In Swingle's work Entanglement Renormalization and Holography, he mentioned that a black hole in AdS bulk corresponds to a finite temperature boundary state. With the MERA picture of the entanglement renormalization, the coarse grained state at the IR side becomes a completely mixed state.

I am not clear why the coarse grained state is a 'completely mixed state'. If the MERA structure is regarded as a unitary quantum operation (where the disentanglers are unitary and the isometries can also be regarded as unitary by adding ancilla $|0\rangle$ states). Then the coarse grained state of a thermal boundary state is a mixed state with a diagonal density matrix, but not necessarily a completely mixed state (unitary operation does not change the spectrum of the boundary state).

Why the coarse grain operation will increase the temperature so that the coarse grained state has an infinite temperature? What if the coarse grained state is a normal mixed state with a diagonal density matrix, what should we call it? Something like a black hole but not exactly?

PS: Constructing holographic spacetimes using entanglement renormalization is a more detailed version. It seems the RG operation is not exactly unitary so the correlation in the non-completely mixed state with a diagonal density matrix is also removed by RG operations. This looks like a whitening of a colored noise.

But still, why whitening the noise? What if I keep this colored (Gaussian) noise? What's the difference between the thing in the center of the bulk and a black hole? A highly curved spacetime just before forming a black hole or a generalized black hole? From the state complexity point of view, the two systems have the same complexity.

Another related issue is the eternal black holes in AdS, which correspond to entangled thermal states, where the temperature is not infinite and the BH entropy is just given by the thermal entropy of a system with a finite temperature. Maybe the size of the eternal black holes are restricted by $S_{BH}$ so it corresponds to a subset of the DOF of the boundary states and the coarse grained system still has an infinite temperature.


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