I think it is computing the fine-grained entropy. However, I am confused by the case that when there is a black hole in the bulk. The Ryu-Takayanagi surface may include the horizon of the black hole in this case. However, the Bekenstein-Hawking entropy is a coarsed-grain entropy.
I have also heard that the RT formula is computing the coarse-grained entropy of the complementary region. What does this mean?
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I cannot tell you how stoked I am to see a fellow AdS/CFT-er. For the sake of the answer and physicality, I will not work with Ryu-Takayanagi but instead the covariant description, given by the Hubeny-Rangamani-Takayanagi (HRT) prescription. That is, the entanglement entropy of a boundary subregion is computed by the area of a (minimal) extremal surface $\mathcal{X}_{HRT}$. To answer your question, HRT is a fine-grained quantity (now see the case for RT), and the essential way to see this is that this does not involve coarse-graining on thermal d.o.f. The Jaynes' approach to holographic entanglement entropy describes the maximization of some von Neumann entropy to find a coarse-graining procedure, so when you have to make sense of black hole entropy in a varying state, e.g. finding a holographic second law (using holographic screens), you would have to coarse-grain. This is referred to as the Engelhardt-Wall prescription, which is a very fascinating result. There are some other aspects to this, but I think this should be a good motivation for RT as well.
Edit: Let me also point out that RT + coarse-graining would not be useful. Primarily, since RT is the statement that the for a black hole, the RT surface $\gamma _{A}$ "wraps" around the black hole horizon, giving the entropy of the black hole. However, it does not have the bulk interpretation with different kinds of marginally trapped surfaces (called minimar surfaces; extremal surfaces like $\mathcal{X}_{HRT}$ are a class of minimar surfaces), meaning that one cannot give a nice generalized second law interpretation of the black hole. See Engelhardt and Wall and Engelhardt and Bousso's works on holographic screens inside AdS black holes. RT will not give such kind of a description (why?), and does not have any physical importance like HRT does.
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$\begingroup$ Thank you for your answer. Can you recommend some papers on Jaynes' work? $\endgroup$– gshxdSep 30 at 5:09
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$\begingroup$ Well, I am aware of coarse-graining from Jaynes from two papers, Information Theory and Statistical Mechanics I and II. I would suggest seeing Engelhardt and Wall (and subsequent papers) works to actually get a feel for this. $\endgroup$– VaibhavKSep 30 at 11:09
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$\begingroup$ I thini RT is not useful for coarse-graining as it ia not covariant, right? By coarse-graining, do we mean by finding an entropy that obey the second law? $\endgroup$– gshxdSep 30 at 18:47
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$\begingroup$ @gshxd Well that is right. RT is a picture where one has to explicitly deal with a minimal spacelike geodesic joining boundary subregion endpoints. On the other hand, HRT does the covariant calculation for bulk surfaces, due to which we can coarse-grain it. Since RT is something that exists on a particular Cauchy slice (as opposed to HRT which has a maximin prescription), it does not make sense to coarse-grain it. Well, by coarse-graining we mean one that could be used to find a second law. See Engelhardt and Wall's paper for this. Are you aware of calculating outer entropy for minimars? $\endgroup$– VaibhavKOct 1 at 5:58
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$\begingroup$ Yes. One last thing. I have found in HRT's original paper that the HRT surface of a bulk with BTZ black hole is its apparent horizon, which is proposed by Engelhardt and Wall to be proportional to the outer entropy. Why is it? Also, Engelhardt and Wall say that for a black hole formed by gravitational collapse, its HRT surface is an empty set. Why is it? I think this fact alone is sufficient to argue that the HRT gives fine-grained entropy $\endgroup$– gshxdOct 1 at 18:03