  We consider the tensor formulation of the non-linear O(2) sigma model and its
gauged version (the compact Abelian Higgs model), on a $D$-dimensional cubic
lattice, and show that tensorial truncations are compatible with the general
identities derived from the symmetries of these models. This means that the
universal properties of these models can be reproduced with highly simplified
formulations desirable for implementations with quantum computers or for
quantum simulations experiments. We briefly discuss the extensions to
non-Abelian symmetries and models with fermions.
