Matrix product states with long-range localizable entanglement

We derive a criterion to determine when a translationally invariant matrix product state (MPS) has long-range localizable entanglement, where that quantity remains finite in the thermodynamic limit. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3.

Figure 1

Spin chain with $N$ spin-$S$ particles and one auxiliary particle of spin $S^{\prime }$ at each of the borders. The matrices of the real particles are $A_i$ and those of the auxiliary particles ${\mathbb{P}}^{†}$ and $\mathbb{Q}$, respectively.