Entropy and Entanglement in Quantum Ground States

TL;DR

This paper explores the relationship between correlations and entanglement in gapped quantum systems, demonstrating that some systems have exponentially large entropy but can still be efficiently represented by matrix product states, impacting numerical simulations.

Contribution

It proves the existence of gapped 1D Hamiltonians with large entropy and provides evidence for systems with arbitrarily large entropy, while showing efficient representations are possible under certain conditions.

Findings

Existence of gapped 1D Hamiltonians with exponential entropy.

Strong evidence for systems with arbitrarily large entropy.

Efficient matrix product state representations are possible under certain density of states assumptions.

Abstract

We consider the relationship between correlations and entanglement in gapped quantum systems, with application to matrix product state representations. We prove that there exist gapped one-dimensional local Hamiltonians such that the entropy is exponentially large in the correlation length, and we present strong evidence supporting a conjecture that there exist such systems with arbitrarily large entropy. However, we then show that, under an assumption on the density of states which is believed to be satisfied by many physical systems such as the fractional quantum Hall effect, that an efficient matrix product state representation of the ground state exists in any dimension. Finally, we comment on the implications for numerical simulation.