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Multi-scale entanglement renormalization ansatz (MERA) includes two kinds of isometric tensors, disentanglers and isometries. Thus tensors in MERA are by definition composed of isometries. Meanwhile we have another tensor network called PEPS. Reading this post (https://physics.stackexchange.com/a/188302/141894), it seems MERA can be represented as PEPS. When constructing MERA from PEPS by reducing tensors, how can one make tensors of PEPS isometric even though PEPS, which includes loops, does not have any canonical form?

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  • $\begingroup$ Having isometric tensors is part of the definition of MERA. What is your question? $\endgroup$ – Norbert Schuch Sep 12 at 7:05
  • $\begingroup$ Usually a TN with loops does not have a canonical form and can only be approximately canonical. (If I am understanding it correctly,) This means at least one tensor in the TN does not satisfy orthonormality. Doesn’t this contradicts the existence of a MERA? or does it mean MERA is not sufficient to express a completely general state even if the bond dimension is taken to be large enough? $\endgroup$ – Amplituhedron Sep 13 at 17:28
  • $\begingroup$ Having isometric tensors is part of the definition of MERA. The same tensor network without the isometric tensors is not called a MERA. --- What is your question? (And please edit the question to make that clear.) $\endgroup$ – Norbert Schuch Sep 13 at 18:58
  • $\begingroup$ And of course MERA can express any state with sufficiently large bond dimension, you can just take all isometries to be the identity. $\endgroup$ – Norbert Schuch Sep 13 at 19:15
  • $\begingroup$ I added the last paragraph to make my question clear. So when you want to express an arbitrary state with MERA, its top tensor is important? MERA is the top tensor itself? $\endgroup$ – Amplituhedron Sep 16 at 4:19

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