Filtering preserves existing frustration freeness
= Filtering preserves existing frustration freeness
If each local term is already minimized by the global <ground state>, subtract its minimum $a_Z$ to obtain a <positive operator>. Nonnegative normalized <spectral filtering of Hamiltonian terms> preserves positivity and leaves the ground eigenvalue $a_Z$ unchanged. Thus the same state minimizes every filtered term. This preserves <frustration freeness> but does not create it for an initially frustrated decomposition.