An Area Law for One Dimensional Quantum Systems

TL;DR

This paper proves an area law for entanglement entropy in gapped one-dimensional quantum systems, revealing a rapid growth of entropy bounds with correlation length and implications for state approximation.

Contribution

It establishes an area law for 1D gapped systems and explores bounds on entropy, including a conjecture on quantum expanders and implications for matrix product state approximations.

Findings

Proves an area law for entanglement entropy in 1D gapped systems.

Shows entropy bounds grow rapidly with correlation length.

Demonstrates ground state can be approximated by matrix product states.

Abstract

We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy for sufficiently large $\alpha<1$ and implies the ability to approximate the ground state by a matrix product state.