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The multiscale entanglement renormalization ansatz (MERA) provides aconstructive algorithm for realizing wavefunctions that are inherently scaleinvariant. Unlike conformally invariant partition functions however, the finitebond dimension $\chi$ of the MERA provides a cut-off in the fields that can berealized. In this letter, we demonstrate that this cut-off is equivalent to theone obtained when approximating a thermal state of a critical Hamiltonian witha matrix product operator (MPO) of finite bond dimension $\chi$. This isachieved by constructing an explicit mapping between the isometries of a MERAand the local tensors of the MPO.

Entanglement compression in scale space: from the multiscale entanglement renormalization ansatz to matrix product operators