We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with projected entangled pair states (PEPS) as a function of the bond dimension (D), temperature (beta(-1)), and system size (N). First, we introduce a compression method in which the bond dimension scales as D = e(O(log22 (N/epsilon))) if beta < O (log(2) N). Second, building on the work of Hastings [M.B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, D = (N/epsilon)(O(beta)). This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with D = N-O(log2 N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice

We analyze entanglement in the family of translationally-invariant matrix product states (MPS). We give a criterion to determine when two states can be transformed into each other by SLOCC transformations, a central question in entanglement theory. We use that criterion to determine SLOCC classes, and explicitly carry out this classification for the simplest, non-trivial MPS. We also characterize all symmetries of MPS, both global and local (inhomogeneous). We illustrate our results with examples of states that are relevant in different physical contexts. arXiv:1901.07448v2 [quant-ph]