  We prove that every injective Matrix Product State is the unique ground state
of a simple hopping theory. We start by studying the low energy spectrum of
parent Hamiltonians of injective Matrix Product States in a particular long
range and system size limit under the validity of an asymptotic regime with low
particle density. We show that in this limit a natural first quantization
arises. This allows us to compute a specific type of low energy spectrum. This
spectrum depends solely on the properties of a quantum channel, i.e. transfer
matrix of the ground state, and not on any other details of the ground-state.
We also review normal quantum channels for which the expression is more
simplified. The construction possibly has some interesting uses for the study
of quantum and classical Markov processes which we briefly expose. As an
application, we revisit the notion of (many-body)-localization with our
framework. Our calculations revealed that translational invariant Matrix
Product States can be interpreted as a stationary sea of particles. As a next
step rather than starting from some local Hamiltonian with random potentials,
we consider fluctuations of the local tensors of a continuous one-parameter
family of Matrix Product States. Localization in 1-dimension, is then
understood from a simple study of spectral and mixing properties of finite
dimensional quantum channels.
