Detectability lemma
= Detectability lemma
{title2=$\|K|_{\operatorname{supp}H}\|\leq(1+\Delta/2)^{-1/3}$}
In a nearest-neighbour <frustration-free Hamiltonian> chain, multiply the bond <orthogonal projections> in two disjoint alternating layers. The resulting product $K$ fixes the common <ground state> space but contracts its orthogonal complement by a factor bounded away from one in terms of the <spectral gap>. A bound $\|K|_{\operatorname{supp}H}\|\leq(1+\Delta/2)^{-1/3}$ implies exponentially convergent powers. The product is generally neither Hermitian nor normal.