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 two subsystems A and B, where A is a block of L consecutive spins and B is its complement. In that setting we compute the partial transpose density matrix with respect to 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 AB, we find an expression for the reduced density matrix of the A and B blocks, (A, B). With this expression (in the thermodynamic limit), we compute the entanglement spectrum and other several entanglement measures, such as the purity P = tr((A, B) ²), the negativity and the mutual entropy.