Entanglement of disjoint blocks in the one dimensional Spin 1 VBS

Abstract

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.