 So first I'll thank the organizers for giving me the opportunity to come and give these lectures. So the title of my lectures was Gravity and Entanglement. And so the context of that is trying to understand gravity quantum mechanically. So this is the major challenge for theoretical physics for the last several decades, trying to understand gravity within the framework of quantum mechanics. And fortunately by now we have at least some examples where we think we have some complete description of theories of quantum gravity. And the examples that I'm talking about are in the ADS-CFT correspondence. So I imagine that most of you have some familiarity with this. I'll just mention as we go along, I'll mention some of the things that I'll need. It won't be all that much. But the basic idea is that we can define certain theories of quantum gravity starting with non-gravitational quantum systems, just ordinary quantum systems on some fixed spacetime backgrounds. The simplest ones are actually just, they don't have any space at all. They're just quantum mechanics, matrix quantum mechanics models. But mostly I'll be talking about quantum field theories. So some, for example, certain conformal field theories on some fixed spacetime background B. And this actually, it could be Minkowski space or a sphere times time or actually anything you want. But it's fixed. OK, so we consider different states of the CFT on this fixed background. And the idea is through this magical equivalence that Maldesena taught us about, then this theory somehow describes non-perturbatively completely a theory of quantum gravity, first spacetimes with certain asymptotics that are determined by this geometry B and the UV behavior of the field theory. OK, so mostly I'm just going to be talking about ADS CFT where we're actually talking about a CFT. And in that case, the geometries that we're talking about, the states of the quantum gravity theory that we're talking about are asymptotically anti-decider spacetimes. And the boundary geometry of those is the same geometry that the CFT lives on. So let's just mention a few examples. So maybe the most common example would be to think about B to be equal to Minkowski space. And then, for example, if we take the vacuum state of the field theory on Minkowski space, then this is supposed to describe in this dual picture, pure, the state of the gravity theory has a spacetime, which is pure ADS spacetime. If I need to use a specific metric for that, I'll often use this Pfeffermann-Gram description. So I'll write down the metric. So there's some dimensionful parameter that sets the curvature scale of this ADS spacetime. There's a coordinate Z, which starts at zero at the boundary and then increases as you go away from the boundary. And the metric for pure ADS spacetime looks like this, but then you could consider some excited states, so I could add some excitation to the vacuum. And this would correspond to then not just pure ADS, but perhaps pure ADS with some gravity waves in there. I could do something which is a larger excitation, so for example, if I considered a thermal state, so now I think of my field theory at finite temperature, so then there's lots of excitations. It's understood that this would correspond to, again, an asymptotically ADS spacetime but not pure ADS, actually a spacetime that includes a planar black hole. So at some value of Z, we reach a horizon and then we have this black brain. So incidentally, all of these excited spacetimes, we could also describe at least close enough to the boundary in the same gauge where we take Z as some special radial coordinate, and then the difference would be instead of just this Minkowski metric in the directions orthogonal to Z, then we have some general metric. Okay, so the black brain would be some particular choice for this tensor, and other geometries that would correspond to other states would correspond to other choices for that tensor. Another example that will come up, so in understanding quantum gravity, one of the most interesting types of states that we would want to study would be black holes. Okay, so instead of considering the thermal state of a field theory on Minkowski space, we might consider the thermal state of a field theory on a sphere. And, okay, so then it's quite similar to this example, except that now we have a spherical symmetry instead of a planar symmetry, and you end up with a black hole. So you can have an asymptotically ADS space whose boundary is spherical, and if you look at the thermal state of the field theory on the sphere, then you get a black hole geometry as the gravity side of that correspondence. So that was a lightning review of just roughly how ADS CFT works. I mean, the important thing is that you have different states in the field theory. These correspond to different geometries on the gravity side. And this is now, well, 18 years old. But there are many big questions that I think we don't really understand. So these questions are what will motivate many of the things that I'll talk about. So some of the questions would be, well, why does this work at all? How is it that we start by considering some, say, a field theory system? And, okay, so I should mention, I think I mentioned that the field theories for which this works, they tend to have large numbers of degrees of freedom. So if there's a Lagrangian, often at some matrix, field theory, a UN gauge theory where n is very large, or in one plus one dimensions, we would have large central charge. And often these are strongly coupled theories. So they're somewhat difficult to study. But somehow out of this strongly coupled dynamics of large numbers of degrees of freedom, there's some interpretation that emerges, that's equivalent to a spacetime geometry, a classical spacetime geometry. And somehow from the dynamics of this field theory, you end up with the dynamics of gravity. So how and why does this work? How do you get, how does spacetime emerge, how does gravity emerge? One of the things that I probably should have, that you're surely familiar with, but I should mention, these spacetimes are higher dimensional than these ones. So because that's the boundary of this spacetime, really an entire extra dimension has emerged from the physics. And so the recent work that I'm going to be talking about is giving some answers to some of these questions. And it's quite surprising. So there's various recent work that suggests that in order to understand how spacetime emerges here and how gravity emerges, you actually have to think of, so even to understand just the classical level on this side, classical spacetime and Einstein's equations. To understand that, it seems that you have to think about intrinsically quantum mechanical things on this side. And specifically what we are now seeing is that the structure of entanglement, so how different parts of this field theory system are entangled with each other, that seems to be crucial for understanding the spacetime structure or the spacetime geometry. And we'll also see that just various constraints on entanglement, things that quantum information theorists had come up with in just understanding fundamental properties of quantum systems. So these constraints on entanglement seem to be fundamentally related to constraints on spacetime, or in other words, the dynamics of spacetime. Einstein's equations are sort of a constraint on what possible spacetimes that you can have. Okay, so that's roughly an introduction. I'm gonna start by mentioning that all of this recent work, there were actually hints of it starting in the 1970s, okay, so even before ADS CFT, there was this really interesting work in the 1970s where Jacob Beckenstein and then later Hawking and other people started to realize these connections between black hole physics and thermodynamics. Okay, so that's actually going to be intimately related. All this recent work is in a sense a generalization of that. Okay, so let me remind you of some basic statements there. The observation of Beckenstein was that if you consider black hole horizons and you think about the area of those horizons, then the horizon area of these black holes actually has various properties that are quite similar to entropy in thermodynamic systems. So specifically this combination, black hole horizon area over 4G Newton. The 4G Newton is for some quantitative reasons, but to get something dimensionless, so the horizon area has the same dimensions as G Newton, and so to get something with the dimensions of entropy, then we have this ratio and the 4 is for some detailed quantitative agreement. So the idea is if you look at this quantity in the context of dynamical gravity, then it obeys various relationships, very similar to laws of thermodynamics. Very similar entropy appears in laws of thermodynamics. So there's a first law that says if you consider some black hole of a certain mass and a certain horizon area, and now we make a perturbation. So we maybe add some matter so that the mass of the black hole increases. Then there's a first law that says that the change in mass is proportional to the change in this horizon area. So that's like the statement that the change in energy is proportional to the change in entropy of the system. What's playing the role of temperature is this quantity called surface gravity, which is related to how much curvature there is at the horizon. And then there's a second law, so in classical general relativity there were results that the area of horizons should only increase with time. If you consider general dynamics or throw things into the black hole, this should be increasing. And again, this sounds like a property of entropy in thermodynamic systems. And so Beckenstein suggested that this horizon area of black holes actually should be interpreted as an entropy of the black hole, an entropy associated with some degrees of freedom of the black hole. By that time, this was very mysterious because we're talking about a classical theory of gravity. It's not really clear what the microscopic degrees of freedom are or how you would count the number of states of a black hole. Okay, but 30 or 40 years later we now have examples of quantum theories of gravity where we talk about black holes. And we know that what he was saying is exactly realized in ADS-CFT, okay? So in ADS-CFT, I mentioned that the black holes, so say a spherical black hole in anti-dissidus space, this corresponds to a thermal state of a conformal field theory on a sphere. Okay, so think about a field theory on a sphere. This has a discrete spectrum of states. Actually, in the previous lecture, there was a discussion of the spectrum of operators in a CFT. And so many of you might know that the spectrum of energy eigenstates for a CFT on a sphere is actually the same. It maps over to the spectrum of local operators in a Euclidean CFT. So there's this discrete spectrum of states and just like any other theory we could consider the thermal ensemble. I'm talking about you, so here we're you, Lorenzian, okay? So this is the CFT on, right? So it's, I'm talking about this example, again, okay? Okay, so it's a black hole in ADS, Lorenzian, CFT on a sphere times time, a thermal ensemble. And so what that means is I'm considering this ensemble of energy eigenstates with probabilities given by the usual Boltzmann. Okay, so when I said before that the thermal state corresponds to the black hole, this is precisely what I mean by the thermal state. This particular mixed state of the quantum system, this ensemble corresponds to the black hole. And now we can understand exactly what's going on. Why does this horizon area on the, on the gravity side have properties that are like entropy? Well, it's because in this fundamental description of the black hole through CFT degrees of freedom we actually have a microscopic theory. And it's true in that theory that the horizon area on the gravity side maps over to the thermodynamic entropy in the CFT side. So in ADS CFT, the statement would be that, yes, this area on the gravity side does have a microscopic interpretation. It's counting some states and it's precisely the usual formula for entropy. So if you take those probabilities given by the Boltzmann factors, then this is the standard formula for entropy. And this corresponds to the area of the black hole horizon. Okay, so, so now if I, if, if I consider, you know, going to a larger black hole, then I'm, I'm, I'm taking this ensemble and I'm changing the temperature and I could tell you exactly how much the entropy changes and just tell you how much the mass changes. And you can just verify all these things. Yes, right, so, yeah, so we can consider more general kinds of, of black holes with charges and so forth. And again, the idea would be that, so if, if I start from, if I start from this one and, and say, add some charge or some spin, then we should still, this type of formula would still be understood to hold in the CFT. And sometimes in the limit of, you know, for certain very special black holes, you have limits where the area goes to zero. And in this case, this is understood as that you're, you're taking a limit where the ensemble of states in the CFT is, is a sort of a small number of states. So that the entropy also would, would go to zero. I should say when, right, sometimes it's important to distinguish these CFTs often have a parameter like the central charge or the N in a gauge theory, often when we're talking about classical things on the gravity side, we are really talking about these being equal to the leading, the leading contributions to the field theory quantities in an expansion and one over the large number. So, so sometimes there could be zero classical area, but then there could still be some small entropy, which is instead of going like N squared or like the central charge, it could go like some smaller power. All right, I'm going to, so that's, that's how we understand this connection between black hole areas and entropies in this modern context of ADS CFT. The entropy of the CFT in the thermal state maps over to the area of the black hole. Yeah, oh, yeah, wait, so not talking about entanglement entropy yet. This is just thermal entropy. Yeah, this is ordinary entropy, yeah, but that's the next thing. So I'm going to jump right to the most important, the most important formula or development in, in the recent, in all this recent work. So there's a generalization of this. So 35 years after this, this work in the context of ADS CFT, it was understood that there's an amazing and massive generalization of this connection where instead of just considering thermal states, we can consider any state of the CFT that would have some kind of arbitrary gravity dual, so it, it could be not just a black hole, but any kind of, any kind of space time, even, even empty space. Okay, so we're going to generalize to the case where the CFT is in some much more general state. And we're going to generalize to the case where we're talking about the entropy, not of the entire CFT, but of an arbitrary subsystem of the CFT. And then what these guys Ryu and Takiyanagi did was to say that, that much more general kind of entropy, so the entropy of a sub, an arbitrary subsystem of an arbitrary state. They said that also has a geometrical interpretation as some kind of area in the, okay, so, so now this is an arbitrary state. We consider some arbitrary region. And here's the dual, okay, dual space time. And this entropy has some interpretation as some kind of an area in the dual space time. But I have to, I'm going to, I'm going to wait a little bit. I have to tell you more about what I mean by the entropy of a subsystem. So we need to understand that better and, and then we'll understand exactly what the statement is. But this is, you can think of this as just, you know, one, one particular tiny example of this much more general statement, which has kind of opened the door for a huge leap in our understanding of, of this ADS CFT correspondence. Yeah, well, it's, yeah, I'm going to, I'm going to talk a lot more about this. But so far just think of, there doesn't have to be a double. It's just a mixed state. We can talk about entanglements without actually purifying. But, okay, so I want to review what, what do I mean by subsystem entropy? So people, by the way, people talk a lot about entanglement entropy. It's really just the entropy of a subsystem. It's not, it's not a diff, it's the same notion of entropy, but just applied to subsystems of general quantum systems. So that's what I want to remind you of now. So the context of this is that I want to say, consider any quantum system with, with multiple parts. Okay, so I consider a quantum system where the Hilbert space factorizes into a tensor product. So this would always be the case if I have individual, like, different degrees of freedom. If I, if I can isolate some, some degrees of freedom, which are a subset of the whole system, then according to general rules of quantum mechanics, the Hilbert space has this tensor product structure. And we're going to consider a subsystem, I'll call it A, of a quantum mechanical system. And so we'd ask the question, if we have a state of the whole system, let's say we have some general state of the entire quantum system. What is the state? How do we describe the state of just the subsystem? We're only interested in, say, expectation values of operators that act on the subsystem. What is, what is the description? What is the most efficient description of just the subsystem where I don't have to worry about the entire state? And, you know, in the classical world, we just say, well, okay, this is the configuration of sub, of the A part. This is the configuration of the B part. In the quantum world, it's not possible. There's no, there's no state psi A, which reproduces all the expectation values for that subsystem, starting from some general state. So what you need to do is actually consider an ensemble. So, okay, so there's no, well, let me just write down the statement. So the state of the subsystem is an ensemble of pure states. So if I consider all of the operators O, A acting on that subsystem, and I wanted to try to find a state psi A such that the expectation value of O in psi is equal to the expectation value of this full operator in the full system, then I would not be able to find it in general. Okay, but what I can always do is find some ensemble, some set of states of the subsystem, and some probabilities so that, so that this is true. Okay, so I can always find, I can always find an ensemble of states where the calculation of the expectation value in the ensemble, according to these usual rules, would reproduce the calculation of that expectation value in the full state that I started with. I'll tell you. Yeah, so I'll just write it down explicitly. So the question is, how do I write down this ensemble? And so this is the procedure. You start with a system, the state of the original system, you can write in terms of the basis elements of the tensor product and then you calculate what people call the reduced density matrix. Okay, so I'm going to write down now an operator on the subsystem A, which is people describe this mathematical operation as tracing over A bar or the partial trace over the subsystem, the complement of A. Okay, so I can take these coefficients, I can just write down this operator and then you can always diagonalize this operator. So it's Hermitian. I can always diagonalize it so I can write it in this form. And then the ensemble that I'm talking about is actually just this one. Okay, so it's probably more familiar to you to say, the state of a subsystem is described by this reduced density matrix, which I get by this procedure of tracing out the rest of the system. But I want to emphasize that when we talk about density matrices, we're really talking about, it's equivalent to be talking about ensembles, just sets of pure states with certain eigenvalues, with certain classical probabilities for being in those states. Okay, so in particular this calculation that I did over here, where you calculate the ensemble average of this operator, this is the same as just taking trace of row A times OA. So this is the more familiar thing. Take the density matrix and evaluate the trace of the density matrix times the operator. Okay, so given any quantum system, you choose a subsystem and you can always either calculate the density matrix or find this ensemble of states, which completely captures the state of the subsystem. Now in very special cases, you might just find a single state with probability one. So that would be if the subsystem is itself in a pure state. But in general, the set of these probabilities is not just the set one. There are multiple probabilities. And this is the definition of what it means for something to be entangled. So when you have your subsystem and you find the ensemble of states that describes it, and it's not just a single state, if it's actually a number of states with probabilities less than one, this is what we mean by saying that A is entangled with the rest of the system. So A and A bar are entangled. The fact that you have these probabilities, where it's not just one and zeros, in the thermodynamics context, we would say that's saying that we have some classical uncertainty about what the state is. So there exists classical uncertainty about the state of A. And so then we're back to the situation, exactly the same situation that we have in the thermal ensemble. When we talk about the thermal ensemble, you have a set of energy eigenstates, and you have a set of probabilities for those eigenstates. And when we talk about the thermal state, you compute expectation values exactly by this procedure. In that case, we quantified the uncertainty, the classical uncertainty about the state by using entropy. So in this case also, when we realize that our subsystem is now not defined by a specific state, but actually by an ensemble of states with these probabilities, then again it's useful to characterize, to quantify this uncertainty. Are we very close to this case where you have a pure state describing the subsystem, or do we have a large amount of uncertainty? So being entangled is not, you could be very entangled, or just a little bit entangled, you could have just a little bit of classical uncertainty about which state your subsystem is in, or a lot. And we quantify that by entropy. So quantify. And it's exactly the same formula as I wrote down before. This is the formula for entropy. It works just as well for these general ensembles as it does for the thermal ensemble. In the quantum context, we can write down an equivalent formula in terms of the density matrix that I guess was written by von Neumann. So sometimes you'll see this as a formula for entropy, but it's the same as this, and it's the same as the usual formula for entropy. So this is just the ensemble describing the subsystem. What is the entropy of that ensemble? And it's called entanglement entropy, but it's not really any different than the other entropies that we talk about. It's the same definition. And so this is really the same quantity that we're talking about. And this I think motivates more this Ryu and Takenagi formula that we're going to write down later, that if the entropy of the whole CFT corresponds to some geometrical quantity, the area of the black hole horizon, and now we look at the entropy of just a part of the CFT, well maybe that should also correspond to some similar quantity. So summary. If you start from the state of a whole system, you can define this density matrix or this ensemble, and then you can characterize how much classical uncertainty there is using this entanglement entropy. But there are actually a few other quantities that come up. This is one very useful combination of the set of probabilities that characterize the ensemble. But sometimes we're interested in more detailed information. So if we have a thermal system and we calculate the entropy, I could find another system which is not thermal, which has exactly the same entropy. I could have a different distribution of probabilities. Or I could have the canonical versus the micro-canonical ensemble, and I could find two ensembles with the same entropy, but they could be different. So sometimes you want to look at in more detail how do you characterize what are these probabilities. And so just the set of probabilities in this entanglement context is called the entanglement spectrum. And then the other thing you'll hear about are some other combinations of all of the eigenvalues. So this is the entanglement entropy, but in the literature often you hear about Renye entropies. So this is a quantity that you can compute from trace of the density matrix to the nth power, or the alpha power. So if I have a density matrix, the trace is always one, but if I take some power of it and then I take the trace, well this is some quantity which is given by the sum over the probabilities raised to that power. And then this particular, if you take the log of that and divide by one minus alpha, this defines the Renye entropy. The reason for all of this is that if you take alpha to one, then you get the entanglement entropy. So if you take a limit. So in some, in a lot of context, the main reason why people talk about these ones is that in a lot of contexts it's easier to calculate these Renye entropies. And so if you can calculate these and find some analytic formula as a function of alpha, often you can calculate it for integer alpha, but if you can find an analytic formula, then you can take the limit where alpha goes to one, and this gives you a way to calculate entanglement entropies. So we've talked about going from a big system and describing the state of a subsystem, but you can also think about going the other way. So say you start from a density matrix. So given rho, we can ask is there a state, is there a larger system, and can I find a pure state of a larger system where the state of the subsystem is rho? Yes. So if, yeah, I mean it's considered to be some more generalized measure of entanglement. You see that if all of the probability, if there's just one P and it's one, I guess you get zero, it's more of a formal quantity, but it's often described as a more general measure of entanglement. It doesn't have some of the nice properties that entropy in general has, but so I'm not sure if it has a really nice direct interpretation. It's more used as a calculational tool, but I guess you can consider it, I mean you can consider anything that you build from these probabilities as some measure of entanglement, because if it's not entangled, then it's just one, zero, zero, zero. So any other quantity that you compute from those probabilities, it's sort of some measure that will tell you are you entangled or you're not entangled. Okay, so given a density matrix rho, we can go the other way and try to find what are called purifications of rho. So generally, I can describe the most general purification. So say I start with a Hilbert space for a system. So what I can do is take some other Hilbert space, I can consider this tensor product, and then starting with my ensemble, I'm assuming these are orthogonal. So if I take another Hilbert space whose dimension is at least as large, then I can write down the following state. So I choose some orthogonal states in the other Hilbert space. So again, these are orthogonal. So I need a dimension of the other Hilbert space to be at least as large as the number of states. Appearing in my ensemble. And then I can write down this state. So this is a state in the larger Hilbert space. And it turns out that any purification can be described in this way. So anytime where you have a state in a big Hilbert space and you find that the state of a subsystem is equal to this, then the whole state you can write in this way. You can always find sort of orthogonal states in the two subsystems where I can write the full state as this kind of the superposition of the product states. So you can check if you start with this one and then you calculate the density matrix using the procedure that I described. Remember I had these coefficients c and then we had the c and the c dagger. So the square root of p will appear twice and then we'll end up with just getting the p i and the states appearing. So you can check that starting from this state calculating the density matrix and getting the ensemble you get exactly this. So this is sort of the general way you can go from a density matrix to a pure state of a larger system. This particular way of writing it is known as the Schmidt decomposition and you can prove that it's always possible to do that. So not... Actually generally not unique. So once I fix the larger space I'm actually able to choose any orthogonal say I have 12 different states appearing in the ensemble. Just choose any 12 orthogonal orthonormal states here and it gives you a purification. So it's very nonunique actually. One of the interesting things that you realize though an interesting thing is if you were to consider the other part of the system so instead of looking at A look at B and calculate the entanglement entropy or the spectrum of the density matrix for B you're going to get the same answer. So this formula is completely symmetrical between A and B. If I find a purification and I say what is the entanglement spectrum for the B subsystem what is the entanglement entropy for the complement it's exactly the same. So it implies row A and row B have the same... row A and row B have the same entanglement spectrum the same entanglement entropy and so forth. So that will come up later. It's a little bit counter-intuitive. Yes, that's a really good question. So I was going to talk about that specifically. Let me postpone the answer to that. But just for two minutes I'll postpone the answer to that question. So let me just do an example. If you take the thermal state so you have some general thermodynamic system in the thermal state with temperature T well we actually often think about a purification of that when you're learning about thermodynamics or statistical mechanics you say the thermal state is really the state where you have your system and you have it coupled weekly to a heat bath. And then you say the whole system is in some pure state and we have if you just consider the subsystem then it's in this canonical ensemble or this thermal ensemble. So in that context we think about the purification as being the pure state of the system plus. And in that context also the thermal entropy of the subsystem is the entanglement entropy between the system and the bath. So the thermal. So again this entanglement entropy is considered all the time when you were thinking about thermodynamic systems as systems coupled weekly to a heat bath. Okay yeah I should hat or something. So yeah the well oh so good B could be much larger but I've only chosen I'm only summing over the non-zero probabilities. So I guess I should say PI is not equal to zero. It's a special case of this purification that comes up a lot in these discussions. So one nice choice that I can make which is rather different here we normally assume that the heat bath is a much larger system but if I want to I can write down a purification where the other system is exactly the same system. So if I take B equal to A and I take the state of the combined system as this is for a thermal state instead of considering a big heat bath I'm going to write a purification where the other system is exactly the same and I choose these states to be exactly the same as these states so then I've defined this state where I have the square roots of the Boltzmann factors and it's symmetrical between the two sides so this is called the thermo-field this is called the thermo-field double so this is an example where you start with a thermal state and choose a particular purification where the other one is this the B system is the same as the A system actually when do I start do I have five minutes? okay so in the last five minutes what I want to do is go back to ADS CFT now answer your question and motivate what I'll do in the next lecture this afternoon okay so back to ADS CFT so I said earlier that the CFT in a thermal state on a sphere is equal to is dual to the black hole and that the entropy is equal to the horizon area but I wasn't very precise when we talk about black holes in general relativity there's the part of the black hole outside the horizon I'll draw these conformal diagrams so in ADS that corresponds to this so there's a part of black hole outside the horizon but then you can sometimes extend the geometry past the horizon so you can do that and actually maximally extend it there's a canonical way to do that and you find that there's this two-sided black hole that has a Schwarzschild geometry on the outside in both sides and two different asymptotic regions in ADS this would be asymptotically ADS with some spherical boundary this side is asymptotically ADS with some spherical boundary if I look at a time slice through that it looks like this wormhole geometry with two asymptotically ADS regions so when we talk about the thermal state what is it dual to is it dual to this or is it dual to the whole thing it's not totally clear but Maldesino wrote a paper in 2001 with a specific proposal that ties into what we were just talking about so Maldesino said the maximally extended black hole because it has these two asymptotic regions and therefore two different boundaries which are spheres he said that should correspond to something which is not just one CFT but actually two different CFTs and in this in the case where we're talking about a thermal state of a CFT we just realize there's a canonical system and state where you start from a thermal state you double it you double the whole system and you write down this symmetrical state so there's this very natural thing from the point of view of quantum mechanics where it's a symmetrical system with two copies of your original system and each side looks the same just like this extended black hole from the geometrical prescription so Maldesino said conjectured that this thing was dual to the thermo-field double and I'm just drawing the spatial this is the spatial slice and he says this is dual to this specific state where you take two CFTs which are not interacting at all with different systems except you put them into this particular state and this hints at something really dramatic what we've really done in the context of what I've been talking about the rest of the lecture is we've taken two different systems and we've just entangled them in this particular way so I took two completely different CFTs just two copies of the same thing no interactions the only connection at all that says that those are related in any way is that we've added we've gone into this thermo-field double state which is a particular purification of the thermal state on one side so it's entangled between the one side and the other side in this symmetrical way and apparently doing this entangling on the gravity side according to Maldesino what it's done is it's taken your two empty ADS space times that you might have thought you had if you had to say the vacuum state of those two CFTs and it's provided some kind of connection geometrically between those this wormhole going from one side to the other okay and so this is I think the first suggestion that and this was before we were attacking everything this is a suggestion that somehow entanglement just by adding this entanglement you've connected the two sides up and the next lecture I want to argue that even in the context where you just have one CFT we'll see that entanglement between different parts of that CFT is kind of intimately related you might say it's the reason why you have some connected space time that's emerging as the gravity dual okay but we'll get to all of the details in the next lecture