Absolutely maximally entangled (AME) states are quantum states where every bipartition of the system into two equal halves produces a maximally mixed reduced state. They are the most entangled states possible — maximal entanglement is not just achieved for one partition but for all partitions simultaneously. These states are foundational for quantum error correction, secret sharing, and multipartite entanglement theory.
Graph states — constructed from simple graphs via local Clifford operations — provide the most tractable route to building AME states. The stabilizer formalism makes them computationally manageable and experimentally preparable.
The result: AME states of 4k qudits with even local dimension cannot be realized as graph states. When the local dimension is even (qutrits with d=4, quhexas with d=6, etc.) and the number of parties is a multiple of four, the graph-state construction is provably incapable of producing maximal entanglement.
The obstruction is algebraic: it arises from the interaction between the parity structure of even dimensions and the graph-state stabilizer structure. Odd dimensions (like standard qutrits with d=3) do not face this barrier. The same entanglement target — AME states — is reachable via graph states in odd dimensions but unreachable in even dimensions for these system sizes.
The structural point: the failure is not about entanglement being impossible. AME states may still exist in these parameters via non-graph-state constructions. The limitation is specifically about the stabilizer/graph framework — the most efficient construction tool — being insufficient. The tool fails where the target does not.
(arXiv:2603.18193)