  We prove that all R\'enyi entanglement entropies of spin-chains described by
generic (gapped), translational invariant matrix product states (MPS) are
extensive for disconnected sub-systems: All R\'enyi entanglement entropy
densities of the sub-system consisting of every k-th spin are non-zero in the
thermodynamic limit if and only if the state does not converge to a product
state in the thermodynamic limit. Furthermore, we provide explicit lower bounds
to the entanglement entropy in terms of the expansion coefficient of the
transfer operator of the MPS and spectral properties of its fixed point in
canonical form. As side-result we obtain a lower bound for the expansion
coefficient and singular value distribution of a primitve quantum channel in
terms of its Kraus-rank and entropic properties of its fixed-point. For unital
quantum channels this yields a very simple lower bound on the distribution of
singular values and the expansion coefficient in terms of the Kraus-rank.
Physically, our results are motivated by questions about equilibration in
many-body localized systems, which we review.
