Classical restrictions of generic matrix product states are quasi-locally Gibbsian

We show that the norm squared amplitudes with respect to a local orthonormal basis (the classical restriction) of finite quantum systems on one-dimensional lattices can be exponentially well approximated by Gibbs states Hamiltonians (i.e., are quasi-locally Gibbsian) if conditional mutual information (CMI) any connected tripartition lattice is rapidly decaying in width middle region. For injective matrix product states, we moreover CMI decays exponentially, whenever collection operators satisfies 'purity condition'; notion previously established theory random products. furthermore violations purity condition enables generalized error correction virtual space, thus indicating non-generic nature such violations. make this intuition more concrete constructing probabilistic model where typical property. The proof our main result makes extensive use products, and may find applications elsewhere.