MUB flatness rigidity for arbitrary numbers of qubits

Prove that, for arbitrary numbers of qubits, every pure state that is flat on its support in every basis of a complete stabilizer mutually unbiased basis (MUB) is a stabilizer state; equivalently, establish that all centered supports of such a flat state are additive subgroups up to translation.

Background

The paper defines an all-MUB-flat pure state as one whose nonzero measurement probabilities are equal within the support in every basis of a complete stabilizer MUB. It proves that stabilizer states have this property and that, within the additive-support branch, all-MUB-flatness is equivalent to stabilizerness.

The authors establish the conjecture for one and two qubits and exclude centered supports of cardinality three in arbitrary dimension. They also report numerical evidence for three qubits and argue that the algebraic flatness constraints are increasingly overdetermined for larger support dimensions. A proof for arbitrary numbers of qubits remains unresolved. The conjecture would establish that the optimized witness value M(|ψ⟩)=1 characterizes stabilizer states rather than merely all-MUB-flat states.

References

Conjecture 1 (MUB flatness rigidity). For arbitrary $n$, a pure state that is flat on its support in every basis of a complete stabilizer MUB is a stabilizer state. Equivalently, the set of equations (\ref{Qflat}) does not admit nonadditive physical solutions, i.e., all supports of a flat state are additive subgroups (up to translation).

— Optimized discrete Wigner representations and non-stabilizerness in qubit systems  (2610.10471 - Klimov et al., 7 Oct 2026) in Conjecture 1, Section IV.A, subsection “Flatness, phase matching, and the rigidity problem”