Entanglement, fractional magnetization, and long-range interactions

Based on the theory of matrix product states, we give precise statements and complete analytical proofs of the following claim: A large fractionalization in the magnetization or the need of long-range interactions imply large entanglement in the state of a quantum spin chain.

Figure 1

Given the tensor $A=(\langle \alpha |A_i{|\beta \rangle )}_{i\alpha \beta }$ which defines the MPS, and with the usual convention that rotating means complex conjugation, we can represent the map ${\Gamma }_n^{*}{\Gamma }_n$ as the map in the figure from systems $ac$ to systems $bd$ and the map ${\mathbb{E}}^n$ as the same figure but now mapping systems $cd$ to systems $ab$.