Anshu, AnuragArad, ItaiVidick, Thomas2024-02-072016-05-23Anshu, Anurag, Itai Arad, Thomas Vidick. "Simple proof of the detectability lemma and spectral gap amplification." Phys. Rev. B 93, no. 20 (2016). DOI: 10.1103/physrevb.93.2051422469-99502469-9969https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37377727The 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 new, 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 O(γ−1/2), one gets an Hamiltonian with an Ω(1) spectral gap.en-USJournal Article2024-02-0710.1103/physrevb.93.205142