Anshu, AnuragArad, ItaiVidick, Thomas2025-11-242025-11-242025-11-212016-05-2310.1103/PhysRevB.93.2051422-s2.0-84970950224https://infoscience.epfl.ch/handle/20.500.14299/256277The detectability lemma is a useful tool for probing the structure of gapped ground states of frustration-free Hamiltonians of lattice spin models. The lemma provides an estimate on the error incurred by approximating the ground space projector with a product of local projectors. We provide a simpler proof for the detectability lemma which applies to an arbitrary ordering of the local projectors, and show that it is tight up to a constant factor. As an application, we show how the lemma can be combined with a strong converse by Gao to obtain local spectral gap amplification: We show that by coarse graining a local frustration-free Hamiltonian with a spectral gap γ>0 to a length scale Oγ-1/2, one gets a Hamiltonian with an Ω1 spectral gap.entruetext::journal::journal article::research article