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What is a good summary of the results about the correspondence between matrix product states (MPS) or projected entangled pair states (PEPS) and the ground states of local Hamiltonians? Specifically, what "if and only if" type of equality/approximation hold?

I know about this review-like paper by Verstraete, Cirac, and Murg but I feel some of its results are superseded by new ones.

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