 The radiation emitted by a black hole, and he found that the radiation is thermal, and now we want to investigate the possibility that small corrections to the computation Hawking can restore unitarity. And the claim we mentioned before we stopped is that exponentially small corrections with simple observables are sufficient in order to restore a unitarity. And we were about to argue why this is a very general property of statistical systems with many degrees of freedom. So we'll try to investigate this statement now, which is that if you have a large quantum system, so a quantum system with many degrees of freedom, then most pure states look almost identical when probed by most observables. And moreover, not only are different pure states look identical to each other, they also look almost identical to the maximally mixed state. So this is the statement, and you see all these underlined words I have to make more precise, and that's what I'm going to do by giving you the derivation of this statement in terms of some equations. So what we'll consider is a finite dimensional Hilbert space. So we consider a quantum mechanical system, and it could either be a finite system, or you could take a system like a quantum field theory, and consider the subspace of the Hilbert space, which is spanned by energy eigenstates in a particular energy window. So in that way, you get a finite dimensional Hilbert space, and we denote the dimensionality of the Hilbert space by n, which we can also think of as e to the s, where s is the entropy. And then we consider a linear observable, a linear operator, a, acting on this Hilbert space. On this Hilbert space. Now, this Hilbert space has many different pure states, and what we're going to understand is, in what sense is the expectation value of this operator different among the different pure states. So let's denote i, an orthonormal basis of this Hilbert space, where i goes from one to n. The most general pure state that we can write down is a vector of this form. So this is the most general pure state. And we want to consider the expectation value of this observable on this pure state, and we want to investigate how this expectation value depends on the choice of the coefficient Ci. And what we want to argue is that for most choices of these coefficients, this result is going to be almost the same, provided that the dimensionality of the Hilbert space is very large. In order to prove the statement, first of all, I have to remind you that these coefficients have to obey the condition that sum of Ci squared must be equal to one. And these are complex numbers, so you can split them into the really imaginary part. So you can think of this equation as defining a sphere of dimensionality two n minus one of unit radius. So the set of pure states of this Hilbert space can be thought of as points on a sphere of very large dimensionality. So we have this very large sphere, I mean sphere of very large dimensionality, and every point on the sphere represents one particular pure state. And then we have some observable A, some operator, whose expectation value we want to calculate on a given pure state. So this defines a function, if you fix the observable A and you look at different pure states, this defines a function of this sphere. And we want to know whether this function fluctuates a lot as you move on the sphere or whether it has more or less constant value. Now, so this is the set of pure states. Now, obviously, this result will depend on the coefficient Ci, so we don't want to know the details of this function, how it behaves in detail, because the details will depend on the particular observable. We want to make a statistical statement. So to do that, we want to introduce a notion of a measure of the space of states. So we will introduce a measure on the set of pure states, which is going to be the most natural one, namely, each pure state is equally likely. So this is called a micro-canonical measure. Geometrically, what it means is that if you select a particular region of the sphere and you want to know what is the probability to select the states which are inside this region, then you declare that the probability is proportional to the volume element of that region when we select the round sphere of this metric. So we define this micro-canonical measure in equations. We can write it in the following way. D mu is equal to a coefficient A that I will fix in a little bit, times dC1, dC1 star, so the dot dCn, dCn star, so these are all these complex coefficients which define the pure state, times a delta function which restricts this measure on a sphere of unit radius, and then this coefficient A, this constant A, is fixed by the requirement that the integral, the integral of d mu must be equal to one. So by imposing this condition, we can fix this coefficient A, yes. Yes, you can also do it by modding out by the phase, but it will only introduce some overall multiplicity factor. So this is called the micro-canonical measure on the set of pure states, and it's also called the hard measure because you can get the same measure by starting with one particular reference pure state on the sphere and then map it around with all possible unitaries where you select the unitaries of the hard measure of the unitary group. So this measure is called either the micro-canonical measure or the hard measure, and statistical mechanics is the most natural measure you can write down in the space of states. But we usually do statistical mechanics when we declare that all micro states are equally likely. So it's a very natural measure, and what we're going to do now is we want to take this observable that we want to study and calculate the statistical properties of this expectation value on the set of pure states. So the first thing I want to calculate is the average value of this quantity over all possible pure states. The first quantity we will calculate is this one, which I will also denote as psi A psi average. And to evaluate it, we introduce the matrix elements of this operator, and we can write this quantity as sum over ij A ij times the integral over the mu C i star C j. Now, if you look at this measure, it should be obvious that this measuring is invariant under independent rotations of the seeds by phases. So if you take C i and you rotate it by e to the i theta i C i and C i star by e to the minus i theta i C i star. So if you rotate each of these seeds by different phase, the measure remains invariant. This condition is a condition that the sum of the C i squared is equal to one so that the state has unit norm. It must be a pure state with unit norm. So this measure has this invariance which in particular implies that this integral will give you something known zero only if its i is equal to j. Because if i is different from j, you can rotate one of the two guys by a phase, but not the other one. The measure is invariant, so the integral will be rotated by a phase, but that can only be consistent if the value of the integral is zero. So what we learn immediately by the invariance of the measure is that this quantity is proportional to delta ij. Sorry, what was the question? Say it again. In front of the delta function, there's a constant. Is that your question? This C i, this is a change of variables. It's a phase. I'm saying that this measure is invariant under a redefinition of the seeds by phases which can be different for each of the seeds. It's obvious because from this one, this one star, right? This is two, this is two star and so on and so forth. So this means that this quantity here, d mu C i star C j is equal to some constant. Let me call it a small kappa times delta ij. And then if you want to evaluate this constant, it's very easy. You notice that this kappa is equal to the integral over d mu of C i squared. I mean, in principle, this could depend on i also, kappa i times delta ij. So kappa i would be the integral of C i squared. However, all the seeds are entering the measurement in an equivalent way, so that measure is also invariant under permutations of the seeds, which means that actually this quantity is independent of i, it's a constant, so I can drop this index i. And then I can sum over all the i's to get that n times kappa is equal to the integral over d mu of one, which is one. So this kappa is one over n. So we learned that this quantity here is one over n times delta ij. All right, now we take this result and we plug it into this formula. So we find that the average expectation value of this operator over all possible pure states is equal to one over n times a ii, sum over i, which is nothing else, but the trace of rho m, the micro canonical density matrix times the operator a, where rho m is defined as the identity operator acting on this Hilbert space divided by n. So what we proved is that the average value over all possible pure states of the expectation value of an operator a is exactly equal to the micro canonical, to the expectation value of the same operator is the micro canonical mixed state, the maximally mixed state. We didn't use it yet. So this is true regardless of the value of n. Just hang on for one minute and I will explain what is the importance of large n. So this is an exact statement. Of course, what we have proven is that, so this quantity we have, depending on how this will be different, this will have a different value and when you integrate over all possible values you get this result. Now, this by itself is not telling you that different pure states look almost the same because it could be that the variance of this observable is quite large and only if you take the average over all possible pure states you get something which is related to micro canonical density matrix. So to make the statement, to prove the statement above we need to consider not only the average of the expectation values but actually the variance of this quantity. So I need to do one more calculation of this type which is the following one and here is where large n will start to play a role. So now we want to calculate the expectation value of the operator A on the same side minus the average, the whole thing squared and then we take the average over all possible pure states. Now I'm not going to do it in detail. If you want to do this calculation you need a little bit more complicated integrals of this form. So you need something like integral over d mu C i star C j C k star C l. You need to calculate this quantity which comes, which appears when you take the square of the first term but it's very easy to, it's an easy exercise to evaluate this quantity as a function of n is some group theoretic computation. You can also think of it in terms of random unitaries and integrating random unitaries and computing products of matrix elements of unitaries. And I'll just write down the final result which is that this variance is equal to one over e to the s plus one. I remind you e to the s is equal to n, right? I just write it in this way to make it more clear that it's explanation in the entropy times raise rho m a squared minus trace rho m a the whole thing squared. So this is the final formula we want to emphasize. So this equation is telling us that if you have a system, a quantum mechanical system with many different pure states and you consider how the expectation value of an operator varies among different pure states, you find that the variance is exponentially suppressed in the entropy of the system times a factor that depends on the operator a but as I will explain, usually this operator is this quantity is bounded. Hence, we find that the variance is exponentially small. So this shows that most pure states will look exponentially close to the maximally mixed state. And sometimes we write it in the following way. Psi a Psi is equal to trace rho m a plus corrections of the order of e to the minus s. But this is the more precise statement. Yes, say again. What else goes like one over n, I'm sorry. Yeah. No, no, the average is, the average, there's one over n here but you are summing over n indices. So this order one. So the important thing here is that this formula is an exact mathematical identity. It's sort of trivial to prove. It just takes, you just need to calculate that integral and you're done. So it's a trivial mathematical identity which is true for any observable a. So the nature of the operator a of the observable a does not enter the proof. It can be a very complicated observable in principle. The only thing we need to know is that it's a linear operator acting on the Hilbert space. It's just an identity in linear algebra. Moreover, the Hamiltonian of the system did not enter the calculation. We have not made any assumption about the dynamics. So this is a general identity which has a very important physical implication which is telling you that no matter what the quantum system is that you're considering, it is exponentially difficult to distinguish different pure states. Most pure states will look almost identical up to exponentially small corrections provided that the entropy is very large. Well, yes. I mean, if the operator a is not bounded, then... Well, in the way I did the derivation, the Hilbert space had final dimensionality. So every operator is bounded. So if you want to do the same thing for an infinite dimensional Hilbert space, there may be subtleties. There is no condition, absolutely no condition. Good. So let me make two comments. First of all, somebody could say that, all right, here we have found that there's an exponentially suppression, but perhaps there are some observables, A, where this quantity will be exponentially enhanced and will cancel that factor. But I think that most observables that we want to study quantum mechanics can be phrased. Most questions we want to study quantum mechanics can be phrased in terms of projectors. And for projectors, these quantities are bounded by one from above. So if you take this A to be a projector, then it is guaranteed that this quantity here is bounded. So then you definitely get exponential suppression, no matter what this observable A is, or this projector A is. So you shouldn't worry about the possibility of getting enhancements because of the large eigenvalues of this factor, because if you want to calculate a probability in quantum mechanics, we want to calculate probabilities for things to happen, right? So any probability you can phrase in terms of projectors. And here we notice that this quantity is always bounded if A is a projector. Now to come to the second part of your question, which I think is what I'm going to say now. Of course, this theorem may, now you can ask, does this mean that there is no way to distinguish different pure states? No. If you know, for example, that your quantum system is in a particular pure state size zero, you can define the projector P zero, which is equal to size zero, size zero. And then the expectation value of this operator, all the state size zero is going to be one, while on most other states, it's going to be exponentially small. So if you know that the system is in the state size zero, you can identify a specific fine-tuned observable that will click with that state and you will give you a very large deviation from the average. That's why in the slide, there is this almost observables. So the statement of the theorem is that you take a fixed observable, which you do not select in relation to a particular state you want to study, it's just some specific observable. And then you calculate the variance of a fixed observable over all possible pure states. If you start correlating the observable with a state, you can sort of avoid this result. So okay, this is telling us that most pure states look identical to each other and also identical to the maximum mixed state, the macro canonical mixed state. So this means that if we have a large quantum mechanical system and we want to compare the result of expectation values in a situation where we are in a mixed state or in a pure state, we expect the deviations to be exponentially small. Okay, are there any questions about this before I move on? Let me say a few more words. First of all, for those of you who are familiar with the eigenstate thermalization hypothesis, this does not require, I mean it's a more basic theorem, does not rely on the ETH. Also, please keep in mind that here we're comparing the typical pure states to the micro canonical density matrix. Sometimes we want to compare it to the canonical density matrix. In that case, the deviations are suppressed by inverse powers of the entropy, not by exponential factors. So here it was important for the proof that this was the micro canonical density matrix. If I want to consider a thermal density matrix, then these deviations can be larger. Also, this theorem does not tell you that, this is a statement about most pure states. You can always find atypical states where the variance from the average is quite large. So the way you should think about the statement is that by introducing that measure that we wrote down before, we introduced the notion of a typical state, of a typical pure state. And what we're saying is that a typical pure state looks exponentially close to a maximally mixed state. All right, so usually statistical mechanics, canonical and micro canonical, differ by one over S corrections. Here, we are finding something which is exponentially small, so it's even better in this theorem, right? But if you compare a typical state to the micro canonical ensemble, the agreement is exponentially good. Yeah, so if you compare typical pure state to the micro canonical, you are making exponentially small mistakes. If you now want to go further to the canonical one, you introduce one over S corrections in addition to those, yeah, I mean, those are sub bleeding. The difference between these two assemblies of order one over S, yeah, yeah. I didn't say anything about typical observables, right? I only talk about typical states. Well, if you have a statistical system, like if you take the gas in this room, and if you assume for a moment, it is in a pure state, and if you believe that statistical mechanics works, then it is a typical state. That's the statement of statistical mechanics, that if you have a system, and you want to study its properties, it's very convenient to assume it is in a typical state, and most of the time it will be in a typical state. But definition of typical states occupy the, most of the volume of the set of pure states. Now, I mean, I don't know, are you talking about the ground state of the system? Yeah, then yes, it's not typical. Now we are talking about, we're doing all of these to talk about black holes, which are thought of as thermal systems, right? Systems with temperature. So, for the low-lying states of the quantum system, this is not very relevant, but we're not interested in those states. Well, since we'll talk about the fastball proposal a little bit later, let me make a remark now. The fastball proposal is the idea that different pure states of a black hole, microstates of a black hole, correspond to different geometry. This theorem imposes a very strong, let's say, condition on that statement, because it's telling you that even if different microstates correspond to different geometries, somehow the expectation of values of observables that an infalling observer would detect have to be almost the same for all of those microstates. So, this suggests that even if there were different geometries for every single microstate, different fastball configurations, somehow they should have some universal behavior captured by the maximally mixed state. So, there should be a universal geometry. Okay, let's move on. So, we have this basic statement that I already explained, and let me say one more time that you can prove the statements just by using linear algebra without assuming anything about A or the Hamiltonian of the system. So, it's a very powerful result. So, what does it mean now about the Hawking radiation? It means that, well, we have this quantum state that Hawking predicted, which is mixed, and we want to see what should be the size of the corrections necessary in order to unitarize the radiation, and this theorem is telling you that if we can think of a black hole as a typical microstate of a quantum mechanical system with entropy given by the entropy of a black hole, then based on general arguments, we expect that these corrections to the computation Hawking are going to be small by exponential factors. So, in particular, this means that the calculation Hawking is reliable for the approximate computation of low point functions up to these corrections, but it may not be reliable if you start looking at high point functions, and the reason is that the number of possible high S black hole point function you can write down is quite large. So, you can always select a particular combination of these guys that clicks with a particular microstate and gives you a very large signal. So, you can write down a fine tuned observables like projectors on a particular black hole microstate if you start using high point functions between the Hawking particles. When I say high, I mean that the number of particles that you are involving in the correlation function scale with the entropy of the black hole. And another way of saying that is that the quantum information of the black hole is encoded in correlation functions between Hawking particles where the number of particles must scale with the entropy. So, the information of the black hole is encoded in high point functions in the Hawking radiation. Which is the thing that cannot be reliably calculated by the Hawking computation. Okay, so this is the scenario for unitarizing the black hole by small corrections. Yeah, this is true for any quantum mechanical system. Like if you take a spin train, a spin train in a pure state and you want to identify the particular microstate, you need to be able to calculate correlation functions between many spin operators. I mean with the number of spin operators that you consider is comparable to the total number of spins. If you only look at two point functions, you will never be able to reconstruct the microstates. So, we expect the same to be true for the black hole. Well, yeah, yes. This is also important. For example, as we will see later in ADSFT where we can make it more clear, we can define those ensembles more clearly. Even if you take the black hole to be in the micro canonical ensemble which you can do in ADSFT because the system is in a box, right? You can see that even though the state is in the micro canonical ensemble fundamentally, the quantum fields around the black hole are thermally populated. So, this issue is not so important. But maybe it will become clearer tomorrow when we'll talk about black holes in ADSFT. Yes. Can you say it one more time a bit louder? Yes. You're asking me, when do we apply this relation? The both one. Yeah, yeah. So, yes, both of these relations you can apply them after the black hole has evaporated and you just have this cloud of radiation flying out. So, this Hilbert space in that context would be the Hilbert space of the Hawking radiation. In that Hilbert space, Hawking predicted that the final state is mixed while we want to see, well, what would happen if the state had been pure? Now, so the question was how do we see that the information is encoded in this corner in this S black hole point functions? I should have emphasized that it's encoded not in a single, I mean, not just in one correlation function of many photons but in the totality of all possible correlation functions with S black hole points. So, you don't see it directly from that theorem, but you can convince yourself that if you look at all possible products of operators in this Hilbert space where the number of operators case like the entropy, you can find a complete basis of operators. So, in particular, any projector you want to construct, you can make it out of those guys. So, in particular, you can make the projector on the microstate size zero and then you can check whether you get one as the expectation value or not and you can repeat the same process for all possible microstates and then you can finally, after a lot of calculations, you can identify the microstate. But you cannot do the same thing with two point functions. That's what I was saying. Okay, so let me also mention one more result which is of the similar flavor. So, it's a result in quantum statistical mechanics. So, this result here told us how different the expectation values are in a pure state versus a mixed state. And another, so now we want to assume that the Hawking radiation is in a pure state and we want to understand the pattern of entanglement between the different Hawking particles. So, we have many Hawking particles after the black hole has evaporated and they are in a pure state. If you take all of them at once, they're in a pure state. However, is one of those, if you look at it individually, it seems to be in a mixed state. It seems to be thermal. That's perfectly consistent, right? That's what we try to argue there, that it could be that expectation values of observables are very close to the maximally mixed ones even though the state is pure. Now, what Pays analyzed was the following question. Suppose we have a system with many degrees of freedom and we divide it in two parts. So, here we have plot of this big system and we have selected the subsystem A, B is defined as a complement. So, we have a large system A times B and we imagine that the whole thing is in a typical pure state psi. So, typical is precisely what I already defined. So, we define the notion of typicality by using the hard measure. So, what Pays did is he considered this setup and then he asked the following question. Suppose we assume that the full system is in a pure state and we want to calculate to consider the reduced density matrix of the small system. And it's an easy exercise to do using these techniques and what you found is that if this system is small enough, then this density matrix is very close to the maximally mixed one and if you calculate in particular the entanglement entropy of the subsystem on a typical pure state, you find that it's proportional to the logarithm of a dimensionality of a system. So, this is telling you basically that if you take a big system and you divide two parts and you start making this guy bigger and bigger, you find that the entanglement entropy of the subsystem starts to increase with the size of the subsystem. However, this calculation that Pays did is reliable only in the limits where A is small compared to B. When the size of the two systems starts to become comparable, then certain corrections that were ignored in this calculation start to become important and in particular you can see that, well, I don't know if it does, you already talked about it, but if you have a bipartite system in a pure state, the entanglement entropy of A is equal to entanglement entropy of B, right? So, if we start making this guy A bigger and bigger and bigger and at some point, if A becomes larger than B, then B will start to play the role of the small system, so the behavior of the entanglement entropy has to be sort of symmetric around the midpoint and then what Pays conjectured is that if you plot the entanglement entropy of the subsystem A as a function of the size, you get this curve which goes up to the midpoint and then it starts to go down again. So, this is called the Pays curve and then if we apply this idea to the black hole, what we expect is that as we start collecting the Hawking particles in the beginning, the entanglement entropy will keep increasing, whether the process is unitary or not, then if the evaporation is unitary, we expect that the entanglement entropy of the Hawking particles will start to decrease and eventually it will go to zero when the black hole has completely evaporated contrary to what Hawking computed where the entanglement entropy keeps increasing forever. So, this point in the middle of the evaporation is called the Pays time. So, this is a point where the black hole has emitted half of the particles that it's going to emit, it has lost half of its entropy and we can see from this diagram that in the beginning, the Hawking results and the exact results seem to coincide which means that the black hole does not emit any information, the state of the Hawking particles is almost the same as the thermal one and only after the midpoint you start to see a difference. So, you start to see that the entanglement entropy of the Hawking radiation starts to go down which is a signal that the state is actually pure. So, in that sense, the information of the black hole starts to come out only after the midpoint of the evaporation. So, sometimes we will also call this black holes young black holes and these old black holes, those that are after Pays time. You can also check the following thing. If you have a big system and you divide two parts, let's say you divide it in these two parts and you take, let's say, one of these qubits which is in neither of the two parts, you take this guy and if you take the system being a typical pure state, then this system will be mostly entangled with a larger of the two Hilbert spaces. This is what follows based on similar arguments where you integrate over all possible pure states to calculate the average entanglement and the mutual information and so on and so forth. So, what this means is that during the black hole evaporation, at the early stages, we have two Hilbert spaces. We have the Hilbert space of the black hole and we have the Hilbert space of the radiation which was emitted in the past. So, in the beginning, this Hilbert space is larger than that one because there are a few particles here. This is a big black hole. So, any new Hawking particle that is emitted based on this analysis, this new particle is going to be mostly entangled with a black hole because it represents the larger part of the Hilbert space. So, before phase time, any new Hawking particle which is emitted is mostly entangled with the remaining black hole. However, after phase time, where the black hole has lost half of its entropy, the size of the Hilbert space of the radiation is larger. So, any new Hawking particle is going to be more entangled with the earlier radiation. So, there's a flip in the entanglement of the Hawking particles depending on whether you are before or after the phase time. I think I don't have the time to explain this, maybe during the discussions I will talk about it. All right, so we have this scenario of uniterizing the black hole by the small corrections, but there are some additional problems that we have to deal with. And a very simple way to phrase these problems is in terms of this paradox which is called quantum cloning of the nice slices. So, here we assume that the previous scenario of uniterization by small corrections is true, and we want to see whether it leads to any other inconsistencies. In particular, there's this very simple thought experiment which seems to lead to a contradiction. So, we imagine that the black hole evaporation is unitary and that, for example, if you throw something to the black hole, then the information on this qubit is going to be encoded somewhere in the Hawking radiation. Now, the point is that on this geometry of the black hole, it is possible to select a set of space-like slices which are called nice slices which have the property that they intersect at the same time the qubits that you threw in at early times in the black hole and also most of the Hawking radiation which is emitted in the future. The way this is possible is by taking the slice and sort of bending it, boosting it outside the horizon, making it move parallel to the horizon while making sure that the slice is always space-like. And you can verify, you can write it down explicitly in some limits. You can verify that this is indeed a good space-like slice which can extend very far in the future and capture most of the Hawking radiation. Moreover, the reason it's called the nice slice is you can check that the curvature on this slice, both the intrinsic and the extrinsic curvature is low everywhere, and all particles on this slice, like the star, the qubit that you sent in and the Hawking radiation, are moving with velocities which are not too high. So you would naively expect that effective theory should be reliable on this slice. And what I mean by that is you can foliate your space-time by a set of these slices. So you have this Penrose diagram and you can take a foliation which is this slice together with the next one, the next one, and so on and so forth. And you can make sure that these slices do not approach the singularity, but they still stretch out and capture more and more of the Hawking radiation. So it is expected that, according to effective theory, that we should be able to do a calculation of the slices by starting with some initial state and then pushing it forward using Hamiltonian. However, here's the problem. The problem is that if you send the qubit into the black hole, it is captured on this slice at this point in the interior of the black hole. But also at the same time, it is captured in the outgoing radiation if we assume that radiation is unitary. And this is something which is problematic because it leads to duplication of quantum information and this is not allowed by the No-Cloning Theorem of Quantum Mechanics. So let me remind you that Linearity of Quantum Mechanics does not allow us to have an evolution of a system where you take the state psi and you evolve it into a psi, path of psi. So it is impossible to come up with a physical system or a device that takes us and inputs a state psi and reproduces two copies of the state regardless of the choice of psi. So that's not possible. And the way to see that is very simple. Suppose we could do this. Then we could start with a state A and get the state A A. The state B would give us B times B. Now, if you apply Linearity of Quantum Mechanics, the fact that time evolution in Quantum Mechanics is linear, you can use a superposition principle, and then that would predict that the state A plus B must be equal to this state plus that state. So Linearity would predict that A plus B goes to A A plus B B. On the other hand, the rule that we came up with would imply that A plus B has to go to A plus B, tensor product A plus B. And these two states are not the same because of the cross terms. So this is what you get from Linearity of Quantum Mechanics. This is what you get by imposing the conditions that you have some device which can clone any state. So this type of cloning is not allowed in Quantum Mechanics. It violates linearity. And this geometry seems to be able to do cloning, right? Because you send this particle and it produces a second copy on the outgoing radiation. Now, this paradox highlights the importance of the black hole interior in formulating the information paradox. If you have a star and you throw something into a star, you do not run into a paradox of this type because there is no smooth and empty interior where you can sort of stretch your slice and make it go in and have the information at two different places at the same time. So this type of paradox arises precisely because the black hole has a smooth interior. And as we will explain later, it makes it clear that the information paradox is difficult to resolve if you insist on preserving the smoothness of the interior of the black hole. Yeah, are you asking whether it is an exact copy? Well, no, I'm saying that if evaporation is unitary, then it means that whatever information you send to the black hole has to be contained somewhere in the Hawker radiation, right? No, no, no, not necessarily. So this is not the precise paradox that can be a statement that can be applied here. This is a much more complicated system. Here I only wanted to mention to you what is the no cloning theorem of Quantum Mechanics. But if you work it out, you run into similar paradoxes if you simply assume that this quantum information is encoded somewhere in this Hawker radiation even if it is in scrambled form. There's nothing, well, the black hole evaporates, right? So there's nothing left behind. So everything that, I mean, if evaporation is unitary, then anything that falls in has to get out. Okay, so the usual, so an old idea of how to resolve this paradox goes under the name of black hole complementarity, which was developed by these authors. And it is the idea that this paradox can be resolved by assuming that in quantum gravity, it is not possible to factorize the Hilbert space into the interior and the exterior, into different factors. So why is that the resolution? Well, we run into this paradox by assuming that we start with some Hilbert space and then we produce a copy of this qubit on two different Hilbert spaces, which is a direct product. Now, if this is not the case, then the paradox is not there anymore. And the way that this was motivated in the beginning was that, okay, maybe you have these two copies of the quantum information, but there is no observer who can see both of them. Because the guy who falls in and detects this particle is no longer able to get out and verify that there's a second copy. And also the guy who extracts the information from the Hawking radiation is no longer able to go inside and capture this qubit. So the initial idea was that there was no single observer that can detect both copies. So even if there's some sort of cloning, nobody can see it, so maybe it's not longer a paradox. But the more precise statement, which I already mentioned, is that contrary to what Effect Field Theory suggests, so Effect Field Theory suggests that this region is space-like to that region, so we should be able to factorize a Hilbert space. Then the idea of complementarity is that in quantum gravity, this factorization is no longer correct. And in fact, the interior Hilbert space should be thought of as being somehow encoded in a scrambled form in the exterior Hilbert space. Now, this would resolve this cloning paradox, but there are two issues. First of all, it's not very precise mathematically, so we would like to have some specific mathematical model where this idea is realized. And second, if you start identifying the Hilbert space of the interior with the Hilbert space of the exterior, it's not clear a priori whether this is consistent with Effect Field Theory. Because if you identify these Hilbert spaces, if there's overlap between these Hilbert spaces, what it means is that operators here will not commute with operators in the exterior. And in particular, you could try to use this non-violencing commutator to send information from the interior of the exterior or vice versa, superluminal information, and that might lead to violations of Effect Field Theory. So a priori is not clear whether this idea can make sense. So we will come back to these points during my last lecture, but here I want to mention this old proposal of how to resolve this quantum cloning paradox. Yeah, the last part? Yeah, so here the proposal is that the, I mean, this is a space-like slice and this segment here is a space-like relative to that one. So in quantum field theory, naively, you would say, I can factorize the Hilbert space into this part, tensor, the other part. The proposal of black hole complementarity is that in quantum gravity, this is not possible. So fundamentally, the Hilbert space does not factorize into these two regions, even though it looks like it does from the point of view of low energy Effect Field Theory. But what I said later was that two problems with this idea is that, first of all, it's not precise, right? These are just words that I'm saying now. So it would be nice if we could find a concrete mathematical model where this is realized. And I will try to do it during the last lecture. And second, if you start making this identification of the Hilbert space of regions which are very far away, there is a danger that you will violate Effect Field Theory in a dramatic way, right? For example, you would be able to communicate, two observers would be able to communicate superluminally like within Effect Field Theory that would not be acceptable. It would be a violation of locality at the level of Effect Field Theory which we don't want to have. So it's not trivial to show that this kind of idea can be implemented in a consistent way with Quantum Field Theory. Okay, anyway, so this was the old story. This we can discuss maybe during the discussion session if we have time. But what was realized recently is that there's one more problem with this idea of complementarity which has to do with this properties of entanglement that I discussed before. So remember, I have made in my lectures two contradictory statements. One statement that I made in the beginning was that during the Hawking radiation, these particles are produced in pairs which are highly entangled. That's what follows from the calculation of Hawking. So according to the calculation of Hawking, all these particles which are produced are always entangled, so B is always entangled with C, right? But a little bit earlier, I made a small drawing here where I said that after paced time, the newly formed Hawking particles are entangled with the earlier radiation which represent the larger part of the Hilbert space. So I have made two statements about this particle B. I made a statement that it is entangled with C and I also made a statement that if you are after paced time, it is entangled with the earlier radiation. But that is not okay because it leads to a violation of monodome of entanglement in quantum mechanics. You cannot have a single quantum system simultaneously entangled with two other systems. And this can be made more precise by writing down this inequality. They're strong about the derivative of entanglement entropy, which is saying that if you have three systems, A, B, and C, and if you calculate the reduced tacy matrices, A, B, C, and the for no human entropies, they have to obey this inequality. It is a theorem in quantum mechanics. And if we apply this theorem to this setup for a black hole which is after the paced time, so it's an old black hole, we are looking at this situation where we're taking A to be the early radiation and A, B means A together with B. It means it's the early radiation together with one more Hawking particle, right? Now, since we are in this part of the graph where the entanglement entropy with the Hawking radiation goes down, if we want to preserve unitarity, then it must be that S of A, B is less than S of A because the curve is going down, right? Otherwise we would be still in the Hawking regime, right? So now the curve is going down. So S, A, B must be greater than SA. So after paced time, we need that S, A, B is less than SA. We need, however, this inequality is telling us something else. It's telling us that, well, SA, B is greater than plus this guy, SA plus SA. Now, SC is the entropy of this particle. According to Hawking, this is maximum entanglement B, so these guys together are in a pure state according to Hawking. These are in a pure state, so S, B, C is zero, but SC is non-zero, it's positive. So this inequality is telling you that S, A, B plus zero is greater than SA plus a positive number. So it's telling you that S, A, B is greater than SA which contradicts what we need to restore unitarity. Well, this is a more precise statement of what I already said before that it cannot possibly be that this B guy is at the same time entangled with these two other systems. Now, you can ask, well, what about all these small corrections that you were talking about before? I mean, this statement that these guys are entangled is what follows from the calculation of Hawking, but we already said that we're going to consider corrections to the calculation. Well, Mathur looked at this problem, so he looked at what happens if you relax this condition a little bit by allowing these two guys to be not fully entangled but a little bit away from maximum entanglement. And what he showed is that even if he introduced small corrections, this problem, this paradox cannot be resolved. Well, according to Hawking, so let's say, according to Hawking, B and C are together in a pure state. So SBC is equal to zero, right? But SB is not equal to zero, right? Because this particle is a thermal particle and by symmetry, SC is not zero. So according to Hawking, SBC is zero and SC is positive, right? Well, if you look at this inequality, it predicts then that S of AB must be greater than S of A. But this is in contradiction to what you need in order for the Hawking radiation to become pure after space time. If SAB was greater than SA, that would imply that as you take more and more Hawking particles and you add them to the Hawking radiation, the entanglement entropy of the radiation will keep increasing forever. So then this curve would go up like what we saw before in the Hawking calculation. Well, this is what we need in order to restore unitarity. Yes? No, it would be even worse, right? Yeah, yeah, yeah. Okay, so this is this paradox which shows that, again, the difficulty in resolving the paradox becomes clear when you try to reconcile unitary evaporation with a horizon, which is the same as the one Hawking studied. And again, you could say, all right, maybe there is some violation of this subactivity inequality, but perhaps there's no observer who can detect it. But these authors also try to work it out carefully and see that actually you can find an orbit of a time-like observer who can go through this cloud of radiation, the early radiation and fall into the black hole and intersect B and C. So there's actually a single observer who can detect all these three systems and observe the violation of inequality. So in this paradox, you cannot say that maybe there's a violation, but there's no observer who can see it because there is an observer who can in principle see it. Okay, finally, there is one more problem which is more intuitive maybe, but it will be more precise in ADSFT. When you look at the calculation of Hawking between these two particles, you find that they have to be highly entangled, but not only that, they have to be entangled in a very precise state, very specific entangled state. So what I mean by that is that when you write down an entangled state, there are some coefficients that you write down. So you write, for instance, if you write an EPR pair, you write zero zero plus one one over square root of two, this is an entangled state. But if you put a phase here, e to the i theta, that is a different entangled state with the same amount of entanglement entropy. So all these states with different phases are different pure states. And if you want to look at the calculation of Hawking, it predicts a very specific phase for all of these coefficients. So this seems to be very peculiar from the point of view of statistical mechanics because if you take a system and you divide two parts and you look at a typical state, then you can check that the pattern of entanglement between the two subsystems has some sort of random behavior regarding these phases, for instance. And so it's very difficult to imagine how dynamically typical states would end up having the correct entanglement needed for smoothness of the horizon. This can be made more precise in ADSFT where now I'm jumping a little bit ahead, but I will explain that more tomorrow. There is a very standard example in the ADSFT of the eternal black hole. We have two CFTs which are non-interacting, but they're in specific entangled state, the thermo-free state. And the smoothness of the horizon depends on having the correct entanglement between the two sides. But if you consider more general typical states, not the so-called thermo-free states, but more general states, you find that these operators do not have the correct entanglement to describe this geometry. And I will return to this slide tomorrow. So please don't worry if it's not very clear. And also Shankar and Stanford studied how small perturbations can destabilize entanglement due to quantum chaotic effects. So if you send a small particle at early times, it totally changes the nature of entanglement between these particles. So it's very difficult to reconcile this specific entanglement that we need according to the calculation of Hawking here with the statistical properties of the system. Now, so the paradox came because we had these two particles that had to be entangled, and also they had to be entangled, one of them had to be entangled with early radiation. So if we want to preserve unitarity, we have to respect the entanglement between the new Hawking particles and the early radiation. So what if we give up on the entanglement between the Hawking particles and the interior modes, the guys that we called C before? Then you can check that this will create a lot of problems on the horizon, and the easiest way to convince yourself that that is the case is to consider Minkowski space, in the Minkowski, so quantum field theory in the Minkowski vacuum, and write the quantum field in the Riedler coordinates. We can maybe do it later during the discussion, and there you can see that, so you divide flat space into the right Riedler wedge, which is relevant for an accelerated observer and the left wedge, and here you have this Riedler horizons, and we know that these horizons are smooth because it's just Minkowski space, after all, but then you can look at the quantum field in the ground state, and you discover that this region is highly entangled with that region, so in particular, well, we have drawn these wave packets which are very similar to the ones we had inside the black hole. So an observer falling through this Riedler horizon experiences a free infall precisely because these two guys are entangled in the correct quantum state. So if you modify this entanglement either by breaking the entanglement between the two sides completely, or if you modify the details of the entanglement as I was trying to explain there, you generate some energy density on the horizon that you can explicitly calculate because you can do it for a free field in four dimensions, for example, and you can calculate the stress tensor, and you can see what different modifications to the entanglement, what effect they have of the stress tensor that you would detect when crossing the horizon, and well, then in the case of the black hole, this would imply that if we give up this entanglement with the interior partner, it would generate a very large stress tensor on the horizon which would back react and modify the geometry in a dramatic way. Now, the problem here is that this paradox and this possibility becomes relevant after space time, right? It becomes relevant when the information starts to come out of the black hole. And if we start off with a black hole of large enough mass, then we can make sure that even at space time, the black hole is quite big. So this predicts the modifications of the nature of space time on the horizon even for black holes which are macroscopic, which have a macroscopic large horizon where classically, the curvature would be very low. So this predicts modifications of GR in the regime of low curvatures, which would be very dramatic, right? Okay, I think I need to stop, right? So let me summarize. So I try to argue today that the information paradox from the point of view of an asymptotic observer has a natural resolution which is consistent with genetic expectations of quantum statistical mechanics, namely that exponentially small corrections to the observables that Hawking predicts can unitarize the radiation. I also try to explain that if you want to preserve the smooth of the horizon, you run into more challenging problems because it's very fine tuned entanglement between the interior and the exterior and this seems to contradicts genetic expectations from quantum statistical mechanics. Now, my lectures so far have not been very precise, but from tomorrow on we will address all of these questions in the framework of ADCFT where as you will see, they will become significantly more precise because we will translate all of the statements into questions about the conformal field theory. So they will become mathematically more precise. Thank you.