 Okay, let's start, this is the last lecture, Tadashi. So, let me start with a little more update issue, what is called entanglement wage. Okay, so first start with a quantum correction, to holographic entanglement entropy. This was worked out by Faulkner and Rukwitz and Mardasena in 2013. So, this quantum correction means some quantum gravity, one loop corrections, and in terms of the gauge theory, it's like one of my own correction. So, we have entanglement entropy and the previous classical formula, classical formula, compute reading term, which is like order, this area term is order one over g newton. And there is a next reading contribution, which is order one, and there is a higher, which is like including order g newton and some power, more higher power. So, we are interested in this next reading contribution about one over an expansion, or namely newton constant expansion. And actually, but we can imagine what's happening here in the very intuitive level. So, I'm not going to detail, but we can give some intuitive argument. So, we can think about, maybe we can think about some, okay, so maybe we see some some Poincare ADS and Z-direction, and we have some region A, and we have some geodesic or some minimal surface, which we call gamma A. And the point is that in replicatoric, we have to compute always, place low to the N's power, that means we put some 2 pi N. And previously, I just explained some derivation of holographic formula, and that allows us some approximate computation by some deficit angle surface, right, inside the bulk. So, let's assume this approximation is okay, because we are just taking N goes to one limit, exactly same reason what we did last time. And then, this is just obviously just a 2 pi N periodicity everywhere in the bulk space time, and all localized on this. This is much like just entanglement entropy calculation in full space time, in bulk space. So, there should be just bulk entanglement entropy contribution, and indeed, this next reading term is just bulk entanglement entropy. So, maybe we can call this part of the space, or maybe we can cut this way. This is now, for anchor, this is a global ADS one. We can call this region, this is a minimal surface gamma A, and we can call some region surrounded by A and gamma is the M A, or this is M A. This is also M A. Then, this is entanglement entropy about bulk theory with M A for the subsystem M A. So, we decompose here by space. Let's assume, still we can use some quantum field theory in the gravity description, like supergravity. You can imagine some scalar field and graviter and fermion. They all contributes to entanglement entropy in just regarding bulk theory, the quantum field theory. So, then we have M A and M B. And about this decomposition, we can calculate entanglement entropy for full bulk theory, and that's contribution is here. So, this is actually worked out, this FLM paper. And we can also further, but I'm not interested in this, and the reading term, the next reading term looks like this. And then the question is, what does this term mean? But this term actually arises, even classical level of gravity. This only happens in classical theory with, I mean, spins greater than two. So, if you think just a free scalar theory, or any scalar theory, as a classical level, there are no, of course, entropy. So, this entropy arises because the very mysterious nature of general relativity or Einstein-Hubert action. But the reason is simple. It's topological reason. So, as we explained, this has some localised contribution. If we have a definite angle, and there are no such things in scalar field or derivate of scalar field or fields of things, or anything like this. So, this is very special to gravity, so that's the reason we have this term. But this is a standard, just quantum collection. Just one loop level, we have entropy. And this is true for any matter theory without gravity. So, if we want some entropy, we have to quantify the theory, and we compute one loop function, and then we get entropy. So, this is quite standard. So, in that sense, it's quite natural to have this formula. But this formula tells us something more. That is the things which we call entanglement of H. So, we can look at this. So, entanglement entropy in CFT is written this way, low log low. But we can write... So, this is a... So, low log low. But we write this as some expectation value of something called modular Hamiltonian. So, modular Hamiltonian just defined as a minus log of reduced density matrix. Modular Hamiltonian. On the other hand, this kind of bulk entropy, you can imagine this guy. It's computed in a similar way. You can compute the bulk density matrix. That means MA, bulk modular Hamiltonian, and bulk... bulk density matrix. And the difference is area. And then we expect the following relation. We go into this plus area as this plus area gives first one. So, HA, modular Hamiltonian is described by something called area operator. So, we promote this area. It's originally just classical number into some operator. So, we just measure the amount of area, minimal surface area, if we give some quantum state. And so, we have this modular Hamiltonian in the bulk. Now, of course, we have other... Let's work on this. This level. So, this is actually very impressive formula. So, anyway, this is something universal. If we assume small perturbation, small quantum corrections, so that we can only trust one-loop contribution. So, this is kind of universal part. And then, if you change the state, the point is that entropy is not observable because... So, usually physical observable expectation value of operator is taken to be the average of operator against some DC matrix. But, this all is now depends on the state because low A is just log of low A. So, this is not the standard physical observable. Instead, it depends on the state. But the point is that... So, here, these two guys this part is don't. And then, so these two guys we have some one-to-one correspondence between these two guys. So, this is true for excited states. Weekly excited states. These two guys are depend on the state, but in the same way. And this is what ADS-CFT and holographic entanglement entropy tells us about one-loop. So, that means this allows us to introduce the idea of entanglement. So, many ideas are going, but I can recommend this JLMS Jaffa-Risuru-Kwitsumara-Senasu 2050 nice argument. And also, there are many interesting further results. But this means that we have some discorrespondence. This means that so we have some DC matrix low A in CFT. And it is quite interesting to think about what's the dual of this reduced-density matrix, because this is related to locality issues. If we have some, we know ADS space-time, ADS is dual to the boundary theory, but if we restrict to the particular region of the boundary so this region is dual to some region maybe like this, but what exactly this region. So, this is related to this question, very basic question of location of information. So, correspondence between two different information, but this tells us that there are one-to-one correspondence if we assume some low energy limit. So, that means we have some MA in the bulk-density matrix, reduced-density matrix in ADS. This is gravity, bulk-density like other things in ADS CFT. So, this, because of this, we say that if we have a region A and we now understand precisely what this region means, this is a really region surrounded by this minimal surface and this region A, and so this is the definition of MA and this MA is called entanglement to edge. So, in other words, so this region A is dual to the bulk space within this entanglement to edge. So, this is a related to location of ADS CFT. So, this is a simple example, but we can think of a little more non-trivial example related to the phase transition to disconnected curve system. We were sorry, last time we explained some phase transition phenomena. So, I'll write a picture. This is again global ADS. You know, inside the ADS and the boundary is this circle like ADS-3, global ADS. So, let's assume A and B are far apart. They are not so large. This case, as we explained this disconnected guy gives entropy for SAB. So, we are interested in this case. So, we are interested in the entanglement to edge which correspond to low AB. So, we need some bulk region which is called MAB and we wanted to identify this guy. So, then if they are far apart so that it's a mutual information we'll write it this way. Zero, then this also identical to this guy, MAB. So, these two guys are MAB. So, this is MA and MB, but MAB is just a union of these two guys. It's not interesting. And entanglement to edge is disconnected. This is disconnected. Entanglement to edge is disconnected. We are not so much interested in this. But if they are closer to each other or they are larger, gets larger like AB. Then actually, so here also we can think about this kind of surface. But this surface is definitely larger than some of these two surface. But here, actually these two guys some of these guys is maybe we can write more clearly so that maybe cut it here. So, then some of these two surfaces are smaller than some of individual minimal surface area. So, this entanglement to edge is turned out to be this connected region. This new place, new region. So, this IAB is positive. And then MAB is connected. So, this is more interesting case. Anyway, so what we do according to this claim we run that this low AB in CFT is dual to this region called entanglement to edge. More precisely, in low range space we have to take into time direction but I'm just talking about static state. So, I'm doing some simple description. So, the next question is more recent one. Is this 34 and let's go 35. So, using this idea so this leads to the idea of so-called holographic entanglement of purification. This is the best one of my work with graduate student Uemoto 2017. So, the idea is I mean motivation is very simple. So, anyway we run this nice entanglement to edge. So, we can extract some interesting. Can I ask you in this case is the classical piece different since you're connecting like that before you were connecting like that. So, the classical piece in the two cases is different, no? Classical, sorry, what do you mean? If you connect. Yeah, this classical value of mutual information is different. These two guys. So, this is a phase transition phenomenon. If it's gradually enlarging the size of A and B it suddenly changes. So, mutual information is continuous but suddenly go this way. Deliverative. Good point, actually it's modified. That's a really excellent point. I like this way, but this is kind of operator and if you have some, we are now taking one loop collection, right? And then Einstein equation, right, it's modified. G-Menu and it's like a cosmological constant but I mean, you have energy stresses and this is expectation value, quantum expectation value and this, yeah, little bit this modified. I just didn't mention that. So, this part actually there are contributions from this part because the surface shifts by because of this one loop effect. So, now yes, go back. But now the story is quite classical and so we have some entanglement which and the next question is that we can think of some new interesting geometric quantity but if we look at this geometry, it looks like 1m4, right? And if we assume this part it's eliminated. It's 1m4 which connected and then the natural thing is just look at some cross-section, minimum cross-section of which we let's define this as sigmaab let's call this as sigmaab so you have some 1m4, right, how much they are connected, measured by this minimum area, right, minimum cross-section area. If this is 0 it's essentially same as this mutual information 0, no connection at all. But this is actually independent this area we can now come up with this area of this guy it's independent from this mutual information. If this is 0, this is also 0 but no other connections are quantitatively different. So, it's quite interesting to ask about it. So, we are thinking about this situation and so we have mab is separated into two parts which we call maybe mab but the a part and b part. So, this is a part this mab a part and this is a b part. And this boundary of mab a is just includes so there are some boundaries like this this region is a boundary, this plus, this plus, this plus but includes a and the sigma ab and b is similar, right, if we say we just replace this b. And we are interested in this minimum area. So, you can think this way also you can separate but this is definitely this has a smaller area and we keep this ab has a minimum area. So, this is a kind of cross section. So, we can define this quantity. We can introduce this quantity. This is related to some density matrix low ab in safety side it's given by four times g newton and the area of this sigma ab. So, always it is good to divide the area by 4g newton. This is just what we learn from Beckinshaw and Hawking formula. So, this is some new quantity. This is independent from a holographic entanglement. And we define this as an entanglement let's call this a cross section. So, this seems to be one of the most natural quantity we can associate with entanglement wage. So, the next question, what does this mean? And I will give some answer to this question. So, a and b are connected through this entanglement wage. Then it's zero. So, we have some conjecture of this conjecture this conjecture is like this e w low ab equal to some nice quantity called entanglement purification, which is purely defined in safety side. And I will explain the definition of this guy. So, this is called EOP entanglement of purification. So, it's written there in the title. So, this is the idea. So, anyway, we are talking about mix state as already mentioned before mix state we cannot use phononema entropy as the counting of EPR pairs. We cannot be easy to get some entanglement measure. But we can think of the following kind of intuitive argument that maybe if we purify the system, once we purify the system, we can measure amount of entanglement by phononema entropy. So, let's do this. So, purify means, so we have some low ab. So, we start with Hilbert space low ha times hb, right? But, so anyway, this low ab. So, we some enlarge, we consider larger Hilbert space is called maybe you can put hc. But let's decompose the hc into ha prime times hb prime. Let's decompose this is quite useful for later purpose. So, of course, the result depends on the composition. But anyway, let's ask that slater. And then if you have enough larger space, always you can purify the mixed state. So, it's like, it's written in terms of some a prime b prime trace, but some single wave function. So, this, this Hilbert space includes one sum wave function, so ab a prime b prime. And then we can have this. And then, so under this condition, that we define entanglement purification as follows. E p is defined as minimum of something more precisely infinite. But you can regard this as some minimum. So, this is the minimum of quantity, just von Neumann entropy s a a prime computed for this state psi. So, is this condition for any choice of psi purification a b a prime b prime which satisfy, of course, this should be satisfied. And this problem is that here, there are many different purifications. So, there are many candidate of purification. There are infinitely many. Because we just require this condition, and you can allow any hg. But, so, but we want just one quantity, so we minimize it. This is a basic idea. And it is known in quantum information that this has a nice property. But unfortunately, I should mention this is not rigorous entanglement measure. This is a measure of correlation. So, that means includes some classical contributions. Nevertheless, it has some nice property. And we also have more, I mean, better measure of entanglement, for example, squash entanglement. But this also involves minimum of something. So, this is kind of toy version of such a minimum minimization procedure. So, it's quite interesting to look at this quantity. But actually, this turns out to be the same property as this entanglement which cross-section. So, that's the reason why we argue this equality. Maybe I can give some to basic. In fact, there are many properties, but I don't have enough time. So, please look at paper if you are interested. But the first one is a trivial one. This is if you assume low AB is pure, your state is always any measured reduced entanglement entropy. EP, low AB is just reduced SA which is equal to SP. And this is well known for this formulation. This is very obvious because we don't need to purify it. And then, this is actually related to cross-section, but this is that because we have A and B and they are next to each other, no other space. If they are extra parts, then low AB is not pure. So, then of course, this is a full space is an entanglement range. So, the minimal cross-section, we are standard minimal surface. So, that way this property is obviously matched. And second one is a non-trivial inequality which is quite interesting. So, we know EP, low AB is bounded from above as a half of mutual information. This is easy to prove from this formula by using sort of strong sovereignty kind of argument. And the holographic formula should satisfy this, but this is very easy to see this. So, you can imagine a similar setup, AB and we have some connected surface and we have some cross-section which calls sigma AB. Maybe, let us call so we have some minimal surface here, which is gamma A gamma B and we call the other part like this part is maybe gamma 1 gamma 2, gamma 3, gamma 4. And then, obviously, we can understand that area of sigma AB is bounded by this summation of three surfaces. This is a very standard geometrical fact triangle inequality. So, gamma A 1 and gamma 3. And we can have a similar inequality for other part for this part. So, it is bounded by low AB as a gamma B and gamma other things 2 and 4. So, we know this gamma AB for full system is given by union of these guys. So, by summing over these two guys and then what we get is yeah, so we get sorry, I am very sorry but this is the same logic but we can think about this gamma A is bounded by this way so sorry, this is sigma AB so this is again triangle inequality. This surface area is smaller than this summation. This area is smaller than this guy so we have this. Otherwise, we get different inequality. Anyway, we take some of this and this summation is just given AB. So, that means so A sigma AB is bounded this is an EOP is bounded by half sum and take half so then it is really gamma A plus gamma B and gamma A gamma AB so this proves this inequality so it is very easy to show this and of course we need more evidence and we have more inequalities which we can confirm and they also work well and even though we don't have enough time to explain we have derived this conjecture by connection between ADS safety and as an argument is like connection between ADS safety and tensor network tensor network and another argument is we can also compute this quantity in particular case using CFT using as an integral optimization I don't have enough time to explain this but as an integral optimization so this is my work with CAPTA MIRAG and just last December if you are interested please look at this so there are also in particular case we can derive the same result from CFT so this is a story of this entanglement to edge and entanglement to cross section and this gives some new quantity because this surface don't attach at the boundary so this sense is quite new minimal surface is really end on the boundary but this is don't attach so it's like more bulk quantity so it's quite interesting to look into more and using some rest of time I just quickly want to explain some dynamical aspects of entanglement entropy this simplest example is evolution of entanglement entropy and quantum quenches, local quenches this is a special kind of quantum quench so you can regard basically this as a local excitation at some particular time we suddenly locally excite the system and we can think about three different local quenches so first one the most simple one is a local operator local operator so we just in that local operator at t equals 0 t less than 0 we just back in but we act some local operator at t equals 0 and see time evolution another one this is first introduced nozaki and masawa and myself it's like you can look at this and then another one it's a local this is local quench but this is kind of joining procedure so that means we start with some semi-infinite 2-copy of semi-infinite 2D conformal fuselage and attach end point at t equals 0 suddenly attach end point t equals 0 so again this has some local excitation and also we can think about opposite procedure which is this is first concept of my collaborate in 2007 so we recently considered the opposite procedure it's like splitting actually these three are quite interesting for the holographic dual so that's I'm finally would like to mention so this is t less than 0 so it's just connected just backing state but suddenly they split out just cut at some point and have individual time evolution by individual Hamiltonian and rate so we see and if we compute entanglement entropy high entanglement entropy evolves that's the question basically how space time evolves in ADS safety actually but we have to be careful that result depend on conformal fuselage we are looking at even though we are looking at one single interval entanglement we just let's focus on simple case subsystem A it's bounded by describe some interval but this is not grand state but excited state result highly depends on the class of conformal fuselage and especially the cell is integrable or chaotic that are so different especially first one is so different rational conformal fuselage I'm not going to details but if you compute this it behaves like this way so we excite some point at t equals 0 so we excite this point then once you insert some local operator then you create some EPR pair source of EPR pair details depend on the cell then this propagates this EPR pair propagates so once this guy one is here this goes at the speed of light the other somehow come into region A then non-trivial entanglement is observed so that sense time evolution, non-trivial entropy and if they pass through then nothing so only non-trivial entropy is observed within this time range time region so then if you plot for rational conformal fuselage like the Ising model of Friskera it's always like this like this this is just finite amount finite amount and this is actually equal to something called log of a quantum dimension the operator for each operator in 2D safety we can associate in rational conformal fuselage we can associate some quantity called quantum dimension and that's appeared I think in Shahogan's talk few days ago but we are not actually interested in other theory more holographic theory which is more I mean chaotic so that's case this is we just have a very small amount of entanglement increase because theory is integrable and each sector is a decouple but if you think about holographic theory one sector excitation in one sector affects other sector and so on there are strong strong interactions and then result is so different so then so we can think about the so this is a setup now we can think about CFT calculation so I'm not going to detail but just want to say few words on this so for number one this is a local operator case then reduce density matrix looks like this again we have some cut here region A and operator we put in a symmetric way so we are thinking about the excitation which is given by vacuum acted by some local operator and we need some always some regularization because we cut it out local operator is so singular if they are close to each other they get two point function get divergent so we have to regulate this parameter let's call alpha so this is a setup in the local application we can compute applying replica for this case for this setup so it's like in the end you end up with two n point function right for replica computation two n point function on this manifold which you call sigma n right for in this linear and so we can do also the second case this joining case this looks like now this looks like like this so it's at some point it joins so initially they are separated like this so we put some this kind of cut and again we put some regularization alpha so we are considering this path integral on this region and with some cut A as before always cut ways here and for this we can apply conform map into upper half plane because it's a boundary so we have some just upper half plane it's a boundary and so third case we can have a similar situation but a little bit different maybe I should write this here so third case is like splitting it's the opposite so this case path integral looks like this we cut in the middle so we suddenly separate the two regions so this is path integral and this is again region A is here and also we can apply some conform mapping up to some upper half plane so this for example this map looks like this so this coordinate is W and this is a exact coordinate we have a similar map here and this you can look at the latest paper by our graduate students and Wei and myself this also includes many reviews including this area works so now we go to holographic side so in principle you can compute using three-stop operator language and by using conform map you can compute entanglement entropy and now we go to the holographic side which is maybe more interesting so how geometrically we have this kind of setup so first one local operator quench local operator this is actually simply written this way so we have some Poincare ADS and z-direction and this actually the previous map between global ADS and Poincare ADS comes into play so there are parameter alpha there are regularization parameter and that's actually related to energy scale if we put some massive particle here energy scale depends on the location because the metric looks like this right so actually this position is alpha we put massive object massive particle here and see some time evolution this is as already explained in the first lecture today so it's like we have trajectory there are strong gravitational force towards this horizon so they're falling this falling particle this explains the energy stress as well as entanglement entropy behavior because anyway we act local operator but we smeared little bit we smeared roughly scale alpha otherwise it's very singular so this is exactly same as putting massive object here so we can compute entropy by using this and now we go to the second and third setup for that actually we need to introduce boundary so we need this we can discuss in the same time we need we need boundary conformal boundary we need some conformal boundaries so what we consider is conformal field theory on some manifold with boundaries so this is called BCFT boundary conformal field theory so we apply the formalism of something we call ADS-BCFT so this is developed with Fujita and Eric Tony here and myself it's like you can look at this so this is you can use this holographic construction is boundary so the idea is very simple so originally we have ADS-BCFT so we have let's assume some Poincare ADS but we just want to restrict particular region particular region of this space and there are some boundary this is boundary here and we are interested in conformal field theory on disk then one effective way to describe some bottom approach to describe holographic dual space time is just focus on this boundary and extend in the bulk so we have some surface which we call Q you might think this looks very similar to holographic entanglement entropy but this is so different because this boundary surface is very important and this is not like probe approximation this is really boundary surface back reacts it's dynamical this is dynamical and we impose here we impose Neumann boundary condition of gravity in gravity ADS gravity so namely that is the following expression this is extrinsic curvature this K is extrinsic so this K mu nu and the side stress is K and H mu nu is the induced metric on this Q and this is simple cases we can set this 0 but there are some one more parameter if you want to put some tension parameter which is like this tension parameter is just tension of this boundary surface but once we admit this formula we can actually show that this preserves half of the conformal symmetry that gives boundary conformity so this is also same as boundary condition we have in brain world set up like Randall, Sandor and so on just come from the Einstein Hilbert action plus Gibbons Hawking term this is the only there are some two consistent boundary condition one is this Neumann boundary condition another one is just deletionary boundary condition we fix particularly this is usually we take in ADS safety so if we have at this ADS boundary we normally take this deletionary boundary condition but on this dynamical surface we impose Neumann boundary condition then many things works well including some G-seolems such as monotonous G-seolems but I don't have time to look at that but anyway so by using this framework so we can apply this framework to these guys easy because we know the answer here upper half planes just dual to of Poincaré ADS we just cut it into half and we can do we can get back to this complicated geometry by conformal mapping there are analog of conformal mapping in calligraphy called the Bernadoff's map anyway we do this then what we find it's a little bit interesting and first of all I want to say these two guys give the exactly same metric you can find this metric in the bulk same metric but of course they are so different physics like one is separating one is joining and so different and difference come from the different choice of Q so let's focus on first second case which looks like which looks like this so we have so the idea is using again so we start with somehow Euclidean pass integral and at T equals 0 we go on a low range time evolution so the physical space is upward this is like Hartle Hawking wave function construction so we take this real time evolution so we are interested in this upper region in this setup so then what we find this boundary surface is quite dynamical and it's like this way so this is the x direction this is the x direction and this is the time direction and this is the z direction and boundary surface is something like this something like this and going like this in the end the approach is null direction it's null curve so this is a case second case this is quite natural because after rate of time so initially they are separated and at some T equals 0 suddenly join so in the bulk space time looks like this there are some cut here physical space is like here and this is the z direction and this guys disappear at rate of time because we took a long time after we took a long time after joining procedure and the third case exactly same metric so here metric is same as the second case but choice of surface is different so what we find is actually we have same surface actually like this this is also very important same surface there but actual boundary surface is here it's like this way so it's like this it's attached to here there are two boundary surface this guy is another guy just next to each other so this is so you can understand this following way so initially we have separate this is a kind of splitting quench and initially we split the space right and then there are some surface which connect this guys this is opposite to this picture here it's like going this way but this is a boundary and this splitting after just split it this propagates as a speed of light into the bulk so in the end completely disconnected space we have so this really again this tip is propagates as a speed of light so this tip is like z is like t and we know this precise curve this is like you can write this way to t square and plus alpha square and the square root alpha square plus 4t square so we can actually work out and here it's interesting thing is that this surface has some meaning so if we these two points are identified if we go reach somehow observe reach this point this is same as this this is empty inside is empty suddenly waft into the other side this surface means that and the boundary surface q is here so there are two guys because we have this part we can call q plus and the other guy q minus so this is so different so if we think about entanglement entropy for example let's take a subsystem A to be very large this is actually quite a simple center so we started some interval like region A, A, B but let's take big of infinity so this covers half of the space time and then we see some very characteristic but simple behavior so if we think that way this end point and we have end point here and end point go this way and in the end on this region q so it's like this so we can end on this region q to compute entanglement entropy and here if you some point here like boundary point then it can end on this boundary of q but this is not correct actually naively if we think this way or maybe this way just this way we don't get correct answer just finally I mentioned and just finish so this 4, 4 this holographic entanglement entropy calculation under local range and let's take the following limit region A is the positive but T is the intermediate but B is so large so that it covers almost half space region A subsystem A is almost half space and take time is intermediate this is most interesting point interesting time zone and then this local operator case we find entanglement entropy is actually computed as a connected surface and it's like log 6 C over 6 and log log T T over alpha this is a characteristic log T behavior logarithmic growth so different from rational conformity and C and this is a quite normal term static entanglement entropy and second case this is a join join in case we have actually entropy is dominated by disconnected geodesic and T over epsilon but this case a double log we get a double log time evolution alpha epsilon is a standard lattice constant alpha is a regulator special to this local operator so local quench local excitation and then so here we have this we have one more term B over epsilon and finally for this splitting quench actually we don't have a logarithmic divergence this again it's related to disconnected but we have two contributions connected and disconnected and we should choose minimum one just following holographic entanglement entropy this case just constant but the ratio appears but it's constant so finally I'd like to explain why this kind of behavior appears in this holographic center so it seems for example if you compute energy stress tensor which is really behavior near the boundary this three theory is really same energy stress tensor you excite some two one point then energy increases and time evolution after time evolution you get a two peak because one of the excitation got left and right energy stress tensor is all the same but the entanglement entropy are so different so this is because we are of course different state we are considering but also number of log has some simple interpretation in terms of holography so I just want to come back so let's focus first start with the second case we can go back to first case but this second case is quite nice so we have a boundary so we have ADS boundary and this is some right and we have entanglement surface boundary surface Q by ADS B CFT and we have some A is here and zero is here and B is let's take B is here then we see this disconnected geodesic we see one of them is going here another one is going this roughly speaking this is not so interesting but this guy is quite interesting if you go back to this geometrical picture let us remember that Poincare patch is not complete that actually global patch is given by this global ADS metric so we can write this way this is the globe so this previous picture Poincare picture only covers part of here but actually this heavy object so we introduce this path through in the center and we can easily go out of the horizon so let's talk about this geodesic first so this is an A point actually going here it's very close to the boundary so this is T half and here it's like please remember previous and that case and the tip the boundary surface is go this way this same same thing as this trajectory so this trajectory is essentially equal to this trajectory trajectory tip of this guy tip of this guy and then it pass through the center so if we look at this picture so we can maybe just we need to compute this one maybe it goes to infinity but just compute this that's only explain first contribution so we are missing the second contribution and if this extended picture actually this is very similar so very easy so this is going to end on the boundary this is a tip of Q and we have two regions inside Poincare patch and outside of Poincare patch and this guy gives log T over epsilon first term and the second guy gives log T over alpha so that's the reason we have double log contribution we have B but it's just end on here but it's not so interesting this doesn't have any it's inside Poincare patch so this guy is not interested this guy gives these two two contribution and if you go back to first case because of the similar issue because of the back reaction we see that these two points so we connect this this case is of course no boundary so we have to think about connected geodesic between these two boundary points and then so we see that so we connect these two points this is A and B is here and it's actually going out of Poincare patch and coming back so because of the existence of this region we have one logarithmic contribution this gives actually you can see this gives logarithmic contribution but no other contribution and finally if you see the splitting case then so we can write this similar picture so this is zero and A and B is very very large and so this boundary surface is sitting here this is a boundary cube and we have this is a trivial this surface is very trivial nothing interesting and the boundary surface of B is like end on this point this tip maybe it looks interesting but we can see this only generates logarithmic enhancement but no time evolution just just produce this term so this is just explain this contribution sorry this is alpha so this basically come from this picture sorry sorry this is A this is just come from this part this length is just log of this standard result and so this part is related this guy maybe you can think this way so we start with 3 divided by 2B epsilon that is a full length full length just standard static formula and we have some time evolution which maybe come from here here of alpha but however we have to subtract this contribution right this so this is like cancels you can think this way anyway so this gives contribution which is related to alpha so in this way we can explain the time evolution of this local quench and it's characteristic behavior is especially first case include all cases but the first case includes logarithmic close but this is missing in an integrable conformal few series quickly just for example free boson, free scalar CFT, 2D CFT right so this case you can think about operator quench operator quench which is given by this following state so we have some exponential i5 plus exponential i-5 and acted on the vacuum you can compute entanglement entropy and in this case there are no logarithmic close because this is nothing related to holographic CFT this case you find that at some point A you have just constant shift and if you compute carefully this is precisely equal to log2 log2 maybe we can come up with a simple example like EPR pair, indeed this is EPR pair the reason why we have EPR pair here is you can regard as we are now decomposing Hilbert space in the A to B, roughly this way so that's case this left moving right moving particle going here and left moving particle going here so it's like left right entanglement so you can regard i5 as a i5 left moving part right moving this is usually conformal few series then so this first one let's call this guy is up spin and minus phi is down spin then this state is just equivalent to EPR state this is square root of 2 and up to up spin and down down spin indeed this log2 come from this entangled state but this is very nice very simple because we can we can decompose into many different sectors and they don't talk with each other but in holographic CFT it's quite more chaotic and in the end we find logarithmic close it's very much larger and also depends on the cut-off scale so really this holographic CFT is so different from integral CFT sorry I extend the time but I'm sorry I think I'll stop here