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 $P=\mathrm{tr}(\rho (A,B{)}^2)$, the negativity $\mathcal{N}$, and the mutual entropy.

28 pages, 4 figures

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• Exact Excited States of Non-Integrable Models
• The entanglement negativity in random spin chains
• Entanglement negativity after a local quantum quench in conformal field theories
• Macroscopic Quantum Entanglement of a Kondo Cloud at Finite Temperature
• Long-range entanglement and topological excitations
• Negativity in the Generalized Valence Bond Solid State