Area law in one dimension: Degenerate ground states and Renyi entanglement entropy

TL;DR

This paper proves an area law for the Renyi entanglement entropy in one-dimensional gapped quantum systems with degenerate ground states, providing bounds and efficient approximations by matrix product states.

Contribution

It establishes an area law for Renyi entanglement entropy in degenerate ground states and offers bounds on entanglement and matrix product state approximations.

Findings

Renyi entanglement entropy is upper bounded by ilde O(rac{ ext{constant}}{ ext{gap}})

Ground states can be approximated by matrix product states with subpolynomial bond dimension

The results apply to systems with constant degeneracy and energy gap

Abstract

An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems. Suppose in a chain of $n$ spins the ground states of a local Hamiltonian with energy gap $\epsilon$ are constant-fold degenerate. Then, the Renyi entanglement entropy $R_\alpha(0<\alpha<1)$ of any ground state across any cut is upper bounded by $\tilde O(\alpha^{-3}/\epsilon)$, and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension $2^{\tilde O(\epsilon^{-1/4}\log^{3/4}n)}$.