An absolutely maximally entangled (AME) state is one where every bipartition of the system into halves yields a maximally entangled pair. For four qubits (dimension 2), AME states exist — the GHZ and cluster states provide examples. For four qutrits (dimension 3), AME states exist. For four six-dimensional qudits, the question has been open.
The paper proves that no stabilizer AME(4,6) state exists. The proof reduces the existence question to a combinatorial constraint on orthogonal Latin squares. A stabilizer AME(4,6) state would require a set of mutually unbiased bases with specific algebraic structure, and that structure forces the existence of a combinatorial object that Euler conjectured does not exist — and which was proved nonexistent in the 1960s.
The connection: stabilizer states are algebraically constrained, and those constraints translate directly to classical combinatorial objects. The quantum state's entanglement structure maps to the existence of orthogonal Latin squares of order 6, which is exactly the combinatorial configuration that doesn't exist. The quantum impossibility inherits the combinatorial impossibility.
Non-stabilizer AME(4,6) states might still exist — the result blocks only the algebraic construction, not the general case. But the blocked construction is the one that systematic search would find first, and the reason it fails is not quantum mechanical but combinatorial. The obstacle to perfect entanglement in this specific case is a 200-year-old theorem about arranging symbols in a grid. Quantum information theory discovers its limits by running into classical combinatorics.