Approximate entanglement area law and robust quantum coherence

Entanglement bootstrap is the idea of using entanglement-based conditions to derive nontrivial constraints on quantum many-body systems. It provides nontrivial physical insights into the emergence of anyon species and their fusion spaces in gapped many-body systems. Previously, rigorous proofs were given in the context of exact entanglement area law. In this work, we study the same question under an approximate area law. Assuming an approximate version of entanglement bootstrap axioms and a Markov state condition on a closed manifold, we derive an approximate quantization of the maximum entropy. (The latter condition is expected to hold for Abelian anyon models.) The exponential of the maximum entropy is shown to be close to an integer up to a vanishingly small error. We also show that the set of states locally indistinguishable from the reference state is coherent.