 Okay, we can start. So it's a pleasure to have Irosha Guri from Caltech and the APMU, and he's going to talk about entanglement and geometry. So please. Thank you. Can you hear me? Okay, good. Thank you. Good to have a response. So it's very, I'm very happy to have this opportunity to give my set of lectures here. I have given many lectures at this spring school and topological string theory and supersymmetric queue theory. But this time the organizers asked me to talk about entanglement and geometry. And the study of ADS-CFT correspondence in the last several years revealed that the quantum state that described a gravitational system seems to have very interesting entanglement properties and exploiting that we have gained quite a bit of useful information about quantum gravity in general. And then I would like to review some of these development in the past few years. I think I have one hour for my first lecture which means until 1240 or something like that, right? One hour. Yes, thank you, great. So today I will basically talk about the very elementary thing about information theory, entropy, mutual information, et cetera, so that we know what we are talking about in the rest of the lecture. And then tomorrow I plan to talk about primer of ADS-CFT and then introduce notion of holographic entanglement entropy. And then we'll go on to talk about the entanglementary construction, symmetry in quantum theory of gravity, et cetera. So let me start with a concept of entropy which I believe you are all familiar with. In start mech, we learn that entropy S is given by log of the number of microstate in the micro canonical ensemble. I'm setting Boltzmann constant K to be equal to one. On the other hand, in information theory, in the information theory, so suppose for example there is an event A that happens with probability P. Can you read my handwriting on this side? Okay, good. So suppose that there is some event A that happens with probability P, okay? And then you don't know whether it happens or not but if you learn that it happens, like for example, at the presidential election last year, we didn't know which one is going to win and then on the day after, we learned, oh, this guy won. And then that's some information, right? So then you gain some information. So there is a way to quantify that information which is minus log P. So this is a notion of entropy in the information theory. So for example, if P is one, you're sure that it's gonna happen and it happens, you are not gaining any information. So then the amount of information is zero. So I hope that this lecture is going to be actually P less than one, but we'll find out. But on the other hand, if a very rare event happens and you are surprised by that event happening, I have had that such experience last year, then this is going to be very large, right? So if each microstate is equally probable and if the amount of information, so in that case, if all the event, suppose we have omega events and if all the event happens with equal probability, P is equal to one over omega, in this case, this minus log P is equal to log omega. So these two notions coincide. There are various good reasons why we are considering logarithm. One of the primary reason is the following. So suppose you have two events, A and B. And suppose these are independent and suppose these probabilities are given by P of A and P of B, these are the probabilities that these two events, A and B, happens. And if these two events happen independently, then the joint probability is the product. So if I take log of this, then it's a sum. But what this means is that information is additive, that if you learn that both A and B happen simultaneously and they are independent, the amount of information you gain is additive. And in fact, you can prove that the log is the only function which has this property. So this is why we use logarithm. Now, suppose you have n events that are mutually exclusive with probability P1, P2, Pn in such a way that they sum up to be one, okay? So then you can ask, well, we are going to do some experiment to find out which one of these n things happens. But we don't know yet. So what is the expected amount of information you gain by doing that? Well, the amount of information that you gain by seeing that event I happen is minus P, minus log P. So that means that the expectation value would be sum of all of these, right? So this is denoted with S and it's called Shannon entropy. So this is the expected amount of information you gain by doing this experiment and try to find out which one of n events happens. There are a couple of important property that this Shannon entropy has. So this is going to be exercise for you. Both are very simple and algebraic. The first is to show that S is, of course, non-negative and the question is when S is going to be zero. So that's the first question. The second question is that S is maximum. If all the events happens, and if and only if, in fact, all the events happens with equal probability, namely that P i is one over n for all i. Okay, so these are simple algebraic exercise that I ask you to try doing that. Now, one of the notions that come up very often in the rest of the lecture is a notion of mutual information. So I would like to discuss that. So first I'd like to introduce a notion of a conditional entropy. Okay, so suppose we have, here I considered one set of events, so you do one experiment and to find out whether one of n things happens. So suppose you have two sets of events. So you have a one to a n, b one to b m. Suppose you have two sets of events happening and then they are not necessarily independent, so they can be correlated, for example, and so there is a joint probability which I denote P of a i, b j. So that is a probability that event a i and b j happens simultaneously. So then we have this conditional probability that supposing that you know that a i happens. What is the probability that b j happens? Well, that would be given by, so this is something you probably learned in the high school that this should be given by this kind of ratio. And similarly, this is given by this type of ratio. Okay, so this leads to the definition of conditional entropy, which is the following. That suppose you know that event a i happened. Suppose you know that event a i happens. What is the amount of information you gain by knowing that b j happened? Well, that would be given by just substituting this into that formula there. So that means that the S of a i b j is given by sum of all j's, a i b j, log of P of a i b j, right? So namely that, so this is a probability of b j happening subject to, you already know that a i happened. So therefore this is the amount of information you gain by knowing that b j happened under that circumstance. But suppose you didn't know, you haven't done that measurement to measure a i either, but you know that you are going to do the measurement. And then you want to find out how much uncertainty you have about b j. That you can define by averaging this over all possible a i's. So that means that you sum over all a i's with this probability. And then it's a simple algebra to show that this is given by minus sum of i j P of a i b j, log of P of a i b j plus sum of i P a i log of P of a i, okay? So this is called conditional entropy. So this is conditional entropy. That is that entropy for doing the discovery, entropy about information for b j's subject to the condition that you have already done the experiment a i. Ah, excuse me. So I actually put the wrong. So this is the information, I apologize. So here I have already summed over j's. So this is the amount of information you gain about b when you already know that a i happened. This is the amount of information you gain about b after you did this experiment a i but you don't know the outcome of it yet. So that's why I average over a i's. Okay, so this leads me to define mutual information. Is there any questions so far? Is this clear? What I'm talking about? So this leads me to this notion of mutual information. So this is a following notion that suppose you want to know about b, what you want to know, what is this b? So for example, who is going to win the election or something like that? But then you are curious, well I don't know what, it's very hard to know what b was, what kind of b happens. But suppose I find out about a, how much can I learn about b? So the general uncertainty about b is just s of b. Yes, let me see. Ah okay, so you're saying that this can be b. Okay, I might have made a, it's very hard for me to double check in real time so can I collect it later? Okay, so this is an uncertainty I have about s of b. But now suppose you know you are going to measure a and then you ask how much uncertainty you have about b. So that would be s of a of b. So that means that this is the amount of uncertainty you can reduce by doing the experiment a, right? So namely, this is a measure of how useful it is to do the measurement a when you want to find out about b. So this is a mutual information. Interestingly, this mutual information is symmetric in a and b. That is that if I write this out, then I can express that in terms of Shannon entropy. So one of the important properties is symmetric and b. Namely, so this mutual information tells you that how much you learn about b by learning about a. But it's mutual, that is that this is the same amount of information you learn about a if you measure b. So the information is mutual. So this has been rather abstract. So let me give you some example to think about this concept. So suppose you are going to school from your apartment, but then you forget when you woke up. Where is my key? Where is the key to the apartment? Okay. So you discover that there is a 50% probability that is in your pocket. Okay. But then there is another 15% probability that is in one of 16 drawers in your room. This happened to me very often. Actually, it happened more often than I would like. Okay. So what is the uncertainty of the location of the key? Okay. So key location uncertainty. Well, so there is a one half probability that is in my pocket. So that's this one. And then there are 16 possibilities. There are 16 possibilities that is one of the drawers. The probability that is in each of the drawers, I assume that they happen in the equal probability is 32, one over 32. So it's one over 32. And just make the calculation easy. Let's compute a log with base two. You know that different base just give you a different normalization of the entropy. So here I'm using base two. So then this is three. So when you wake up and then you are in this situation that you don't know where the key is, the uncertainty you have is three. Okay. So that's how you quantify your uncertainty. Now I'm going to present you some paradox. The paradox is as follows. Suppose you actually check because you have to go out and then check your pocket first and then you discover that it's not in it. Okay. So what's your uncertainty now? So you check the pocket and then you discover that it's not in my pocket. And then what is the uncertainty about the key location? Okay. So we can use this conditional entropy to evaluate this. So A is not in my pocket and the key location, right? And so if you evaluate it, so it's 16. Now there are 16 possibilities. One over 16 log two of one over 16 is actually four. So it's greater than three. So why is it that you actually, so this means that you lost information. You did something and you lost information. So that doesn't seem to be fair. Uncertainty has been increased by knowing that key is not in my pocket. So what's going on? Actually, this is not the correct calculation to do. The correct calculation is the conditional entropy that you obtain by checking your pocket but you don't know your outcome. Okay, so check the pocket, check my pocket and then uncertainty of key location. So let's calculate that. Well, so we can use that formula. So that probability is one half for four. So let me just mention the calculation. So there is a probability one half that it's in my pocket. But if it is in my pocket, now you know where the key is. So uncertainty is zero. And then there is another probability that there is another half probability that it's not in my pocket, which is in that case, it's in one of these drawers. I have already calculated that to be four. So this is four. So that means that the conditional entropy for the key location is two or good. So it's less than three. So this is what's going to happen to you in the morning. As far as the key location uncertainty is concerned. So you wake up and you don't know where your key is. You figure that there is a 50% probability that it's in your pocket. There is a 50% probability that it's one of the 16 drawers. Your uncertainty is three. Now you are going to decide that, well let's first check my pocket because there is a 50% probability. You are going to do that because you know that the uncertainty will be reduced to two. So you do that. That's why you do it, right? But then you discover that it's not in my pocket and you really get upset because uncertainty is now increased to four. Okay, so it actually agrees with your intuition about what uncertainty should be. Okay, so far I considered classical information theory that I was dealing with information that are classical that have already always have fixed value. But of course I'm going to talk about, since I'm going to talk about quantum gravity. So I should talk about the quantum information. So that can be measured by von Neumann entropy. Okay, so suppose you have a Hilbert space and suppose you have an ensemble which is a set of states. And I say that each one of these states happens with probability P i. So then let's take this to be also normal. So you have ensemble of also normal states each happening with probability P. So this can be described in terms of density matrix like that. The von Neumann entropy is defined, the von Neumann entropy for this density of state is defined as log of a trace, so excuse me, what am I writing? I just came to Japan last midnight, arrived here so I have an excuse. It's trace of minus log of a log of a log of a log of a log of a actually I just learned this morning that Nima Al-Khan-Hamed gave a lecture last week discussing inequality property of this quantity. But I briefly saw the beginning part of the video and so what I'm going to do is actually give you explanation of why this is a good quantity to measure. Okay, so in particular if you know that the row already have this basis like that, then it's easy to show that this is actually equal to sum over i, pi, log of pi. So in this basis, so when you know that density matrix can be presented in this also normal basis, then it agrees with Shannon entropy. Now let me talk about, so here I talked about joint probability and conditional entropy, etc. So we can repeat the discussion quantumly. So we can consider a notion of joint entropy, which is to say that suppose we have a density matrix, rho of AB acting on the tensor product of two Hilbert spaces, HA and HP, then I can define the notion of joint entropy S of AB simply as minus trace of rho AB log of rho AB. So this will be a quantum analog of this type of quantity here. So this is a joint entropy of A and B, classically, and this is a quantum counterpart of this. And again, when rho can be also diagonalizing also normal basis, these two agrees, okay? Now this joint entropy has various important inequalities. Nima, I saw, gave a very nice surprisingly creative proof of various inequalities, so I'm just going to state those things. So the one important notion is sub-auditivity, which is to say that S A plus S B minus S AB is positive, but you see that this is actually nothing but the mutual information. So the mutual information being positive is one of the property of the von Neumann entropy. And so that's a good thing to know because that means that when you do something, you always gain some information. You never lose information by measuring other things, right? Because it's zero. When they are independent, two events are independent, the mutual information, of course, is zero. So that means that learning about B does not reduce the uncertainty for A, but when they are correlated, when you measure B, you always gain some information about A. That's what it means, that's what sub-auditivity means. Another useful inequality is the concavity of entropy. So the concavity is the following notion that if I consider entropy for some linear combination of density matrices, can you actually distinguish my P and rho? Should I write them more distinguishably? Is this okay? Then the concavity means that it's actually always greater than or equal to this thing. You can take PI out. And then there is another side of the inequality, which says this, where this guy here is a classical Shannon entropy, which is minus log PI log PI. So the concavity means that this entropy for the linear combination superposition of density matrices is bounded below this and bounded above by this. This also has some nice interpretation. It actually is useful in many ways, but one place where it appears is the notion called horrible information. So horrible information is defined in the following way. And this shows up in many circumstances in information theory. So for example, suppose somebody send you some mixed state of this kind of density matrices and you are supposed to do some measurement to figure out which one of these raw eyes was being sent. And horrible information gives you an upper bound of your ability to figure out which one that was sent to you. And it's actually have a very nice interpretation in many circumstances, including holographically. And I might, if I have time, I would like to discuss those. But anyway, so the concavity in this case means that this horrible information is bounded both from below and above. So those are two inequalities. And there is a third inequality that I would like to also mention, which is a strong subadjectivity. And I believe this is what Nima talked about in his lecture, which is S of A, B plus S of B, C is greater than S of B plus S of A, B, C. And this is slightly more difficult to prove. We'll discuss holographic proof of this inequality maybe tomorrow. But for now, let me mention that if I use the notion of mutual information that I erased, oh, no, I have not. So if we use this mutual information, then I can actually write this strong subadjectivity in an equivalent form as S of A and I of A, mutual information between A and B, C, is greater than I of A, B. And I leave this to your exercise to check this. This is a very simple algebraic. You just substitute the definition of this and then check it. But this gives you a very natural information. What it means is that the more you learn, the less uncertainty we are, which is encapsulated in this inequality. So this is the amount of information you know about A by knowing about B. This is the information you know about A by knowing B and C. So that means that more you learn about subjects that is not directly related to A, nevertheless you learn more about A also. So this sounds like a profound life lesson, but this is stated in the form of strong subadjectivity. So now let me come to the main sort of concept of this set of lecture, which is an entanglement. And one of the reasons that entropy is useful is that it gives you a way to quantify the amount of entanglement that a given state can have. So let me first start with a very simple example. So suppose you have two Hilbert spaces, both two-dimensional. So suppose you have HA, which consists of two states, zero and one. And another Hilbert space, which also consists of two states, zero and one. So then you consider tensor product of these two Hilbert spaces, and you consider various states. So for example, you can consider a direct product state. In such a state, we consider really regard as having no entanglement. On the other hand, you can have a state which is sort of deeply entangled, is of this form, for example. And this is actually a particular state that Einstein, Podovsky, and Rosen considered in their famous 1935 paper, where they exhibited the surprising property of entangled state. The term entanglement itself was coined by Schrodinger, I understand, after their paper. But for this reason, let me call this as EPR state. So this is entangled state, and this is a state which is not entangled. So I would like to use entropy to quantify the amount of entanglement. So let me define entanglement entropy. So suppose you have a state in a tensor product of a Hilbert space, and you want to find out how much entanglement between A and B, this particular state, is coding. So for that, we first evaluate what is called partial trace of this state, which is, you start with this pure state density matrix, and you take trace over Hilbert space of B. Then you obtain some state which is acting on the Hilbert space of A, and then you compute the entanglement, sorry, von Neumann entropy for this density matrix in the Hilbert space, of course, of A. So this is the definition of entanglement entropy. So as an exercise, let's evaluate this for these two states that I introduced. So again, to simplify our calculation, let's do that with base two. So if I do that, and if I substitute this and this into this definition and the work out, which I encourage you to do by yourself, sorry, yeah, I keep doing this. Maybe I should do the global replacement, but then this is going to be changed, too. So that will be a problem. Thank you for, please point out any errors. I appreciate Mukund and telling us, Mukund and Atish are collecting, and I encourage you to collect, and maybe I'll give prize for people who collect my mistake. Anyway, so suppose I evaluate this for this pure state which is just a tensor product, then you can actually check that this quantity is zero, whereas if this is EPR, then this is one. And in fact, this is a maximum amount that you can have for this particular entanglement entropy. So in that sense, EPR pair is actually maximally entangled. In fact, in some sense, entanglement entropy defined in this way is measuring how much EPR pair you have between these two Hilbert spaces. And it is because you can actually prove the following theorem. I'm not going to prove it, but you can actually find the proof in the literature, which is the following. So suppose you start with this state that I have. So suppose I pick any state in this tensor product. This can be any superposition of this state in this Hilbert space. But then I consider n tensor product of this. So I compute n or consider n copies of these Hilbert spaces, and then I consider n tensor product. And then what I'm going to do, what information cell is called, local operation and classical communication, which is the long word, sometimes called L O C C, which is an operation which you are permitted to do unitary transformation on A only or B only separately. But you don't do any quantum operation mixing A and B. So for example, if you can do arbitrary unitary transformation, certainly there is a unitary transformation on this total Hilbert space which maps this into here. You can find that unitary transformation because both are normalized to be one. So you can do that. That would destroy your entanglement. That would change the amount of entanglement. But suppose I do unitary transformation on A side only. Suppose for example, I just define this to be one and change this to be zero. The amount of entanglement is not changed. So if I do the unitary transformation on A side only, then it won't change the amount of information. So L O C C is sort of formalization of this kind of concept, the kind of operation you can do in this joint Hilbert space which won't change the entanglement property of this state. But then you can show that if you consider any tensor product of this, it's actually always you can always map that into tensor product of EPR pairs. And the amount of EPR pair you have is given by the entanglement entropy times n. Where this symbol, this mathematical symbol just means that it's an integer part. In general, entanglement entropy is some real number. So you multiply n and then you take the integer part of it. So this is true asymptotically, so for large n. So for large n asymptotically, this can be mapped into, so this roughly saying that the entanglement entropy with base two is, so this two means that I'm using logo base two is roughly the amount of EPR pair that this particular state can have. So this is sort of useful, sort of intuitive way to think about entanglement entropy. Now I should say that, so this was about pure state. And for mixed state, you can start, you can certainly start with a mixed state. So instead of this, you can also consider row A to be trace of HB of row AB for more general mixed state. And then one can define the entanglement entropy for this one. So this actually is not symmetric in A and B. Yes. So suppose I put minus here, so this is still one. So this one has the same entanglement entropy as the EPR pair. And so in fact you can relate all this by LOCC. So one would consider this as having the same amount of EPR pair as this or this. They all have the same amount, same number of EPR pair, modulo LOCC. Now coming back to the mixed state, so far I talked about pure state, but for mixed state you can still consider entanglement entropy, but that quantity is actually not symmetric in A and B. So may not be necessarily be a good measure for quantum correlation between A and B, because you hope that notion of quantum correlation would be symmetric in A and B. So in that case it's actually maybe more useful to consider mutual information. And this has added a benefit that if you consider quantum field theory, which I'm going to do next, then this quantity is actually ultraviolet finite. So it's more well defined than the entanglement entropy itself. So any questions so far? So I review the notion of shadow entropy, and then I discuss various inequalities. I introduce a notion of phononema entropy, which is a quantum counterpart of it, and discuss various inequalities. And they use the phononema entropy to introduce a notion of entanglement entropy as a measure of entanglement of state in the tensor product of Hilbert space, and explain that it's actually basically count number of EPR pair in it. Any question? So let me present to you one particular example of how entanglement entropy is calculated. So this is going to be an example that will be used in the next lecture. So this is the primer of information theory, and the next lecture I will discuss, I will cover some basic relevant aspects of ADS-CFT correspondence. So naturally we should consider entanglement entropy in the context of conformal field theory. The entanglement entropy in conformal field theory in two dimension has a very nice expression obtained by Karaburese and Cardi in the beautiful paper in FTH 0405152. So let me, in the remaining time, let me review their calculation. So it's a two-dimensional conformal field theory, so it's defined on the circle in the space-like section, and then you have a time in this direction. And then you consider some particular state. I'm going to actually consider a vacuum state, but I'm interested in entanglement between different spatial points of this conformal field theory. So what I'm going to do is to separate this space-like part into two parts. So you have this segment A, and it's complement A bar. So we have quantum field theory, so that means that we have some kind of local degrees of freedom on this space-like section distributed. So intuitively, you would think that the total Hilbert space is decomposed into tensor product of the Hilbert space associated to the degrees of freedom on the sub-region A, and it's complement. It's not quite precise because you have to actually specify how you regularize when you cut the Hilbert space into two parts. One way to think about it is that you first consider a lattice model where you discretize space and you put some degrees of freedom on each lattice. There it's quite clear how you separate things. Of course, even in the case of lattice, there is a subtlety if you have a gauge theory, the degrees of freedom are not on the lattice point but in the link. So then there is an issue of how to separate. So you have to specify the boundary condition here. But in fact, in all known examples, there is a way to separate the Hilbert space into two parts if you add some degrees of freedom in the ultraviolet. But the result is that the way that is separated depends on how you do this. So namely that there is some kind of ambiguity or some freedom to choose in introducing some ultraviolet degrees of freedom in order to do the separation. So that would change the amount of entanglement entropy by some amount which is additive. So for example, if you consider mutual information like this, then that subtlety is actually canceled. So that's another reason why mutual information is sort of more well-defined quantity than the entanglement entropy itself. And this has been discussed in various recent literature. For example, in a particular case of two-dimensional conformal field series, there is actually interesting paper by Tachikawa and Oomori, and this is the number. So if you are actually interested in this type of issues, then this might be one of the place to start. And then there are also some literature on how you do it in lattice gauge theory. They discussed that in the context of actually CFT2 that I'm going to talk about, and parameterized how much uncertainty you have when you try to separate the Hilbert space into two. So anyway, so let me assume that you can do this separation, and then start with the vacuum state of conformal field theory and define the density matrix by taking trace of a complement A bar of this space, and then calculate the entanglement entropy. The procedure that was employed by Karaburese and Cardi is first start with evaluating a quantity called the rainy entropy, which is defined as SN of A, which is 1 minus N log. So I'm going to from now on take the base to be E, and then trace of rho HA to the N. And you can easily see that if I take the limit of N goes to 1 of this rainy entropy, then this gives you the desired entanglement entropy that you wanted to evaluate. So density matrix is of course a positive definite operator. So you can take a logarithm of this, and this is sometimes called modular Hamiltonian. So knowing a rainy entropy is quite closely related to understanding the spectrum of this modular Hamiltonian. Now this quantity can be calculated in conformal field theory, because this is related to computation of a correlation function in fact of operators that generate branch cut on the worksheet. The reason is the following. So here, so pictorically what we are doing is that we have the space like section here, and we separate the space into A bar, A, and it's complement. And we are taking trace in order to define rho A, what we are doing is taking a trace of a complement of the vacuum state. So we have this vacuum state coming here, coming here. I'm taking trace over here, but I keep this thing open. We have in state on this side and out state over here. That is this. So we have in state here, out state here, which is in state here, out state here. Now this quantity is obtained by taking n trace, n product of this. So that means that this is now glued to here and then glued here, glued here, etc. You do this n times. That's a rainy entropy. So if you have a vacuum state here and here, this can be evaluated with a sphere function on the sphere with cut over A. And if I take n product of this thing, what you are doing is taking product of this n times. So that means that you are taking n spheres and then cutting, introducing, adding n slice over A and then gluing them together. So this is the same as doing introducing branch cuts here, where it's n for the branch cut. If you go over n times, you come back to the same sphere. Okay? So such quantity can be defined in conformal field theory and then basically this amounts to calculating the correlation function of this operator, where I call this point u and v. So the distance between them is u and v. And so I actually only have five more minutes, so I don't have time to discuss more detail. But this quantity can be evaluated very easily by knowing how energy momentum tensor of this conformal field theory transforms under coordinate transformation. Suppose I do the coordinate transformation in two dimensions from z to w, then it is known that the energy momentum tensor of conformal field theory transforms like this, where this is Schwarzian derivative. And using that, we can actually evaluate this quantity, this correlation function as u minus v to 1 over 6 times n minus 1 over n, where u and v are locations of these branch cuts. And then from this, you can actually calculate this rainy entropy. And then if you take the limit, so this amounts to, so I'm sorry, this is actually this part. And then if I take log and multiply 1 over 1 minus n, then the rainy entropy is obtained as c over 6, 1 plus 1 over n times log of distance. So let's call this length of this to be small l. So this is small l, then you have log l plus some constant. And this constant is related to the ambiguity that you have in separating the Hilbert space into two parts, the one associated A and the one associated A bar. You can see that there is actually such, there should be such ambiguity because l is actually dimensionful quantity. On the other hand, we are considering conformal field theory, so the quantity that you can actually evaluate should be dimensionless. And in particular, it's funny to have dimensionful quantity in log because if you change the scale, you would change the value of log by shifting some constant and that is absorbed into this constant part. So from this, you can also see that somehow this constant part has to be related to the way that you regularize and separate the Hilbert space into two. So at any rate, so from this, you can actually show by taking the limit of n equal 1 that this is given by one-third of log of l plus constant. So actually, this formula is true when the size of l is much smaller than circumference, the radius, the circumference of the entire sphere, the entire circle. The more precise formula when the circumference of S1 is l is given by l sine i small l over capital L plus constant. So I had to run a little bit quickly to do the calculation. If you want to know more detail, I encourage you to consult the original paper where this is actually explained very clearly and nicely. But I just wanted to give you an appetite for getting into this reference. And also, I'm going to use this formula in the next lecture, so I wanted to have something to refer to. Okay, so today basically I covered the concept of entropy in information theory and introduced various inequality for both classical Shannon entropy and quantum for Neuma entropy and explained that the entanglement entropy defined in this way count the amount of EPR pairs and then discuss how you can actually evaluate this quantity in the case of conformal history. Okay, any question here? Yes, please. You said that the LOCC doesn't change the entanglement entropy. And on the left-hand side, it's a product state, so the entanglement entropy should be there. And how could the right-hand side be there? There is also this state belongs to tensor product of this Hilbert space and this Hilbert space. It is true that it's a product state for this Hilbert space, but there is an entanglement between here and here. So it depends on what you mean by product state. So, namely, suppose, for example, suppose you have a product state of HA and HB, but HA itself may be a product, tensor product state, and there can be entanglement within it. So when you say it's entanglement, you have to tell me entanglement between what? So this is a tensor product, so that means that it doesn't have entanglement between different components of this product, namely, HA times HB, so there are n copies of this, right? There are no entanglement between different copies because it's a product state. But there is an entanglement inside of each copy. So the whole state, so there are two n components in it, right? There are two n Hilbert space in it. There is an entanglement in some part of that Hilbert space, okay? But of course, it's a product state for this, so there are no entanglement between the copies. Any other question? Down there? Just trying to understand your construction of taking the spheres, letting it along the interval A and joining n copies of that together. Is the resulting thing an n-fold cover of the sphere branched at the two points, UV? Yes. Okay. That's what it is. Yes. So if you think this through, so you know that if you consider the vacuum amplitude with some object, it's the same as the sphere with that operator you started. And what I'm doing here is taking, so there is a subset, sub-region A, and then I'm taking trace on the complement. But I'm not taking trace over A, so that means that I'm opening up this so that you can have state coming in and coming out. And then I'm gluing this together here. And I think on the covering, the log becomes a single- So this is U and V, by the way, yes? On the covering, the log becomes a single-valued function? Yes. So in n-fold cover, it's a single-valued thing. So what you're supposed to do is to consider functional integral of this conformal fuselage on the n-fold covering of the sphere. And compare that with the partial function of the n-product of the sphere. And that is this quantity. Thank you. And that is actually shown very carefully in this paper. So if you want to know the detail, this is a good reference to look at. Okay? Yes? Can you be a bit more explicit about the assumptions that go into being able to separate the Hilbert spaces on spatial slices? Beg your pardon? When you separate your space in A and A bar, what are the assumptions that go into being able to also separate the total Hilbert space? So let me try to understand you. So you're talking about this separation of the Hilbert space? Yes. What is your question? What are the assumptions that go into being able to separate the Hilbert space? A would be, sorry? There are assumptions going into being able to separate the Hilbert space in H A and H A bar, right? This is not true in general? You're asking under what assumption, so if you can have, actually depend on the kind of UV regularization, you can have, for example, if you can define conformal field theory as an infrared limit of some lattice model with degrees of freedom residing on each lattice point, then you can separate the Hilbert space. But for example, there can be a case where you have degrees of freedom on the link, then in that case, you have to actually have prescription of how to separate the link. What they did in this particular paper, for example, is use open string field theory to define separation of conformal field theory. Because if you have open string field theory, then you can have a vertex operator which turns closed string into open string, in a way that preserves conformal invariance. So you can have closed string coming in and then you can have open string coming out like that. So this will be separating the Hilbert space or entire Hilbert space into A and it's complement. So in fact, you need two open string vertex operators here and on the other side. But what happens if you do that, is that you have to specify the D-brain boundary conditions. You have open string, so you have a boundary condition here. So you have D-brain, you have to put D-brains on it. Now, you don't necessarily have a unique choice of D-brain. In many conformal field theory, you can have different D-brains. So you can have D-brain called denoted by alpha and you can have D-brain denoted by beta on the other side. That's one of these ambiguities. And what these people showed was actually this ambiguity can be traced back to the ambiguity in separating the lattice model into parts using this. So there's actually a nice match-up in that case that they studied of how the ambiguity in separating the Hilbert space in the UV definition of the theory in the lattice model is reflected on the ultraviolet description in terms of conformal field theory. Okay? If there are no other questions, we can probably thank Hiroshi, yes.