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

TL;DR

This paper demonstrates that classical restrictions of 1D quantum states can be approximated by local Gibbs states if their classical conditional mutual information decays rapidly, linking quantum information properties to Gibbsian approximations.

Contribution

It establishes conditions under which classical restrictions of quantum states are quasi-locally Gibbsian, using decay of classical CMI and properties of matrix product states.

Findings

Classical restrictions can be approximated by Gibbs states under certain conditions.

Classical CMI decays exponentially for injective MPS satisfying a purity condition.

Violations of the purity condition relate to error correction on the virtual space.

Abstract

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