Negativity in the generalized Valence Bond Solid state

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 nonzero 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 ground state.