Entanglement in Finitely Correlated Spin States

TL;DR

This paper derives bounds for entanglement in finitely correlated spin states, proving a conjecture about entanglement conditions and enabling more efficient calculations of entanglement in quantum spin chains.

Contribution

It provides exact bounds for entanglement in finitely correlated states, confirming a recent conjecture and generalizing previous models like AKLT.

Findings

Bounds become exact for single spin and half-infinite chain entanglement

Confirms the sufficiency of a necessary entanglement condition

Enables more efficient numerical and analytical entanglement calculations

Abstract

We derive bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure finitely correlated state (FCS) on a chain of spins of any magnitude. Finitely correlated states are otherwise known as matrix product states or generalized valence-bond states. The bounds become exact in the limit of the entanglement of a single spin and the half-infinite chain to the right (or the left) of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for non-vanishing entanglement in terms of a single spin and the ``memory'' of the FCS, is also sufficient . Our result also generalizes the study of entanglement in the ground state of the AKLT model by Fan, Korepin, and Roychowdhury. Our result permits one to calculate more efficiently, numerically and in some cases even analytically, the entanglement of arbitrary finitely correlated quantum spin chains.