  For the past twenty years, Matrix Product States (MPS) have been widely used
in solid state physics to approximate the ground state of one-dimensional spin
chains. In this paper, we study homogeneous MPS (hMPS), or MPS constructed via
site-independent tensors and a boundary condition. Exploiting a connection with
the theory of matrix algebras, we derive two structural properties shared by
all hMPS, namely: a) there exist local operators which annihilate all hMPS of a
given bond dimension; and b) there exist local operators which, when applied
over any hMPS of a given bond dimension, decouple (cut) the particles where
they act from the spin chain while at the same time join (glue) the two loose
ends back again into a hMPS. Armed with these tools, we show how to
systematically derive `bond dimension witnesses', or 2-local operators whose
expectation value allows us to lower bound the bond dimension of the underlying
hMPS. We extend some of these results to the ansatz of Projected Entangled
Pairs States (PEPS). As a bonus, we use our insight on the structure of hMPS
to: a) derive some theoretical limitations on the use of hMPS and hPEPS for
ground state energy computations; b) show how to decrease the complexity and
boost the speed of convergence of the semidefinite programming hierarchies
described in [Phys. Rev. Lett. 115, 020501 (2015)] for the characterization of
finite-dimensional quantum correlations.
