Matrix Product States: Entanglement, Symmetries, and State Transformations

We analyze entanglement in the family of translationally invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by local operations with a nonvanishing probability, a central question in entanglement theory. This induces a classification within this family of states, which we explicitly carry out for the simplest, nontrivial MPS. We also characterize all symmetries of translationally invariant MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts.