Negativity in the Generalized Valence Bond Solid State

TL;DR

This paper analyzes the entanglement properties of the generalized Valence Bond Solid state using a graphical approach based on $SU(2)$ representation theory, revealing that negativity is non-zero only for neighboring blocks.

Contribution

It introduces a graphical method to compute the spectrum of the reduced density matrix and negativity in the VBS state, applicable to similar algebraic models like Levin-Wen models.

Findings

Negativity is non-zero only for adjacent subsystems.

The method can be extended to Levin-Wen models.

Provides a spectrum of the mixed state reduced density matrix.

Abstract

Using a graphical presentation of the spin $S$ one dimensional Valence Bond Solid (VBS) state, based on the representation theory of the $SU(2)$ Lie-algebra of spins, we compute the spectrum of a mixed state reduced density matrix. This mixed state of two blocks of spins $A$ and $B$ is obtained by tracing out the spins outside $A$ and $B$, in the pure VBS state density matrix. We find in particular that the negativity of the mixed state is non-zero only for adjacent subsystems. The method introduced here can be generalized to the computation of entanglement properties in Levin-Wen models, that possess a similar algebraic structure to the VBS state in the groundstate.