Entanglement of disjoint blocks in the one dimensional Spin 1 VBS

Starting with the valence bond solid (VBS) ground state of the 1D AKLT Hamiltonian, we make a partition of the system in 2 subsystems $A$ and $B$, where $A$ is a block of $L$ consecutive spins and $B$ is it's complement. In that setting we compute the partial transpose density matrix with respect to $A$, $\rho^{T_A}$. We obtain the spectrum of the transposed density matrix of the VBS pure system. Subsequently we define two disjoint blocks, $A$ and $B$ containing $L_A$ and $L_B$ spins respectively, separated by $L$ sites. Tracing away the spins which do not belong to $A\cup B$, we find an expression for the reduced density matrix of the $A$ and $B$ blocks $\rho(A,B)$. With this expression (in the thermodynamic limit), we compute the entanglement spectrum and other several entanglement measures, as the purity $P={\rm tr}(\rho(A,B)^2)$, the negativity $\mathcal{N}$, and the mutual entropy.