What the global gap gives immediately

Likewise, the local operators $I-P_{\mathrm{supp}{\rho }_X}$ annihilate a pure state, but their sum need not have a unique ground state or a positive comparison gap. For a full-Schmidt-rank entangled pair, each one-site support projector is the identity, so all such one-site constraints vanish. Constructing useful positive quasi-local constraints is substantive additional work.