Definition: Sub-capacity and global diagonal faithfulness

A strategy satisfies ${\mathrm{DF}}_{<N}$ if, for every input on $n\le N-1$ qubits and every positive-probability measurement branch,

Nullity ensures that the Lüders residual exists on such a branch. The diagonals are taken on the surviving full computational basis.

The condition ${\mathrm{DF}}_{\le N}$ imposes the same equality on inputs with $n\le N$. Neither condition identifies residual phases.