A simple proof of the detectability lemma and spectral gap amplification

TL;DR

This paper presents a simplified proof of the detectability lemma applicable to any ordering of local projectors, and demonstrates how it can be used to amplify spectral gaps in frustration-free Hamiltonians.

Contribution

It offers a new, simpler proof of the detectability lemma that is tight up to a constant and applies to arbitrary orderings, enabling spectral gap amplification.

Findings

The proof is simpler and applies to any ordering of projectors.

The lemma is tight up to a constant factor.

Spectral gap amplification is achieved by coarse-graining Hamiltonians.

Abstract

The 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 $\gamma>0$ to a length scale $O(\gamma^{-1/2})$, one gets an Hamiltonian with an $\Omega(1)$ spectral gap.