An improved 1D area law for frustration-free systems

TL;DR

This paper introduces a new combinatorial proof for the 1D area law in frustration-free quantum systems, significantly tightening the entropy bounds and providing insights that could impact understanding of entanglement in quantum many-body systems.

Contribution

It presents an exponentially improved entropy bound for 1D frustration-free systems with a constant gap, using a novel combinatorial approach combining the detectability lemma and approximation theory.

Findings

Entropy bound is $S_{1D} o O(1) X^3 ext{log}^8 X$ with $X = (J ext{log} d)/ ext{epsilon}$

Proof is fully combinatorial and tight up to a polynomial factor

Locality considerations suggest near-optimal bounds in 2D cases

Abstract

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings' 1D area law, and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\epsilon>0$ and interaction strength of at most $J$, our entropy bound is $S_{1D}\le \orderof{1}X^3\log^8 X$ where $X\EqDef(J\log d)/\epsilon$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. Incorporating locality into the proof when applied to the 2D case gives an entanglement bound that is at the cusp of being non-trivial in the sense that any further improvement would yield a sub-volume law.