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My question is about the top tensor in MERA -- might be a little technical and naive:

Does the top tensor always transform the tensor from layer-$(\tau -1)$ into a pure state?

If so, suppose I firstly optimize all tensors for a specific off-critical Hamiltonian, then I reverse the RG direction, going from top to bottom (say, roughly, IR to UV), but start from two different initial states at the top (both pure), using this finite-step circuit to re-construct the original-lattice states. Then what's the difference between the two final off-critical states on the original-lattice?

I know when I optimize the MERA, the top tensor should also be optimized. So I suppose that means if I start from another initial (top) state, it would not lead to a reasonable original-lattice state. But does it has any meaning?

Also, can I don't choose to use a state for the top layer, but rather, start with a general density-matrix at the top? Then what original-lattice state does this re-construction lead to? (Also a mixed state?)

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