing foundational resources for distributed quantum computing. The work demonstrates how combining systematic search with group algebra decomposition can efficiently identify high-dimensional entanglement structures previously inaccessible through standard generalized Reed-Solomon approaches. Notably, the research leverages large language models to accelerate the discovery process while maintaining rigorous mathematical verification via exact field arithmetic and graph-state cut-rank computations.
Symmetry-guided construction and exact certification of absolutely maximally entangled states
arXiv:2608.05781v1 Announce Type: new Abstract: We construct Hermitian self-dual maximum-distance-separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. Through the nonbinary stabilizer construction these codes define $\text{AME}(12,5)$, $\text{AME}(18,11)$, and $AMD(18,13)$ states, and one-party projection gives $\text{AME}(17,11)$ and $\text{AME}(17,13)$. The $[12,6,7]_{25}$ code, which is not monomially equivalent to a generalized Reed-Solomon code, is obtained by a systematic search without prescribed coordinate symmetry; its monomial-semilinear automorphism group acts with a regular $\mathbb{Z}_3^2$ orbit on nine coordinates. Imposing that action on two nine-coordinate orbits reduces the length-eighteen search to a nine-element group algebra kernel, whose Hermitian character decomposition splits the invariant self-duality constraints into independent blocks with exact family counts; a norm-one coordinate scaling removes a factor $q+1=14$ at $q=13$. Hermitian self-duality and the MDS property are certified by exact field arithmetic and complete square-minor enumeration. Graph-state cut-rank computations, including all single-vertex deletions of the length-eighteen graphs, provide additional exact checks.
Abstract:We construct Hermitian self-dual maximum-distance-separable codes with parameters $[12,6,7]_{25}$, $[18,9,10]_{121}$, and $[18,9,10]_{169}$. Through the nonbinary stabilizer construction these codes define $\text{AME}(12,5)$, $\text{AME}(18,11)$, and $AMD(18,13)$ states, and one-party projection gives $\text{AME}(17,11)$ and $\text{AME}(17,13)$. The $[12,6,7]_{25}$ code, which is not monomially equivalent to a generalized Reed-Solomon code, is obtained by a systematic search without prescribed coordinate symmetry; its monomial-semilinear automorphism group acts with a regular $\mathbb{Z}_3^2$ orbit on nine coordinates. Imposing that action on two nine-coordinate orbits reduces the length-eighteen search to a nine-element group algebra kernel, whose Hermitian character decomposition splits the invariant self-duality constraints into independent blocks with exact family counts; a norm-one coordinate scaling removes a factor $q+1=14$ at $q=13$. Hermitian self-duality and the MDS property are certified by exact field arithmetic and complete square-minor enumeration. Graph-state cut-rank computations, including all single-vertex deletions of the length-eighteen graphs, provide additional exact checks.
15 pages, 8 figures. Results were found and written up with the help of Claude Fable 5 (Anthropic) and ChatGPT 5.6 Sol (OpenAI). AME(12,5) was found within 3 hours of beginning the investigation, with the other 4 AME states found autonomously while preparing the manuscript
Quantum Physics (quant-ph); Information Theory (cs.IT)
arXiv:2608.05781 [quant-ph]
(or arXiv:2608.05781v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2608.05781 arXiv-issued DOI via DataCite (pending registration)
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