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

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