Existence of absolutely maximally entangled states at specified parameters

For which party counts n and local dimensions d does an absolutely maximally entangled state AME(n,d) exist? The recorded cases are (12,5), (18,11), (18,13), (17,11) and (17,13), using Hermitian self-dual maximum-distance-separable codes and projections.

Why it matters and what counts as progress

Why it matters: These parameters test existence of maximally entangled multipartite states through exact coding-theoretic constructions, without claiming a classification.

What counts as progress: Give exact code or state constructions for unresolved parameters, or prove nonexistence; existence certificates do not classify all such states.

How the problem developed

2004 · Foundational connection [1]
Andrew J. Scott Developed the connection between multipartite absolutely maximally entangled states and quantum error-correcting codes, giving a framework for constructing and excluding parameter cases.
10 Aug 2026 · Constructions [2]
Samuel Bevins and Yunus Bidav Constructed AME(12,5), AME(18,11) and AME(18,13), with projections giving AME(17,11) and AME(17,13), addressing five listed open cases.

AI contribution

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

Samuel Bevins, Yunus Bidav

Three explicit Hermitian self-dual MDS codes with parameters [12,6,7] over F25 and [18,9,10] over F121 and F169 establish AME(12,5), AME(18,11), and AME(18,13); one-party projection establishes AME(17,11) and AME(17,13). The two length-18 matrices come from nine-entry group-circulant kernels; the theorem asserts existence, not classification.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (exact finite-field computation + human)

Human understandingFull