Symmetries and local transformations of translationally invariant matrix product states

We determine the local symmetries and transformation properties of translationally invariant matrix product states (MPS). focus on physical dimension $d=2$ bond $D=3$ use procedure introduced in D. Sauerwein et al., Phys. Rev. Lett. 123, 170504 (2019) to all (including non--global) those states. identify classify stochastic transformations (SLOCC) that are allowed among MPS. scrutinize two very distinct sets MPS show big diversity (also compared case $D=2$) occurring both, their possible SLOCC transformations. These results reflect variety MPS, even if restricted with low dimension. Finally, we non-trivial measure zero for $d = 2$ $D > 3$.