Comment on "Regional Versus Global Entanglement in Resonating-Valence-Bond States"

TL;DR

This paper critically examines previous claims about valence bond states' entanglement, correcting errors in specific cases and linking bounds to classical results, thereby clarifying the nature of multi-body versus two-site entanglement.

Contribution

It corrects prior calculations for the RVB liquid on the square lattice and provides an exact correlator value for the RVB gas, connecting bounds to classical results.

Findings

Incorrect calculations for the RVB liquid on the square lattice are corrected.

Exact correlator value for the RVB gas is provided, confirming the tightness of the bound.

Lower bounds for spin correlators relate to classical results by Anderson.

Abstract

In a recent Letter [Phys. Rev. Lett. 99, 170502 (2007); quant-ph/0703227], Chandran and coworkers study the entanglement properties of valence bond (VB) states. Their main result is that VB states do not contain (or only an insignificant amount of) two-site entanglement, whereas they possess multi-body entanglement. Two examples ("RVB gas and liquid") are given to illustrate this claim, which essentially comes from a lower bound derived for spin correlators in VB states. We show in this Comment that (i) for the "RVB liquid" on the square lattice, the calculations and conclusions of Chandran et al. are incorrect. (ii) A simple analytical calculation gives the exact value of the correlator for the "RVB gas", showing that the bound found by Chandran et al. is tight. (iii) The lower bound for spin correlators in VB states is equivalent to a celebrated result of Anderson dating from more than 50 years ago.