Symmetries and local transformations of translationally invariant Matrix Product States

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