Absolutely maximally entangled states

An $\mathrm{AME}(n,d)$ pure state has maximally mixed reductions on every set of at most half of its $n$ parties. Such perfect tensors connect secret sharing, quantum error correction and quantum maximum-distance-separable codes.

Classical MDS codes and index-one orthogonal arrays construct minimal-support AME states; they are not equivalent to unrestricted AME existence. The established $\mathrm{AME}(4,6)$ construction illustrates the distinction from classical orthogonal-Latin-square obstructions. “Planar maximally entangled” imposes a different collection of cuts and is not a synonym for AME.

For qubits, AME states exist for $n=2,3,5,6$, not for $n=4$ or $n\geq7$. Many higher-dimensional constructions come from classical MDS codes and multiunitary matrices. The four-party row is known for all $d\geq3$.

Preprint progress (August 2026): Shi–Zhang–Zhao–Li claim completion of the seven-party row, with existence exactly for $d\geq3$. Bevins–Bidav report explicit constructions for $(12,5)$, $(18,11)$, $(18,13)$, $(17,11)$ and $(17,13)$. These manuscript results close particular cells, not the full diagram. A parameter table should track proof status and the support restriction for each cell rather than label the entire classification one sharp conjecture.

Source disclosures

Shi et al. disclose GPT-5.6 assistance with the odd-q AME(7,2q) construction and state that they verified it independently. Source

Bevins and Bidav disclose extensive author-directed Claude Fable 5 and ChatGPT 5.6 Sol use for search, code, exact checks and manuscript preparation, and state that they reviewed the constructions and text. Source

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: Claimed completion of the seven-party existence row: AME(7,d) exactly for d at least 3. Evidence

preprint: Explicit code constructions for five previously unresolved parameter pairs. Evidence