 Okay, thank you very much. It is a great pleasure to be here for the first time. And I would like to thank the organizers for their kind invitation. So they also asked me to give a talk which should bear in mind that some of the people in the audience would not be experts from the workshop. So I will try some difficult equilibrium. It is also a pleasure and an honor to give a talk about things in front of many of the people who invented these things. So the title of the workshop is Black Hole's Quantum Information Entanglement and all that. And that's the title of my talk. So what I will try to do is work you a little bit through the series of ideas a little bit historically about how the relation between black holes and information theory, the quantum version of it and entanglement in quantum mechanics came about and finished with some broad brush strikes where I will try to introduce some of the ideas that are recently proposed. So before that, let me just remind you that we have great confidence now that black holes exist in nature for real. Not only in our equations, but they exist in nature. Of course, the evidence from astrophysics has been overwhelming for decades. But this is kind of a, you would say kind of very, you would say indicted evidence, right? But now, as you know, we have sharper evidence from a scattering event, okay? We have seen, we have heard the collision of black holes by the interferometer LIGO via the gravitational waves. And this kind of qualifies as an observation of the existence of black holes by particle physics standards. People who is educated in the particle physics, like me, find it very natural to think that, to say that something exists even if you didn't see it ever with your eyes. After all, we never saw elementary particles. We don't even know really what elementary particles are deep down, but we are confident about the properties and the existence of elementary particles because they are described by precise mathematical framework, which depends on few parameters and that works. And by that, a criterion, you can say that the detection of LIGO, which is also a scattering event, if you like, serves that purpose and brings the existence of black holes a little bit into particle physics standards, okay? And I should acknowledge that the mathematical control over this signal, the required mathematical framework that allows you to be so precise was developed during decades with the very important input from one of our hosts, Thibaut Amour, sitting here. So this is part of the real world, right? But in this workshop, we are thinking more in terms of an idealized situation in which we are free to invent all kinds of crazy situations and natural manipulations of information, et cetera, which allow us to think of black holes as very theoretical machines, okay? So they are very idealized and we will be doing things that are not natural in the real world. The main topic that I want to discuss about is how black holes store and process information, okay? And the fact that black holes have something to do with information is basically geared into the basic definition of a black hole already in the classical picture of what a black hole is because a black hole has a causal boundary. Things can get in and they cannot get out, right? In the classical approximation. So the way black holes treat information, as you know, is that all the information that gets into a black hole because things fall in is forced to fall into the singularity. There is a singularity which is a space-like and in fact the black hole is defined as the causal path, the causal path of that singularity, right? So the processing of information is very simple. It's just destroying information, right? And since it is a singularity and we don't have really control of what's going on in the singularity, the capacity for the singularity to eat up information is basically infinite. Slightly more interesting fact is that when you look at the black hole from outside, you see a finite size object with a finite mass. And you could say that perhaps the amount of hidden information in this object is zero because of this theorem that tells you that black hole solutions in general activity are very simple. They are just characterized by a few numbers, mass, charges, and ground momentum. So they look a little bit like solitons, like individual objects. However, physically you should think of black holes together with all the possible perturbations that deviate from the exact solution which you call technically in textbooks of black hole. And it turns out that you can put arbitrarily complicated amounts of material very close to the horizon at finite cost in energy, right? Just because of the infinite redshift that you have at the horizon. So you can have lots of stuff sitting out in the horizon, Lawrence contracted, at finite cost in energy. So if you consider any finite band, however small of energy around the nominal black hole mass, there is an infinite phase space there. And therefore the classical entropy that you will attribute to a black hole is actually infinite in the classical theory. Now, very soon in the quantum theory, Beckenstein noticed that in fact it is more natural to assign a finite capacity to the storage of information for black hole when you consider quantum mechanics because of the fact that if you imagine throwing bits of information into a black hole that are supported over elementary quantum systems that are like elementary particles, you know that there is a minimum amount of energy that an elementary particle can have in order to be sufficiently localized so that you are sure that it will fall into the black hole, okay? If the energy of the particle is too small, then the wavelength will be too large and it will not fit, it will most likely miss the black hole. So if you want to say that you are going to build a black hole by minimal captures in which you throw one bit of information at a time and you're sure that it gets in, the minimum energy that you need to make the steps on is of the order of one over r where r is the size of black hole. Now for this step in energy, it turns out that the area of the black hole increases by one Planckian unit, by one unit of Newton constant, okay? The area of the black hole is the radius square and the radius is proportional to the mass. So it's a simple calculation to see that the increase in area of the black hole because of these minimal captures is one unit of Planck area, 10 to the minus 66 centimeters square. And therefore it turns out that if you imagine building the black hole by these minimal captures the entropy that you would assign to the black hole is proportional to the area of horizon in Planck units, okay? So indeed you see that when h bar goes to zero you get to infinity, right? So this is kind of a regularized version of the previous idea that you have, you can hide the stuff very near the horizon. Now there are two striking facts at least immediately that come to your mind when you look at this formula, okay? The first one is that the result is extensive in the surface area of the horizon, not in the volume of the black hole. And this is unlike normal entropies that we compute when we understand microscopically the workings of a system because most of the physics in nature is local. Ultimately you can always give a local description about things, then the entropy is turned out to be extensive in volume. But this one is extensive in the surface area of the horizon. And the capacity is such that you would get the same amount of Hilbert space, say. If you imagine that you have little qubits or spins, quantum spins, one per Planckian area in the horizon, right? Like if the black hole was a kind of weird neutron star which was only made of the crust, right? And with all these things here. Now this is a little bit ludicrous to think about that because in general activity nothing can hold on the horizon. Unless you are at the speed of light, you cannot hold there. Anything that has any mass cannot hold to the horizon, right? And in general activity the black hole horizon is just a piece of the vacuum. If you are free falling through it, you don't see anything. It's just equivalence principle tells you that you see just vacuum, right, locally. So it doesn't look like this picture makes any sense. And despite that some people like Toft and Saskine were obsessed with this for years. And this is in a sense the beginning of holography. Holography is the art of making sense of this picture despite the fact that it looks silly at first sight, okay? Now the second striking fact is that when you write the area in terms of the mass of the black hole, in this case like mass square and therefore the effective density of states that you would assign to this quantum states of the black hole, skates like the exponential of energy squared. And this is very hard. This is much harder than anything you can see in any textbook. Anything you can manufacture with quantum field theory, even stream theory. So this is also something that is peculiar about this result of Beckenstein. Now if you have an entropy it's natural to have a temperature. You just use the standard thermodynamic formula and you get a natural temperature for these objects which is one over R, where R is the radius, that's GM. So they get hotter as they get smaller, right? And this is of course something that Hawking famously found directly with an entirely different argument independently, okay? Using extra assumptions, the Hawking calculation is more precise. And in exchange for the extra assumptions you get a more precise result. In particular, he gets the coefficient in front of the formula for the temperature. This is the leading term in expansion in powers of the Newton constant. So this is a number which normalizes in an absolute way the entropy of the black hole, the famous one quarter, okay? What is the extra assumption that goes into Hawking analysis? The extra assumption is that the horizon is smooth, okay? So this is kind of the opposite to this crazy idea that the horizon has degrees of freedom floating there, right? So you really take seriously what general activity tells you. General activity tells you that the horizon as you dive locally through the horizon, you just see a piece of the vacuum, right? And the assumption is that when you form a black hole, all the transients, all the details that remember what was the black hole made of. When you solve the Einstein equations, you see that all this stuff decays from the point of view of someone that is sitting outside looking at it, decays exponentially in time with a characteristic time scale which is of the order of the light crossing time of the horizon, right? So after a while, this settles down to the vacuum very quickly. And the assumption is that the quantum state of local quantum fields that you may define in this geometry also settles down to the vacuum as you measure it locally, okay? So if you look at, for example, local correlation functions at short distances, you should see the correlation functions that you measure in Minkowski space, okay? So under this assumption, you get that because in any quantum field theory, in any local Lorenz invariant quantum field theory, the vacuum is highly entangled, it's a highly entangled state when you partition space in, for example, in two pieces along a surface, you get correlations between the two sides which are dominated by short distances. So that means that there is a lot of entanglement in the vacuum, a lot of quantum correlation, okay? And in this case, you can just analyze this entanglement. You can sort of diagonalize it using a convenient basis which happens to have a physical interpretation, which is very nice. The physical interpretation is that you have to look for modes of the quantum field which are constant energy with respect to an observer which is a static sitting static at finite height, looking at the black hole and seeing the black hole as a static thing, okay? So these are observers that are reporting that the black hole is in a stationary situation. The energy in the frame of disguise diagonalizes the entanglement in this particular state so that when you consider just measurements that involve operators that are outside the black hole, there is an effective density matrix that you get by tracing over half of the other part, the part which is behind the horizon and you get this thermal density matrix, okay? This is just the result of how the vacuum of a Lorentz invariant theory looks like, okay? Now, there is a more symmetrical version of this which is kind of global, right? Instead of local, in a case in which you consider the maximally extended analytic solution of the Schwarz solution. You just, the one that you see in the books, okay? If you extend it analytically, the maximum you can until you encounter singularities, what you find is that the exterior of the black hole is connected to another copy of the same spacetime, I mean another copy of the same metric but a different spacetime, different asymptotic region through an interior which is shared between them, okay? So these are two black holes that are joined at the hip sharing that interior. This point here is a point in the horizon so this is a sphere of radius R, right? And it's shared between the two. So it's the same door to enter from both sides and you enter into a room which is the common interior of the two black holes, right? So this is the famous Einstein-Rosenbridge or wormhole, right? This is a section of this time slice in the middle of the diagram. So here you see the two asymptotic regions and here you see the tube and here in the middle of the tube there is the horizon, okay? Or a snapshot of the horizon at the fixed time, okay? Now it's very interesting. Well, of course, in this picture these two are disconnected in the exact solution they are disconnected. They are totally independent but you can construct approximate solutions in which you sort of bring this around, connect this with a very, very low curvature, right? You make a big detour, right? Like for example, one month of the mouth of black hole would be in Andromeda, the other would be here and there would be two million light years in between. But there is a shortcut in the middle with these two kilometers, okay? So this is the wormhole. You cannot go through here and come on the other side because this wormhole, the doors are black holes. So you only can enter, you cannot get out, okay? Now, an interesting fact is that if the picture I was referring to before about quantum fields in the black hole background, essentially due to the analysis of Hawking, it's true you can detect whether two black holes like that are joined by a wormhole without entering, without having to go inside and check for yourself, okay? You can do that if you just consider, for example, two quantum fields, right? And you measure the two point function at two points which are just outside the horizon of both of them, right? Now, if there is no connection between them, the two point function will be one over two million light years squared. But if there is a connection between them and the quantum state is the vacuum, right? It's locally the vacuum, then there will be roughly a correlation function which will scale like one over two kilometers squared, okay? So that's a way of measuring the presence of the wormhole through the quantum correlation of these fields, right? So this is already telling you that there is entanglement in the quantum state which is betraying the fact that there is a connection, a hidden connection in a space time. Now, if you consider the global state of quantum fields that has the property that it satisfies all the symmetries of the solution, so it's just a thing that has relaxed so that it has the same symmetries of the space time. It's irregular everywhere except at the singularities and it's locally the vacuum of Minkowski space. So if you look at correlation functions at very short distances, you will detect adiabatically the same that you see in flat space, right? Then there is a unique vacuum that satisfies those properties that has many names but one of them is the hard to hot vacuum. And you can just write it down in terms of the composition between these two surfaces left and right, dividing it through this special space-like surface which is connected by the wormhole in which it looks just like the composition I did before, right? Only that these energies now are the multi-particle states if you like, these are energies in the full Hilbert space of the quantum field theory, not just single-particle states, okay? So this looks like a, so the total hard to hot vacuum state is a pure state. When you look at it on a sector, right? On half of the system you get a density matrix which is thermal, right? And one interesting thing which is a technical thing about how this is built is that this energy spectrum is continuous, right? And this is related to this fact that modes with a wave function which is arbitrarily peaked has arbitrarily structured, arbitrarily nodes very near the horizon makes up sort of an infinite phase space. So this energy spectrum is continuous, right? In this formal decomposition. This will be a little bit interesting later. Okay, so this is the third point I wanted to make in this kind of tour, okay? Black holes and quantum entanglement, okay? So from the old times in the 70s we have learned that if you have horizon smoothness and this extends to the quantum state of quantum fields in this geometry, okay? This means that you have lots of entanglement in the local state near the horizon region, okay? So this is the way in which, and there is a way of phrasing all this stuff about the thermal properties of the black hole in terms of this formalism. Whether they converse is also true is a subject of current heated debate, okay? Whether if you lack the entanglement that means that the horizon is singular, right? That's a debatable thing, people fight about that. Sometimes it's referred to as the firewall debate or the firewall paradox, things like that. I will not talk about that, although I'm sure that probably we will talk about it in the workshop. Okay, so we have this evaporation phenomenology, right? The black hole as seen by static observers outside has this thermal atmosphere. This thermal atmosphere is mostly contained by the potential well, but from time to time some quanta escape because this potential well is not infinite, right? And so there is this spitting of quanta with a characteristic energy one over R, characteristic wavelength R, characteristic emission time R. So if you were to deplete the whole black hole by these soft emissions, you would need a number of steps which is of the order of the mass divided by the energy of each emission. In order of magnitude this is S, right? The entropy. So the evaporation is just the time reversal of the procedure of Beckenstein to construct the black hole step by step, okay? So the original Beckenstein-Gedanking experiment is just the time reversal of this procedure of evaporation, right? The evaporation time is very long because it's proportional to the entropy. If you write this in terms of the mass, this is m to the cube. It's huge, right? So that means that if you look at the black hole as a kind of a resonance in an S matrix, in a quantum S matrix, okay, you can do that because you can imagine that you make it from some well-defined prescribed initial state of particles. You do an scattering experiment, you form the black hole, it sits there for long, but then eventually decays, okay? Then it is a resonance which is very peculiar because it is very narrow, right? So the width divided by the mass goes like one over the entropy squared which is extremely narrow. But there are many of them. There's a quasi-continuous spectrum of resonances all together. So it's very easy to miss it by a continuum, okay? So it's very difficult to do an experiment with an LHC type to detect these resonances individually because they are very narrow but they are also completely dense, right? Now, there are some constraints that you get just from unitarity. If you ask what is the effect of this on kind of normal observables? Like say scattering of two protons going to, well, two protons is not good. Let's say E plus and minus going to E plus and minus, okay? So a simple scattering, two to two particle scattering which is a simple object. If you do this scattering at very high energy way above the Planck scale, you should form black holes, right? So since the formation of black holes at sufficient energy is a process that occurs classically, you expect the probability for this to be of order one. I mean, controlled by geometry like cross-section but not suppressed by any parametric, by any small parameter, right? Should be of order one because it's a process that occurs classically. But when you analyze this in terms of what is inside, you get the amplitude to form any microstate of the black hole, a square, and then you have to sum over all the finite microstates which you are not disentangling, okay? So what you call the intermediate black hole generically is just something that could be in any microstate. And this whole thing is of order one, so because this are positive because of the square, this gets you what is the size of the vertex for particle to a black hole microstate. And this vertex is very small, it's exponentially small in the entropy which means that for two to two scattering mediated at high energies by some particular microstate you get a scattering amplitude which is very small e to the minus s, right? Even if you consider all, you sum over all the intermediate states because in general you will have faces here, you will have plus and minuses which they will be in principle random, right? This will look like summing a random walk and the result will still be exponentially suppressing s, right? So that means that the scattering at very high energy is going to be extremely soft, right? It will be scaled like this. And this is unlike anything that you find in quantum field theory and stream theory. Stream theory is also soft in perturbative stream theory but it's not that soft, okay? And this is directly related to this other, to this second funny thing that I mentioned at the beginning that the density of states implied by Beckenstein formula is just huge, okay? And this translates when you ask how this translates into, I mean, what is the consequence of this fact at the level of a simple observable like two to two scattering amplitude, you find that the answer is that you get this e to the minus s effects running around, okay? So in the rest of the talk, what I'm going to do is to try to follow the money, meaning follow the e to the minus s effects, right? To try to see in what other places the detailed quantum nature of the black hole has importance, sorry. Okay, so for example, one thing that you have to forget is to compare anything that involves e to the minus s effects with anything in the context of Hawking calculations, okay? Why? Because Hawking calculations were done in low energy effective field theory. Low energy effective field theory in the presence of gravity is an effective theory in the sense that it has a cutoff, right? Plank scale is non-denormalizable but it makes sense as an effective field theory in the sense that if you expand any observable in powers of energy divided by plank mass and you keep only a finite number of terms, the theory can be normalized with a number of counter terms which is of the same order of magnitude as the number of powers that you are keeping, okay? The same happens in the chiral Lagrangian approach to pion theory, et cetera. So as an effective field theory, it makes sense but it has a finite accuracy. The expansion parameter that effective field theory is of course an expansion in Newton constant but in order to find that dimensionless effective coupling you have to multiply it by the characteristic energy scale of the problem by dimension analysis. And in the case of the problem of for example, emitting particles of the Hawking type that is the temperature square, okay? That's the typical energy of these particles and that gives you g over r squared which is one over s. So the effective expansion parameter of the effective field theory is one over s which is extremely small. This is much better than QED for sure for a microscopic black hole, right? But it also means that anything that goes like e to the minus s in a physical quantity is beyond the realm of perturative effective theory because it's something that goes like e to the minus one over alpha f. It looks like an instant on effect, okay? So that means that you have to master the dynamics of gravity at a non-perturbative level otherwise you are not able to discern any effect of this kind, okay? And despite the fact that this thing is small you need to take care of that if you don't want to fall into traps like for example the so-called information paradox, okay? So let me remind you very fast about the information paradox. The information paradox, okay? Remember that I said that the Hawking evaporation you can describe it more or less by saying that each quanta is emitted as approximately a member of an EPR pair, okay? Because there is this entanglement, this strong entanglement near the horizon between modes that are outside and inside. So the mode that comes out is like a very strongly entangled with something that was left behind. And therefore the increase in entropy in the radiation as you are collecting this quanta that are coming out you are counting them, one, two, three, okay? They come out for a big black hole they come out every hour or so, right? For a black hole like the one in the center of the galaxy they come out every eight minutes, right? So these guys increase the entropy of the radiation more or less by one bit every time you get one, right? So the entropy of the radiation will be mounting proportional to the mass in radiation. So if you make a plot of the fine grain, fine omen entropy of the radiation, density matrix, okay? You should find something that goes linearly with the mass of the radiation. On the other hand, at the very end when you just completed the operation and you have this huge amount of quanta floating around if the initial state of the black hole was pure the state of the radiation should also be pure that means that this quantity should be zero. This is fine grain for omen entropy, right? When you have collected everything. So this curve has to come down and the normal thing is that it should come down more or less from the half time, right? So called page time, which is roughly half the operation time, so it's proportional to S. The entropy is a huge time, okay? But then there is a problem. The problem is that at this point the black hole is still big. And everything I said before about the accuracy of Hawking calculation into the minus S effects, et cetera is still true, okay? Curvatures will be very small. So for every emission, you still seem to add one bit of thermal entropy to the radiation every time you emit something from here. So Hawking would tell you that this thing would continue going up until the end, okay? And this mismatch, which is a further one after this time is the information paradox, okay? It's a very striking thing. But you know, let's say that you want to say, okay, what is the effect of the initial state, the details of the initial state, on the emission process of one quantum after half evaporation time? If you estimate that using the Noher theorem, you will say that this should scale more or less in order of magnitude like e to the minus t times the characteristic relaxation time of black hole, which was roughly the Hawking temperature, inverse. So at the page time, this is a small thing. This is e to the minus t times t page, which is e to the minus S, our friend, okay? So very small, right? However, notice that we are not computing a quantity referred to one quantum. We are computing the fine-grained von Neumann entropy of the radiation, which contains order S particles. So in order to establish, I mean to calculate this, you have to diagonalize a matrix which has e to the S times e to the S entries, which is the density matrix, and then take the log and sum, okay? So you have to do e to the S measurements to measure e to the S matrix elements. And an error of e to the minus S times possible e to the S opportunities to error could give you an order one effect, okay? So you have to be careful because you are asking a very complicated question here. So another way of seeing this more graphically is to say, okay, let me say that I have a pure state living in a Hilbert space which has dimension e to the S. The density matrix is this one in the basis that diagonalizes the density matrix. So it's just one and the rest is everything zeros. But if I give you the matrix the same state in a different basis, which is rotated randomly with respect to this one in this big Hilbert space conjugated by a random unitary matrix which is e to the S times e to the S, then of course, you will be handed a matrix which looks like a pretty random matrix with arbitrary faces, arbitrary looking faces and size of the entries of order e to the minus S, right? And if I hand you this matrix it will be non-trivial for you to find out if it is a pure state or not, okay? In fact, I can just write this as e to the minus S times the identity which is the maximally mixed density matrix maximally different from a pure state plus a correction which is just e to the minus S in the entries which looks just like this, okay? So I could say that this is the Hawking estimate and here is the correction which I could say is some non-perturbative correction which is beyond effective in theory, okay? So you could say that purification at the page time is not expensive because it can be obtained by very tiny corrections at the level of each effect on each quanta because there are so many quanta, okay? The total effect could be of order one at the level of very complicated observables, right? So you could say, where was the mistake by Hawking? Well, there was no mistake, Hawking was right because within the realm of his approximations his result is right, okay? There is a sharper way of getting the same insight which uses a different version of the physical setup when you talk about black hole thermodynamics in which instead of thinking about the black hole evaporating, which is a complicated thing because you have to wait for so long in order to see something happening, okay? So slow a process. What you can do is just forget about this adiabatic very slow process and just force the system to be static, right? So you just force the system to be inside the box which is kept at fixed temperature, right? And then you ask if the black hole inside can be in equilibrium. If the black hole can be in equilibrium then you say that the black hole has a temperature which is the one that you set up with the box. So this is actually a very physical way of defining what you mean by the black hole being thermodynamic, okay? Now, there are many subtleties in the definition of the box. The box cannot be too big because then there are problems with genes instability. There are problems with the fact that the equilibrium is unstable. So the box has to be a little bit, not too much. I mean, it has to be bigger than the black hole, of course, to contain it but it cannot be arbitrarily big, right? You also have to have the box make sure that there is weak gravity in the walls of the box. So you have to control that otherwise you don't know what to do. And in fact, there is a very good box for gravity which was invented by Hawking and Page. I don't know if it was invented by Hawking and Page but in this context probably it was, right? To box gravity, right? We did the thermodynamic summaries. Right. It's a perfect box for gravity. And it's anti-decentral space time, right? So you consider maximally symmetric negative curvature solution of Einstein equations in vacuum so with a negative chemical constant. And this space is very funny because it has a harmonic trap for gravity and actually for everything. If you throw a stone, the stone comes back at you, okay? So it's like living inside a harmonic trap and the spectrum of excitations is discrete. It's a non-compact box because it's infinite and the wall is soft because how much you have to wait and the things come back depends on the energy and would you throw them? You throw them at higher energy, they take longer to come, right? But they come, right? They come. So it's a box. They come at the same time. What? I'd say they all come back at the same time. It's an ideal relativistic harmonic oscillator. No matter how big the amplitude is, the period is the same. Okay, yeah, that's true. But they get, the turning point gets farther away. That sense I say that it's soft, it's soft. It's soft because they bounce from... The amplitude still gets back in the finite time. I know, I know, yeah. Very strange. Thank you, thank you. They bounce from farther away. In that sense, it's a soft box. Okay, so the funny thing about this is that when you compute the thermodynamic entropy of the black holes which are big enough, then the answer is not anymore M squared like before, which was this very funny thing, right? The answer is that in D plus one dimensions, you get this energy to D minus one over D, which is just how the entropy of a quantum field theory scales. If you consider highly excited states living in a box of size L, where L here is the curvature radius of ADS. And there's a pre-factor here, which scales in this form like just the number of species of particles that this quantum field theory would have. Okay, so you have a quantum field theory. Consider very high energy states so that you can neglect the masses. So it's a bunch of massless fields. That's technically, you would say, a conformal field theory, okay? So it's a theory of massless fields at very high energies and with the degeneracy in the number of fields. So this is the number of species. N is just a ratio between, it's a power of the ratio between the radius of curvature of ADS and the Planck scale. So N is very big if your ADS space is large compared to Planck units, so it's a gravities-quickly couple, okay? So this is the beginning, if you like, of ADS CFT, right? Because now you have found a box for which the Hamiltonian is something you know, okay? Something is in the books. And the hypothesis that everything that fits inside ADS, including little gravitons wandering around, gravitational waves, planets, small things, can be found inside the equilibrium space of a conformal field theory with a large number of fields is the hypothesis of the ADS CFT, okay? Okay, so now you can just think for example of taking the box and kicking it on the boundary, right? So that the perturbation propagates inside and gets scrambled by the black hole, right? And think about the correlation for the perturbation to come back, measuring this two-point function of this operator that represents the kick on the boundary in time. Now in general, this will decay, right? As the perturbation scrambles around these many degrees of freedom. And if the spectrum of the black hole microstates is approximated as a continuum, neglecting the little spacing, so for the e to the minus s between the energy levels, then the correlation will continue to decay forever because there is an infinite phase space here for the correlation to just, the wave function has always a piece that is still not coming, not coming, not coming, okay? Because there is still degrees of freedom that has to be visited, all right? So that means that this will continue decaying forever. But this cannot happen in any system that has really discrete spectrum. So if you don't neglect the e to the minus s, then you can prove in general that in any system with discrete energy levels, like a conformal theory in a box, right? These correlation functions have always a tail of quantum noise that you can estimate to be a tight e to the minus s, okay? So there is this long time quantum noise of oscillation, which is afforded e to the minus s, you can just prove it just very simply by doing an estimate of the sums. Now, there are many things here that are interesting. For example, there are also Poincare recurrences, revivals on scales which are double exponential in the entropy. The characteristic scale for these oscillations is, or when these oscillations set in is the so-called Heisenberg time, which is the inverse of the typical spacing between the energy levels. So what you want is for the quantum state vector to have basically made all its faces rotate in energy space, okay? So that you cannot approximate any of them by a continuous spectrum. So in fact, there is very interesting stuff that you can get from here. Like for example, you can see some structures in this quantum noise and related to properties of the quantum theory, spectral properties, like different bands with different densities of states, and the corresponding geometrical properties of the gravity picture in ADS-CFT, okay? This was a proposal. The proposal to use this as a criterion for unitarity was done by Maldefena like 15 years ago. And this is a connection with the theory of quantum chaos, actually. So probably we'll hear about this in the workshop. Okay, so I'm going to go fast to the end, right? So now something remarkable is going to happen, right? First, there is a triviality, right? If I give you a box which is described by some thermal density matrix, some mixed state, I can always represent it as a partial trace of a double version of the system where I put a particular pure state in the double version, right? With the coefficients of the entanglement chosen so that when I trace over half of the system I get the right density matrix, okay? So for a thermal density matrix, these are the coefficients that I have to choose, okay? This is kind of obvious. But now, suppose that I take as my two boxes two conformity theories that are totally disconnected. They live in different boxes. They have, the Hamiltonian has no interaction term between the two, right? But allow me just to prepare a state which is entangled between the two with this particular structure, right? The so-called thermo-fill double. Now, if I neglect the tiny e to the minus as a spacings between the energy levels, I just write this in this way, right? I transform the sums in an integral. And because these very dense energy levels are my representation of states of quantum fields in the black hole background, right? In the approximation of continuous spectrum. Then this formula that I get here is just the Hartel Hawking state of the bulk fields that I wrote before, right? Just that. So I obtain that the energy that's composed, the composition of the Hartel Hawking state of quantum fields in this connected geometry, right? So the dual in the sense of a CFT of the separated disconnected CFTs in this particular state provided I neglect the e to the minus s little discreteness is just the theory of quantum fields propagating in this connected manifold, right? So you could say that by bringing in the e to the minus s details, ADS-CFT implies that the background geometry itself is a result of the entanglement of the two disconnected CFTs. So not only the entanglement buys me, sorry, not only the smoothness of the geometry across the horizon gives me high entanglement in the quantum fields, but it seems that the entanglement, lots of entanglement in the dual holographic picture builds the geometry for me, okay? And this is the reason the slogan for this which is kind of famous is EPR equal ER, which is just EPR stands for Einstein-Pogolski-Rosen to represent entanglement and ER, Einstein-Rosen to represent this idea that the geometry is connected, right? The wormhole. So here is a hypothesis accumulating a density of entanglement of a large number of quantum bits well-separated bell pairs within a transversal size of order g times s to the one half seems to generate a geometrical breach of area gs, okay? So it's like by concentrating entanglement, condensing entanglement in huge quantity so that you form a black hole, then there is a breach form through this black hole, a piece of space. Okay, so what about dynamics, for example? What happens if I just let this state evolve in time? Then I get all these phases here, right? And if I look at what happens in the gravity picture, in the dual geometrical picture, then I get this license of the geometry in which I pick more of the interior of the black hole so it's like the wormhole is becoming longer, right? So complicating the entanglement through these random phases produces a longer wormhole according to this hypothesis, okay? So this is a way of interpreting this as the following. So suppose you have a model with s qubits, which is s is huge, okay? And I consider a totally simple, highly entangled state just by pairing them as bell pairs. Now suppose that I switch on the time evolution operator which what it does is to complicate the entanglement, okay? It's a huge matrix which is just rotating all this entanglement in arbitrary direction in hubris space. So it's breaking the coincidence of the phases of the basis which diagonalize the entanglement and the basis that diagonalizes the Hamiltonian, okay? But the amount of entanglement here is the same. The entanglement didn't change. It's just that it got complicated, okay? So what it means for entanglement to be complicated? Well, there is a definition of this in quantum complexity theory, a technical definition of what do we, what do you mean by that, right? What would be a definition, an operational definition of that? And it's the following. It's the idea that you construct the matrix U by multiplying a standardized little matrices, right? So you make a quantum circuit which is just a set of unitary small matrices that are applied in some sequence to build a big matrix, okay? And this definition of complexity depends of course on the choice of the, a little bit on the choice of the standardized little matrices called gates, right? It depends also on when do you decide to stop, namely how well you can approximate this matrix by this discrete procedure, right? So there are lots of technical issues that go into trying to make contact with all this lore of quantum complexity theory, but the idea would be that this could be a definition of what you mean by the wormhole becoming big. It's just that the bigger the quantum circuit you need to build the unitary matrix, the longer is the wormhole, okay? Now in general to parameterize the complication of entanglement, you can use a language which is a little bit more general than quantum circuits, which is just to imagine that if you have a Hilbert space which is factorizing pieces, say S pieces, right? And you consider a general state here, you can represent it by a big tensor, right? Which gives you all the coefficients of an expansion of your vector in terms of this basis, the basis of, that you get by tensor product of basis of the different factors, okay? Five minutes is possible. So I have these big tensors, right? And a definition of what it means for tensors to be simple or complicated would be that I can just try to decompose them into products of smaller tensors, building blocks which are kind of standardized again, right? And then the complication has to do with the combinatorics of how I combine the products. So this could be, for example, a rather simple entanglement pattern, but this would be a more complicated one, okay? And the art of parameterizing entanglement this way is what's called tensor networks, which is a technique that is being used a lot in condensed matter physics to use, to do variational analysis of ground states of many body systems, okay? Because this is just an answer for wave functions, right? And for example, there are some striking things, like for example, people who constructed tensor networks for approximating ground states of spin systems that are supposed to be critical, that are supposed to be close to the critical point so that they are almost conformal to theory, right? They found that they could get a very good variational ansatz for the wave function of these systems by constructing a tensor network that has hyperbolic structure, it's an spander graph, right? It's like a tree. But this is just a lattice version of the T equals zero slice of ADS, right? So there is this kind of eerie correspondence, you know? It looks just like the structure of the entanglement of the wave function of your theory on the boundary is just building for you the geometry of interior. Okay, so final slide, next to final. So in general, this story of complexity, quantum complexity theory is the idea that you want to look at the Hilbert space like you used to look at the phase space of classical physics, okay? So in classical mechanics, you remember that states are sets and in quantum mechanics states are vectors and this makes a difference because in order to say that two states are an ambiguously distinguishable in classical mechanics, you say that they are characterized by two disjoint subsets of phase space. But in quantum mechanics, you say that they correspond to two or three verticals, okay? And these relations are different. And part of the funny things about quantum mechanics compared to classical mechanics has to do with this different mathematical definition of what a state is. In quantum complexity, you just look at the Hilbert space and you try to desolate it like it was a phase space of classical mechanics. So you talk about, you look at a vector in the Hilbert space and you look at all the vectors that are closer to epsilon in some norm to it and you replace them by a representative which is some tensor network, which will be very big. And then in the next cell, you use another tensor network and just like that, you do a cell decomposition of the Hilbert space, okay? And then the complexity is a little bit like the entropy in the classical mechanics. So for example, there is a maximum complexity for a given course grain in scale which is the number of cells which is a double exponential in the entropy, right? Just like the classical entropy is the number of cells once you have defined the course grain in scale. Now a funny thing here is that in classical physics, ultimately, we found that there is a physical course grain in scale which is H bar whereas here epsilon has no place in physics, you know? This is totally conventional. The people that discusses quantum complexity theory don't ask what is the physical value of this thing. This is just a conventional thing, right? It's when you decide to stop approximating the states. Okay, so the final slogan which in a sense covers a number of things that will be discussed at the workshop is that being serious about e to the minus s effects which appear completely negligible gives you to the astonishing, leads you to astonishing suggestion that the very fabric of space could be a condensate of pure quantum entanglement. This sounds very pretentious, right? Very, very profound, right? But there are, you know, there is circumstantial evidence in ADSFT that this happens, at least in ADSFT models. And there is this emerging dictionary that entanglement is related to the connectivity of the bulk geometries and complexity is related to the amount of bulk geometry that you are generating. Okay? Now, I finish with a list of open questions and problems. This is, if you like, for the experts so I will just read them, right? So we need exactly calculable toy models of ADSFT along the lines of the JK model, for example. So this is because ADSFT is non-trivial despite someone would tell you that ADSFT is trivial because everything is in the symmetry. Don't believe, ADSFT is very non-trivial because we don't even know if most of the CFTs we imagine really exist and they have a holographic dual description. It's not every CFT can generate geometry efficiently. We would like to give a renormalized continuum definition of quantum complexity in CFT because after all, the CFTs are objects, the objects that generate the geometry, not these qubit models that are there just for orientating our intuition. Can tests or networks describe bulk gravitons? You know, can we take the continuum limit of these things and really recover something as fine-tuned to the Lorentz, local Lorentz symmetry of space-time as a graviton? Don't know. What is the space-time meaning of quantum complexity saturation? What is the space-time meaning of this epsilon that was mentioned before? Does it have a meaning? The fact that there is a maximal complexity, does it have a meaning? Can we define approximate local observables for black hole interiors? This is just solving the original problem that I was referring to before when you think of the interior of a black hole as codified by degrees of freedom on the horizon. This problem is still open actually despite much effort in ADS-CFT. This problem is still open. Or at least people doesn't agree about how to solve this problem. And in particular, there could be abstractions related to so-called firewalls or fastballs. Namely, it could be that this problem doesn't have a totally genetic solution, that it depends a little bit on what type of states does the black hole find itself in. Okay, so I think this is all. So thank you very much. I'm sorry for the other time. Are there any questions for the introduction? Yes. Regarding your point about local observables for the black hole interior, if the black hole interior is a gravitational theory, one would expect not to have local observables, not to be well-approxacled by local observables at all. So at least from what is known from classical gravity, so why would one expect that? Well, in general, the notion of local observable has to be approximate, okay? But if the local conditions of gravitational theory are, if the gravitational theory is weakly coupled in a certain region, right, there should be approximate local observables, just like the ones we use all the time here, right? These observables should be defined within errors of order e to the minus s, where s is the entropy of a black hole that covers the region in which you are doing your measurements. So I'm referring to this kind of a more modest aim at defining approximate things that you would call local observables operationally. Even that is non-trivial, okay? I may ask about your epsilon, about this resolution. Now, just to take an example, in string theory, one knows that one has t-duality. So if one tries to make the resolution smaller and smaller, at some point, one gets a t-dual configuration and one goes into a dual theory. So is it possible to expect that something similar happens here, that is one tries to shrink the epsilon? I don't know, I mean, there is no physical intuition about that, I mean, if there was a physical value for epsilon, that would be a new theory, a new fundamental theory, some modification of quantum mechanics, okay? Is to say that the Hilbert space has a granularity, right? So I'm not defending here that epsilon is physical, right? Most likely it is not, right? After all, this epsilon is built on this way of thinking which is algorithmic from the people that think in terms of qubits, right? So most likely all these ideas, in order to see if they have a life in, say, ADS-CFT proper, you have probably to learn how to renormalize them in the continuum limit and things like that, right? And focus on things that do not depend on details of how you coarse-grain things in Hilbert space, et cetera, but that's completely an open problem. I'd very much like to talk and I'd like to take you up on one particular point, which is the following. You've distinguished classical state spaces with quantum state spaces, and you use the notion of Hilbert space, but actually quantum states are a special case of classical states, because once you've taken the phase out from your state vectors, you're on complex projective space, which is a symplectic manifold, and Schrodinger's equation reduces to Hamilton's equations. So now, of course, this works only for finite dimensions, but I'm just wondering if your epsilon is something to do with the dimension of yours, rather than some other sort of cut-off. But from a geometrical point of view, this picture of vectors is very extrinsic rather than intrinsic. Yeah, OK. I don't know what to answer, actually. OK, let's thank Jose again. OK. Thank you.