 into the CFT, which we're also imagining factorizes. And so this is just a qualitative picture of the setup at this point. But the claim of the theorem is that these two statements are equivalent. And the first statement is this reconstruction statement that in this picture says, if you have some operator, phi acting in little b, then there exists some operator on capital B that reconstructs it in that precise sense of, right here, there exists an O acting just on B such that this equation holds psi. And likewise, it's important that for any operator acting in b bar, little b bar, that's reconstructable on capital B bar, sorry, that was capital B bar. And the second one is the statement that if you compute the entropy of capital B, it's given by some formula like this. And this is important for us because in ADS CFT, we believe, we have a formula like this. We know that we can compute the entropy of capital B using some prescription involving the bulk physics that finds some, what we call the quantum extremal surface, which is really the minimal quantum extremal surface. And so this theorem tells us that if you find that, that divides the region, capital B can reconstruct from the region that capital B bar can reconstruct. So that's the nice thing. And I didn't say this, but there's language that people use in this context. So gamma, remember, was the quantum extremal surface. And then little b is sometimes called the entanglement wedge. I actually don't really like this name because it's very long and obscure. So this is the entanglement, little b is like the entanglement wedge of capital B and little b bar is the entanglement wedge of capital B bar. So this theorem goes by the name of entanglement wedge reconstruction. So whenever people talk about entanglement wedge reconstruction, it's the statement that some CFD subregion can reconstruct operators in its entanglement wedge. And now you know what that means and you have seen the theorem that shows it to be the case, at least in this sense that it can reconstruct operators that work on this bulk subspace. I do need to say really the entanglement wedge means, so here I'm just drawing a time slice. If I were to draw the full bulk picture, where this time slice here is now this circle here, so little b is this guy and little b bar is this guy. Capital B, the entanglement wedge is actually the full domain of dependence of little b, which makes, so like if you were to find the full domain of dependence of little b, it would sort of end on the demanded, it would intersect the boundary at the domain of dependence of capital B. So it's like a wedge, maybe it's more obvious if I draw it. So if I were to consider this region, C, and it has a quantum extremal surface there, its entanglement wedge hits the boundary like this, it's like this full co-dimension zero region of the bulk that is really the entanglement wedge, but it's uniquely picked out by how the entanglement wedge intersects this slice, so sometimes people will use this term to mean either one. But anyways, entanglement wedge reconstruction holds in either case, if you have some operator that's not say on this one particular time slice, but is further up somewhere else, still in this shaded wedge, C can still reconstruct it, because in principle you could have, it can act any operator up here, you could have say evolved, you can just do it by reconstructing say any operator you want, even complicated ones on this time slice, including sort of the Heisenberg evolved version of this phi back to this slice. So this is all just to say, even though I've been discussing things in terms of time slices, really when we think about reconstruction, some region say B of the CFT can reconstruct sort of an entire co-dimension zero region of the AES, not just some operators on the one time slice. Won't be super important for the rest, but it is important to hear that. So then I mentioned after showing this theorem that in the assumptions of the theorem we have this H code, which is a subspace, you should imagine is a subspace of HADS, and not the full bulk Hilbert space, where the full HADS, I'm taking to be the Hilbert space that has various black holes of different sizes and the vacuum state, there's a great many states. This H code was one where the QES position for all intents and purposes didn't really change that much. So it's much smaller than this. It's a shortcoming because the operators that are guaranteed to exist on capital B that act like say this phi are only guaranteed by this theorem to act like that phi when acting on a state in the subspace. That's just the fact of the theorem, and you might worry if that's, my wonder is that just a shortcoming of the theorem, perhaps there is a way in general to write on capital B this operator phi in a way that works for all bulk states, that works for any state here, not just this limited fixed QES subspace that the theorem was talking about. But then we saw that that hope would be too ambitious. This is where we started talking about quantum error correction even though the full exploration of quantum error correction and what it means would take a while. The point, I do wanna emphasize, the point was that if indeed some subregion B, let's have B is 2 thirds of the bound. So this is its quantum extremal surface. So it's entanglement wedge is this big part. There is some operator here phi. If B could always reconstruct this operator regardless as if we put some black hole or something somewhere, there would be a problem. And the reason is that this theorem, the reconstructions that were guaranteed allows you to reconstruct phi on many different boundary regions. So not just this B, but also one that's say, I'm outlining here in blue and also different ones. So you can basically cut out any say third of the boundary and still find a way to reconstruct phi. If all of those reconstructions acted like phi on the entire Hilbert space, this would be a problem because as we argued, they would have to be, they would have to commute with every operator near the boundary and therefore every CFT operator. And that would mean that it would have to be trivial. So the upshot was that these reconstructions that you can do of an operator and the entanglement wedge, those reconstructions, you should really only trust them to work in the subspace and not expect them to work more generally outside of that. So that's just life. It turns out that's how the ADS-CFT dictionary works is that this sort of operator reconstruction we want to do is, has this limit, has this limitation that the reconstructions we get sort of work at best nicely on a subspace. But that said, it's certainly not clear from what I've said exactly how big that subspace can be. Maybe certainly they work on this fixed QES subspace, but maybe they work on something much bigger. I can't work on everything, but what's the limit? And I haven't, I would say that's a question we basically understand now, but we'll have to build up some technology to answer it. My goal now is to say, given this, and given that, yeah, given that our goal was to talk about reconstructing operators that measured interesting things, like what happens inside of black holes. Let's now ask if these reconstructions are sufficient. So they only work on some say fixed QES subspace so far, the ones that we're talking about, but maybe that's enough to start answering some interesting questions about black holes. I'm gonna now argue to you that it's not, we actually need to generalize, we need to say a more powerful theorem that tells us what sort of reconstructions we can do outside of these fixed QES subspaces because the questions we wanna ask about black holes can't be formulated in these fixed QES subspaces. So I'll say it this way. There's more to life QES subspaces. To argue this, I will just give you an example of some interesting physics that we would like to understand better and why we can't formulate physics that we can in principle understand through this reconstruction idea, but we can't find a fixed QES subspace in which this physical process lives. Okay, and that is evaporating black holes. So the idea is this. So the basic overview of evaporating black holes in ADS is the following. So let's say we have anti-decentre space that has some star in it, or some, let's say dust cloud that's pretty big. And it's big enough that it collapses under its own weight and forms a black hole. The Penner's diagram for this might look like the following. You can construct these metrics, what are they called? The Oppenheimer-Sneider collapse is an example of this sort of thing. So this is a Penner's diagram where this wavy line here is the singularity. This dashed line here is the horizon of the black hole. This left line is not an asymptotic boundary. It's just the, it's like the r equals zero point. This right side is the asymptotic boundary. So a CFT lives here. So this is a black hole and asymptotically anti-decentre space. This is the asymptotically ADS part. And then this shaded region here is this dust ball that was down here collapsing under its own weight. And then at some point falls inside of its own short-shild radius at this point. And this is the causal structure given by this Penner's diagram. So let's just think about this. So this is a situation where we form a black hole and if we make the dust cloud big enough, it's what's called a large black hole where the short-shild radius is bigger than the ADS radius, that L that appeared in the metric we wrote down for ADS. And so it's a big black hole. And that fact will be important momentarily. So one thing Hawking taught us about black holes is so we have this background, this geometry and let's consider what the quantum fields are doing on top of it. And the important thing is that there are modes entangled across this horizon. So I'll draw it like this. So there's this mode, say in this mode, maybe there are two photons and they're entangled. So I'll draw, I'll represent the entanglement by this dotted line connecting those guys. And this guy is outside the horizon and this guy is inside. And this guy will escape all the way to the asymptotic boundary whereas this guy cannot. He will eventually hit the singularity. And this is what an observer say who's floating out here near the boundary or maybe living at the boundary would observe as Hawking radiation. So he would see all these photons hitting him and that would be Hawking radiation to him. And now usually when we have ADS, we have to choose what the boundary conditions are at asymptotic infinity. And often we choose reflecting boundary conditions. And in that case, this photon would hit the asymptotic boundary and then bounce off and fall back into the black hole. And so even though the black hole would have lost energy for a bit, a bit of time as this photon flew away, the photon hits asymptotic infinity and bounces back in finite time relative to the black hole. This is a fact actually that it's true in ADS and I don't think I said it, which is that if you have some observer say floating in the middle of ADS and then say they shot a laser towards the boundary, radially outwards, then they continued to float in ADS. Even though there's infinite spatial volume, space is sort of crunching in the sense, sort of in the opposite way to the sitter expanding such that this laser would actually be able to go and hit the boundary and in this case reflect off of it because of these reflecting boundary conditions and then re-hit the observer who is staying at r equals zero in a definite finite amount of the observer's proper time. So that same effect means that the black hole is going to radiate to have its radiation hit the boundary and fall back in. So in any black hole that's big enough, so a large black hole where it's short child radius is larger than the ADS radius will be large enough to be effectively stable under this process so that it's not gonna shrink and radiate away like black holes in flat space. That's what usually happens because we usually choose reflecting boundary conditions but now we're gonna choose different boundary conditions because we wanna talk about evaporating black holes. This is a thing that became very popular in 2019 these page curve calculations if you've heard of those. We're gonna choose boundary conditions such that when a photon hits the boundary it has some chance that we can make pretty large to pass through it and into some baths. We'll call this the bath or the reservoir. Maybe the reservoir is a better name and this doesn't need to be gravitational. It can just be some non-gravitational flat space region. So the photon passes through boundary and then maybe when it hits here, it appears here and now it's just gonna live in this flat space non-gravitational region for the rest of its life. And so this is by choosing these boundary conditions this is a way for us to allow this black hole in ADS to radiate, eventually evaporate. So this radiation passes through to this reservoir, the black hole loses energy and we'll get smaller and smaller just like black holes evaporating in flat space. And what's especially nice about this is that more and more radiation will accumulate in the reservoir. So it's not like we're just losing track of it. It's passing into here and then later we can say ask about its entropy or ask about what the state of the radiation is, et cetera. Yeah. Sorry, I guess you still have a problem with unitarity even if you don't put a bath. No, because even if the black hole doesn't radiate if you start from like a pure state and it starts like radiating away and even if the radiation comes back to the black hole, I guess you don't have a pure state anymore, no? So it's just battery. I understand with the bath you can let it evaporate but is this bath really necessary to talk about unitarity? I think you could formulate to the information paradox without the bath. The bath just helps make it very sharp. Okay. Yeah. And it'll make this question of trying to reconstruct operators in the black hole interior that I wanna talk about very sharp as well. Okay. Okay, I would like to ask, maybe I missed the point you said, how can you say that after this process of somehow mirroring the radiation at infinity boundary, how can you somehow say the black hole is stable is there is no somehow in literature that there's this name of the bomb. Yeah, the black hole is unstable and somehow collapses under this radiation. So, is there a super radiance, something like super radiance that these radiation increases and the black hole collapses? Yeah, it's very much worth a detailed calculation to understand which black holes will stick around and which black holes will evaporate. So, you can do this sort of, yeah, this calculation of the thermodynamic stability of black holes of different sizes and you'll find that ones that are larger than the ADS, or ADS will be stable and then this is the picture for why that is, yeah. Is there a CFT interpretation of these other non-reflective boundary conditions? Yes, that's CFT interpretation is that you are introducing some local coupling between the CFT and some other quantum system that might be like you're adding to your Lagrangian some operator like some operator like this, I guess you have to integrate over X or something. Yeah, so some like local coupling with some perhaps small parameter G, connecting points in the CFT to points in the reservoir. Just to go back a few minutes to your earlier pictures, can you reverse the statement and say that if I would want a perfect reconstruction on the full calibrous space, I am not necessarily need the full boundary or something, I cannot do that in any sub-region. Good, so it won't be enough to even have the full boundary and that is actually exactly the point that I want to argue momentarily. So for example, in this case, just to look ahead five minutes, it will be the case that we'll find black hole states where even though you have the entire CFT, you can't reconstruct operators acting inside that black hole. And so noticing that and wrestling with this question of when can you reconstruct what operators even using the whole CFT will motivate us to start talking about generalizations of this theorem and how we can understand what's going on. When do we use these boundary condition is holographic renormalization and so on and so forth still under control or we don't know? Yes, that's a great question. Yes, so I think to do it formally, you have to be careful about this where you actually impose these boundary conditions at some epsilon distance, important and great coordinates. So at some very large distance and radial units and then you're gonna couple something very close, some surface very close to the asymptotic boundary to some surface here and then you're gonna at the end take some limits where it goes to the asymptotic boundary and be careful about these divergences. Yeah, and it's still under control as far as I understand. So we have these evaporating black holes and then there's various paradoxes that show up when you have evaporating black holes like the information paradox. And more recently, 2012, there was this famous firewall paradox which I won't describe in detail here, but I will say there's questions about, after the black hole has been evaporating for a long time, what does the interior look like? Is it still well described by this geometric picture we've drawn where it suggests that some observer could say fall in and basically see the geometric interior that Einstein's equations would have predicted or is something more dramatic happening or maybe there's a breakdown in the space time connectivity between the interior and exterior after the black hole has been evaporating. Maybe there's some wall of fire or firewall that shows up there and this might seem like a very dramatic thing. There's very interesting arguments that suggest that this could happen for all we know and we might be interested in settling the debate by reconstructing some operator that acts here, maybe some number operator to, we wanna consider this guy here that could in principle detect if there's a firewall, reconstruct him in the boundary and then just evaluate his expectation value in the CFT state and then if it comes out to zero, these modes are in their vacuum state, then we would say there's no firewall. So that's the sort of thing we might like to do. So we wanna consider reconstructing operators in these evaporating black holes. Problem is that if I draw a time slice now, you'll see that it's not so obvious but we can reconstruct these operators in the black hole using the CFT. So this is now the black hole. So I'm drawing say this time slice and so this dot here, remember that whole outer circle, this dashed line intersecting this time slice is this horizon. So I could be consistent with that if I made this dashed perhaps. So this is the horizon of the black hole and some operator that I might wanna reconstruct is say this operator right here, five. And the question is if I let capital B be my entire boundary, can I reconstruct this operator five on capital B? The reason this is subtle is that in this situation, the black hole has been evaporating for a very long time and there's a bunch of modes. So looking at this Penrose diagram, all of these interior partners, you all have them forward in time, there's a bunch of partners living on this time slice. So I'll draw them here and the reservoir, maybe I'll draw, I don't know, here. This is the reservoir. It also has a bunch of, okay, so this time slice, maybe I should continue drawing like this, this is a time slice. Maybe I'll use a different color to emphasize the time slice from considering this orange slice. So and this time slice of the reservoir, there's also a bunch of modes. They're all entangled with modes in the black hole, lots of entanglement. And then so now the basic lesson of theorem one was we can reconstruct operators in the interior. We can reconstruct this phi if phi lives in the entanglement wedge of B. So we need to find the position of the quantum extremal surface of B and then see if it includes phi. And now there's some subtlety of that are we in a fixed QES subspace? Let's forget that for now. Let's just ask this more basic question, is phi in the entanglement wedge? That at least needs to be true if we're gonna reconstruct it. So from theorem one, we know to ask, so if phi is acting at, I don't know, some point X, we'll just say is X in entanglement. And the answer is, if this black hole is old enough, if it's been evaporating for half its life, then no. So the answer is no, if the black hole is too old. So the reason is that we just wanna take B and we wanna consider all of the surfaces gamma, homologous to B, that are quantum extremal. And then if there are multiple, we want to pick the minimal one. And so the most obvious surface gamma, homologous to B, because B is the entire CFT, is the trivial surface. We're asking about a geometry on this orange slice here. So if I just pick no surface at all, gamma's empty, that's clearly homologous to B because B already bounds this whole slice. So you don't need to add anything to make it bound to region. So for listing candidate quantum extremal surfaces, candidate one, the first one we would think of is gamma is the empty set. Okay, but what is, but in this case, what would the entropy of B, of capital B be? So remember, it's always equal to the area of gamma over 4G plus the von Neumann entropy of the entanglement which. And so because gamma is empty, this term is zero. That's good. I mean, that already seems like this is gonna be the minimal one. Also, I didn't say this, it's extremal because the empty set has a very easy time being extremal. Okay, so this is good. This term is usually very large. The fact that it's zero means it has a good chance of being the minimal QES, but what is this guy? This guy is actually very large, that black hole is old. And that's because he is counting the entire entropy of the bulk. So the entropy of all the matter fields on this orange slice, which includes the entropy of all of these interior hawking modes which are entangled with these guys here. So all of that entanglement, right, means there's a lot of von Neumann entropy of these guys. I realized I never, I haven't like computed a von Neumann entropy before y'all, so some of you might not have seen that. But the idea is that if you took, say, two systems that are maximally entangled, the von Neumann entropy of both of them together is zero, von Neumann entropy of any pure state is zero. But the von Neumann entropy of one of those systems alone is very large. It's like maximal. It's like the log of the dimension of that Hilbert space. So if the black hole has been evaporating for a long time, there's a lot of interior hawking partners. And so these guys all contribute a lot of entropy to S of little b, so it's large. Okay, so are there any other candidates for what the QES could be? Yes, there is. And it's, the other one is that gamma could basically be the black hole horizon. So technically it's not exactly on the horizon for details I'm not gonna get into, but turns out that there is another, so when the black hole is old, this surface, this green surface, that's basically, I should have drawn it just inside the horizon, just barely inside, is extremal. That's a very non-trivial claim, and it was the, it was the, I'll write it here. So extremal was figured out by these 2019 papers, so the fact that there's this green surface that is extremal very close to the horizon and these evaporating black holes, was argued very nicely by these two papers from 2019, one by Jeff Pennington and then one by Almeri, Engelhardt, Merroff, and Maxfields, they came out the same day. So it's a very non-trivial claim that that guy's extremal, but if you trust me that he's extremal under local perturbations, I can easily argue to you that he's more minimal than gamma at a certain point because if you pick this guy as the QES, then what would S of capital B be? Well again, remember it's A of gamma over 4G plus S of little b. But, so now this guy, A of gamma is large, right, it's like the area of the black hole. So this term is contributing area of the black hole over 4G Newton, much larger than zero, but this guy is now very small, or zero. And the reason is that if the green guy is the QES, then the entanglement wedge of capital B, the region little b, is the region between the green surface and the asymptotic boundary. So it's this region out here, which excludes all of these really entropic modes in the inside. This is a competition when you're, so there's two candidate quantum extremal surfaces and effectively as the black hole is radiating, early in its life, this one is the minimal one and then later on, this one becomes the minimal one because this term S of little b here is growing. Meanwhile, this term is not growing but this guy is shrinking because the black hole is radiating and so the area of the black hole is shrinking in time. So this guy, this is shrinking, this is growing, so at some point they cross. The point is that if you're asking at some time, which is called the page time, which is about halfway through the evaporation process, interior is no longer the entanglement wedge, which already is EW of b. So that's the point of this example. That's very important because if we wanna ask, again, it's after this time that we would be worried there's a firewall but we can't reconstruct the operator phi on capital B at that time, which means we can't directly test it by acting some CFT operator and in fact some people even have asked, does this mean that in fact there is a firewall because it's exactly at the page time? It seems like b is losing the ability to reconstruct this operator, maybe that's because that space time region is destroyed. We're gonna take a different point of view. So what you're doing is that you're erasing the information inside the black hole and you're putting it into a reservoir which we don't have access to and then you say that okay, when I lose information more than half, I don't have access to it, so it's somehow obvious from the first place. Isn't it better to actually scramble the data and put it somewhere inside the bulk and then? Yeah, good, so good, good, good. So I think what you're getting at is, in this setup maybe we should have expected that at some point capital B can't reconstruct this operator because we've been putting a lot of information here in the reservoir. Maybe we should be trying to reconstruct the interior using the reservoir and capital B. And in fact that's true and we're going to understand explicitly what's going on with that. I think this is an informative example because prior to this sort of thought experiment, you might have thought that the CFT can always reconstruct any operator in the bulk. But then now we're seeing if there's enough entanglement between this and some external system then that naive expectation breaks down. And so there's something more sophisticated that's true. We want to understand what that more sophisticated statement is. Sorry, maybe you said it, but where did the homology constraint go for the second surface? The homology constraint is there. It's very important because this green guy is homologous to B because this region little B is bounded by this green guy and capital B. I thought like for example, in the just the original Ryota Kanaki, you would, if you had the whole B you would just stay at the horizon because you could not go inside because the homology constraint is this incorrect? That argument was about the two-sided black hole where it couldn't, you had another boundary that you had to, you had to divide somewhere between the two boundaries. That's why it hooked the horizon. So the point here is that we're single-sided, so this thing. Exactly, yeah, because we're single-sided, this is the geometry of the time slice. So it has no trouble shrinking to zero. Okay. Yeah, thanks. You have a question? So at the page time you jump, the minimal surface jumps from type one to two. So a piece of confusion is that one second before page time, you can reconstruct the operator and one second after, you say, no, I cannot reconstruct it anymore, but probably you can still almost, Yeah, good. Very good accuracy you can still reconstruct it. Good, so yeah, yeah, right. So you're absolutely right. This sharp, it's not a sharp division. There's a, what really happens is that when we talk about reconstruction, we should be careful and say, to what extent does the left side equal the right side? And usually there's some very small corrections, but near the page time the corrections add up and become large in a continuous way. That people have calculated very exactly. So like in this West Coast toy model, they like, they do this very explicitly and they compute the reconstruction fidelity as a function of where you are in the page curve. And so, yeah, what happens is that actually you stop being able to reconstruct phi sort of before the page time. So after the page time, by the page time exactly, you can't reconstruct it. And then so after you can't either. Another thing I want to point out about this, so the main moral was that, it's not just as simple as we, I think at all thought originally that using capital B, you could just always reconstruct any operator in the interior. There's something more complicated. This state can't, you have trouble reconstructing this phi. Perhaps, and it turns out to be true, if you tried to also, if you tried to like act some operator on capital B and the reservoir, you could reconstruct phi, something like that is true. But we want to understand that. It might be surprising because the reservoir doesn't have to be some holographic CFT, it's just an arbitrary system. So we want to understand what's going on here. Another thing that's important is that it illustrates that there's important questions that don't live in what I was calling a fixed QES subspace. Because this is one state, these old black holes, where the quantum extremal surface is the screen surface, there it is. But there's another state, like a younger black hole, or if you, let me, maybe I'll say it this way. You take a black hole of the same size as this old black hole, but instead of having all of these modes be entangled with these guys, you have them all be in some pure state that you choose. So the geometry is the same, to a controlled approximation. And, but because these guys are in a pure state, not in some state entangled with something else, their contribution to the bulk entropy would be very small and the minimal QES would again be the empty set. So this illustrates that, you have some black hole of a given size, if that black hole is in a pure state, the quantum extremal surface would be the empty set. If the black hole were in some highly entangled state with something else, then the quantum extremal surface could be very much not the empty set, it could be this. So this whole situation, I have a black hole, maybe it's in this state or that state, doesn't live in a fixed QES subspace. One state has one QES, another state has another QES. So to handle this, what we wanna do first is just understand if the quantum, what we're gonna do is see a generalization of theorem one. And we're gonna see that the quantum extremal surface is still a helpful guide towards what is reconstructible and what's not. Even outside of the fixed QES assumption, after seeing this theorem, what we'll do next time is we'll see a nice model of how this all fits together. So in that model, is some recent work that goes by the name non-isometric codes. So it'll be a nice model that can explain, by understanding this model, you'll see sort of the physics that's going on here and how the information shifts from B to the reservoir and so on. The first thing's first. And this is a theorem that was written by Jeff, 2021. And it's just the theorem that you get when you take this theorem written in 2016 and you lift the fixed QES assumption. So again, we're gonna let there be one Hilbert space that we'll call H that factorizes like this into B and B bar. This is, remember, morally like the CFT. And there's also, as before, going to be some code subspace that we'll imagine factorizes, say, like this into a little B and little B bar. But now there's something different that we're gonna do here. So instead of being some fixed QES subspace, it's like an arbitrary subspace. This is the big difference. Moreover, what we're gonna do is we're gonna allow, I mentioned this yesterday, but we're gonna allow there to be some more structure here. Let me write this in red. HB can further factorize if you want into H little B one, into H little B two and so on. H little B in. And likewise here, it's all right, ditto. Maybe those I'll call H little B bar one and H little B bar two and so on. So those can factorize more. And the picture for this that you should have in mind, let me draw it here. And you have, saying this would be bar and this is B, so B is this guy. And there's gonna be some quantum extremal surface. I'll draw it there. I'm not yet showing up in the statement of the theorem. And so this is morally like little B and this is morally like little B bar. But we're gonna allow there to be, what we will imagine as little bulk points. So this might be little B one, little B two and so on. Maybe this is little B in. This guy can be little B bar one, little B bar two and so on. They don't have to have the same number of points. That's the setup here. You'll see why this extra factorization is nice to have. You don't have to assume that it has any, but if you do, you get some nice extra statement out of the theorem. And both of these are finite dimension Hilbert spaces. And again, we're gonna let V be some map that embeds this H code into H. So it just takes your ADS states to your CFT states. And then finally, we're gonna let there be some state that we're talking about, which we'll call psi, which is some state on little B, little B bar R. So it's a state and H code tensor HR. This HR is a new system that I'm gonna call the reference system or the reservoir. And it's just there for generality. It's nice to have. You could, if you want, remove any mention of R in a statement, the theorem would still be true, but not as general. You can just have some extra reference system R lying around and the state psi could be a state in which little B and little B bar are say entangled with R. Now before I can tell you the statement of the theorem. These are the assumptions, but I actually need to define something. What I'm gonna call, I'm gonna call U little B prod. I'll call these product unitaries. It's gonna be any unitary, say that's acting on little B, factorizes among this given factorization here. U prime is just indicating that it doesn't need to be the same unitary as U. Just a different unitary possibly. So a product unitary is just, it's a unitary on B1, tension with the unitary on B2 and so on. These can be the identity or not, they can all be non-trivial, but they in particular are not a unitary that's acting on B1 and B2. So I can't say create entanglement between these two factors. All it can do is just adjust them all in a factorized way. So this is what I'll mean when I write. And then the statement, all right here, that again we have two things that are equivalent. And it's very much analogous to this theorem, but it's slightly different. So the first is that, it's again a reconstruction statement. And the statement is that for any, you know, if you take this psi, the psi that showed up that we wrote there, and you act any product unitary on B, little b, and then map it boundary-hopert space, you could have gotten the same state if you had taken psi, first mapped it to the boundary-hopert space, and then acted some operator that exists on capital B. This unitary doesn't have to be a product, it's a general unitary on B. We didn't assume any particular, there's no statement here about a particular decomposition of capital B. This is the statement you get. It's really, I should say, for all product unitaries on little b, sorry this is very small, it just says for all product unitaries on little b, there exists a unitary on capital B such that this is true. So this is like the reconstruction statement in one, except instead of being able to reconstruct an operator given any operator on little b, capital B can reconstruct it. This is saying capital B can reconstruct these product unitaries in this way where it does the right thing on psi. And if I have time, I'm really, I'm gonna emphasize why it has the extra restrictions and why that's okay. So also, this is true if you replace capital B with b bar r and little b with r is here as well. This is another difference. It's here where when you swap out b for b bar, it should come with the reference system. This is very much related to the fact that here whenever capital B couldn't reconstruct something, what could reconstruct it is the entire complementary region, not just its complement in the CFT Hilbert space, which is empty, there's no b bar, but also the entangled reservoir. So here, that's why the r is showing up. Sorry question. So this product structure, this restriction on you, is it a proxy for a locality when the little n goes to infinity or not? Yeah, that's the idea, yeah. It's proxy for locality when little n goes to infinity. Yeah, it's a good way to say it, yeah. So the reason why it ends up being this product structure is that you can't reconstruct a general unitary because that could change the entanglement in a way that dramatically moves the quantum extremal surface. And I'll try and explain that. But the second statement is one that's like the quantum extremal surface. So it says you can compute the entropy of b in, say, the state psi. So you can take psi, which is the bulk state, map it with v to the boundary and you compute the entropy of capital B. It's given by, again, this expectation value, let me write it this way, the expectation value of some area operator, but now one that depends explicitly on little b. So there's a different one for different different little bs plus the von Neumann entropy of little b in the state psi. But condition two is not just this. Turns out you have to add in this further statement that for all product unitaries on little b and product unitaries on little b bar, which I'll just write u to be these guys for the moment. This equation holds for any u acting on psi. So this is still very much like the quantum extremal surface formula that we have in ADS CFT, even though we don't usually talk about their product unitaries and this formula holding for all, not just the state we're considering, but for all product unitaries acting on it. But that is true because these product unitaries don't change the entanglement so they don't change the position of the QES. So this is in some sense a technical detail. Finally, let me say that there's a third condition which is an improvement upon theorem one in the sense that recall theorem one, even though it was supposed to be talking about the quantum extremal surface formula, nowhere did it say anything about extramality or minimality. Now we are going to get this minimality. That's condition three. And it turns out that the correct logical structure is not that there's some equivalence between this three of them about to write and one and two. So these one and two are equivalent but they imply the converse is not true. And so the three is the statement that this like, this a of little b plus s of little b evaluated inside is less than or equal to a of what I'll call b prime plus s of b prime evaluated inside for all b prime. So b prime here is an arbitrary other choice. So in this, if this region defined by this surface is little b, little b prime is just any other choice of these factors. So maybe b prime defined by this purple surface includes this thing we're calling little b bar two and little b bar one, but excludes b two and whatever that guy is. And so maybe all the union of all of these points is b prime. And given this guy and the state psi you could evaluate his area plus his bulk entropy. And this theorem guarantees you that will be greater the one for little b if little b is the region that satisfies this formula and the reconstruction thing. So this is like, this is a lot like the quantum extremal surface formula. The quantum extremal surface formula said if you find the region that satisfies this it will be the minimal QES. And this is saying the same thing if you forget that the time direction exists. So there's so far no statement about the time direction, but this is the statement of extramality in the spatial direction. So that's what we have so far. This is the statement of theorem two and there's two things I haven't really explained to you about it. So the first is what this guy is. For the sake of time I'm just gonna tell you that there's just a natural guy that shows up here. He's uniquely defined, he's unique. It's very nice. He generalizes the a that showed up there but the a that showed up there was just some theorem one was just some area that was just a fixed operator. There was no way to talk about the area of other regions. There was no way to even, the only regions that were defined in the bulk were little b and little b bar. In theorem two there is a way to talk about other regions. That's very important because one state, psi, might have entanglement wedge little b. A different state, psi prime, might have a different entanglement wedge, little b prime. We need a way to talk about the fact that they have different entanglement wedges and the areas of both of those. So that's why there's this new structure showing up. This area operator, there's an area defined for every surface and given more time, I would explain to you why this totally makes sense as the type of reconstruction that shows up. Because I am a little short on time, what I will say instead is that this is the type, this theorem tells you that in general, this is what the quantum extremal surface formula promises you can do as far as reconstruction. And you might be disappointed because you might want to do things other than just use capital B to reconstruct product unit areas on little b. Maybe phi here, the guy that measures if there's a firewall is maybe not a product unit area. But I would say don't be dismayed. There are nice theorems you can prove and I could write one down. Where the idea is that if you can reconstruct product unit, like a, yeah. Let's say you have a subspace, so I don't want to say this. This isn't all you can do. So this theorem isn't telling you that this is the most you can do. What this theorem is telling you is that given that this is the QES of one region and one state, you can do this. But if you know more, for example, if you know that say the interior of the black hole is in the entanglement wedge of capital B for all states in some subspace, then that tells you you can take that statement combined with statement one and learn a more powerful reconstruction statement. So this is like a reconstruction primitive and you can combine it with other things you know to argue that you can do more complicated types of reconstruction. So the upshot is that this, okay, so this is capturing what the QES formula tells you about reconstruction, but it's not supposed to be obvious just from looking at this what all of the operators are that you can reconstruct, say using capital B or using just the reservoir or using both together, say, what operators in the interior you can reconstruct. To answer that question, we'll need to dive a little deeper into this example, which we'll do next time. So for now, yeah, the takeaway is we have a theorem sort of telling us what the QES formula says about reconstruction. Let's use that theorem next time, say in a model that we fully understand to understand exactly what we can reconstruct in the interior of evaporating black hole. For questions? Maybe you said it and I lost it, but so in the candidate QES, like candidate number two, you said it's a surface which is very close to the black hole horizon and so, should I think it's something that nucleates behind the horizon after a finite time or it's just something that is tracking the horizon in some way? It turns out to be something that sort of nucleates behind the horizon and it sort of nucleates this extremal surface, nucleates there after some amount of time, but before it becomes minimal. So its behavior is sort of different in different examples, but generally it shows up as not yet minimal and then at some point becomes minimal. Maybe I can ask a question. So is it true that this H little b in principle might have different factorization? I can try to factor it in some other way. Yeah, you could always, I guess, you know. So my question would be which parts of this theorem would depend on the choice of the factorization or? Yeah, good. So this theorem is always true for any factorization that you write, but in ADS, if we want to apply it to ADS CFT, we need to just note there that this formula holds for little bs that are defined well by some spatial factorization. So the spatial factorization is just important because that's the one that seems to be relevant to this formula in ADS CFT. In principle, it could be a different factorization in a different system. When we do one such computation with black holes, like the one you did before, what kind of black hole we can consider? So any kind of black hole or small black holes or do we have some limitation on temperature or on charges or something like that? Yeah, all of these can sort of change the details, but I would say the, so the first studied examples were these large black holes that would normally be stable with reflecting boundary conditions but can evaporate with this coupling to the reservoir. You could also, I think you could do a small black hole which would maybe be simpler because it will evaporate already with reflecting boundary conditions. I think the reason people talk about that less is that it's harder to do a computation that say gives you this so-called page curve. You know, it's harder to compute the von Neumann entropy of the radiation because you have to somehow isolate the radiation and compute its entropy and if it's not, if you're not isolating it into some other system, that's more difficult. But yeah, so you could have like, you could have black holes that have various charges, et cetera, and that does change the details but not the qualitative behavior. They're very close to extra reality. Yeah, so those, right, you know, how quickly these black holes evaporate, these sorts of details do depend on this fact. The fact, the basic qualitative behavior is the same that they eventually evaporate and, you know, therefore build up this large entanglement between the inside and the exterior radiation. That says the same. What is the A in the last statement? Is it the expectation value of the A? Yeah, sorry, this was supposed to be the same. AC, thank you. Any further question? Okay, if not, let's have a break now and then we start again at 3.30.