 Let's get started. Today is the last lecture by Hiroshi. Okay. Thank you. So I just realized that I went through just about half of my lecture notes. So we'll select some material from the rest of the lecture notes. And one thing I want to remind you is that in the last lecture I argued that this eternal black hole in antiretta space is due to some field double state in the tensor product of conformal fuselage Hilbert space. This was proposed by Marat Sena in 2001. But then I would also like you to remember the causal wedge reconstruction or ADS-Rindler wedge reconstruction that we discussed earlier yesterday, which is to say that if you have just one boundary, and then if you have a causal domain on the boundary of some region A, so you have this causal domain A in CFD. So suppose A is some part of the space-like section of conformal fuselage. You can draw a causal domain, a domain of dependence, please excuse me, associated to this A and denote it D of A. So this is on the boundary of ADS. And suppose this is that causal domain. So now I make this into one dimension. And then there is a causal future and the past. So this is the intersection of the future and the past of this causal domain. And what I was discussing based on HKLL construction is that if you use the Hilbert space on this segment of conformal fuselage, you can reconstruct everything inside of it, as far as local operators concern. If you apply this philosophy here, what this is telling you is that if you utilize conformal fuselage on this side, you can reconstruct everything on this wedge outside of the black hole. On the other hand, if you use the conformal fuselage on this side, you can reconstruct information on this side. So this suggests, the following, that this side does not know over here. So if you trace out over here for the sum of your double state, then you have a density matrix, finite temperature density matrix for, this is a thermal density matrix for the conformal fuselage one. So this reconstruction suggests that this knows about this region one. On the other hand, if you trace out the other Hilbert space of the other conformal fuselage, this side, then you would be reconstructing this. Pardon? I denote this to be one and this will be two. The matter of notation, okay? Okay, so this will be a reconstruction. And this picture fits very well on what's happening on this side and what's happening on this side. Because on this side, you just have a tensor product of conformal fuselage. There is no direct interaction between them. You're just selecting the entangled state. This is reflected on the fact that there is no communication between this region and this region. That these are space-like separated, so there are no communications between them. Nevertheless, they are entangled. Since there is no communication between them, so that means that whatever you do on this side cannot influence events on this side and vice versa. So that is related to the fact that there is no interaction between these conformal fuselage. So whatever you do in this conformal fuselage cannot influence this. However, if you make a measurement, observation of this state, the outcome of the observation can influence the outcome of the observation in the other conformal fuselage. And this is reflected on the fact that you can jump into the black hole from this side and you can jump into the black hole on this side and you can meet over here. So the fact that in addition to these regions associated to these conformal fuselage, specifically for left side and right side, you have two additional regions where you can meet. It's related to the fact that these states, even though there are no direct interactions between them, are nevertheless entangled. Okay? So now, in fact, let's discuss nature of entanglement. So if you compute, let's compute the entanglement entropy starting with partial trace over Hilbert space for the second conformal fuselage. So sum of field double, tensor sum of field double. Okay? So then, of course, by definition, you get back to this. So entanglement density matrix is this. So therefore, if you compute the entanglement entropy for the region one for this sum of field double state, then this is trace over Hilbert space one of rho i log of rho i. And so if you calculate it, you can compute it and then find that this is actually expectation value of Hamiltonian for this finite temperature minus the log of the partial function, finite temperature partial function. So this is nothing but the entropy, thermal entropy of this canonical ensemble. And it's given by the Beckenstein Hawking formula, which is 4G times 2 pi d over 2, gamma d over 2 times r d minus 1, where r is the radius of the horizon of the black hole. And this is basically the area of the horizon. For the black hole in d plus one dimension, for d plus one dimensional space time, the black hole horizon is d minus one dimension. And if the black hole horizon has radius r, then this is the area, divided by 4 times g newton is reproduced in the larger new limit. Gamma function. This is the volume of the unit sphere in d minus one dimension. Now, this has also the following interpretation. As I pointed out yesterday, between region one and region two, even though they are not causally related, but they are geometrically connected. Namely, that if you take space-like section like that, where they are connected, and the geometry looks like this. So this is the geometry of this space-like section. So I'm suppressing the time direction. This is the space-like section, where you have sd minus one on this side, and then goes here, and then this asymptote to hyperbolic space on both sides. This goes to CFD two. This goes to CFD one. And then this geometry is called Einstein-Rosenbridge. So the radius, I draw this picture in such a way that you can see that there is actually minimum area region over here. And the radius r is this radius r, which is related to the inverse temperature beta by this relation. This is a space-time dimension for the conformal field theory. Like that. So that means that this also has an interpretation. The entanglement entropy for the conformal field theory one has also an interpretation that it is 4G Newton is dividing the area of the minimum surface, dividing CFD one and CFD two. You can interpret the same thing in this way. So namely that these two conformal field theories are spatially separated. And then this is drawn here like a point because I projected it in two dimensions, but this is actually d minus one-dimensional sphere. And the area of this sphere is actually related to the entanglement entropy between them. So now these are the very interesting features, namely that the area of this black hole, the area of this Einstein-Rosen bridge is exactly proportional to the entanglement. You can also see that from here too because as area grows, as area becomes small, beta becomes small. So that means that, as area becomes small, there is something wrong with this picture. I must have made a mistake because as temperature grows, the beta should grow, the R should grow, I believe. So this is consistent. So d times R squared. So as R grows, the entanglement grows. And as R becomes small, the entanglement becomes small. So this is related to the fact that as I said, even though these two conformal field theories cannot influence each other directly because there are no direct interaction between them, but nevertheless they are connected by entanglement which is reflected on the fact that they are connected in this case by the size of the space-like section separating them. So this observation is very interesting that somehow it is interesting for two reasons. One is that the entanglement itself is influencing the geometry. That somehow as the entanglement grows, the size of the bridge grows and so therefore geometry connecting the two conformal field theories is growing. So somehow entanglement is building up the geometry in the bulk. And the other interesting feature is that this entanglement entropy is related to the area separating the two things. So now, so this motivated Shinseiryu and Tadashi Takayanagi to propose a more general formula for entanglement of a given state. So let me first remind you a general entanglement causal wage reconstruction or ADF lindra reconstruction that I already reminded you over here, but let me just state it in the following way. So I told you that if you have some region A and then you have a domain of dependence A then you can reconstruct the intersection of the past and future of this domain. But I should also mention that suppose you have a pure state, suppose you have one of these pure states, this certainly is a pure state in this direct product. This is a pure state in the direct product conformal field theory. So similarly, suppose you have just single ADS space and then suppose you have some pure state in this case if you have some region A then you have this causal domain associated to A and then you have a complement A bar and then there is a causal past and future intersection of causal past and future associated to it. So this is this region. So if I draw it this side on this side then you have a causal domain of dependence for D of A bar and then they cover this side. So for the space like section say t called zero suppose this is t called the picture t called zero then the entire bulk geometry this is any pure state in the code subspace which is describable by smooth geometry doesn't have to be ADS some smooth geometric solution for the gravitational theory then this is again divided into two regions. So even though this is a single geometry but naturally divided into two domains separated by the fact that you have this causal patch on this side and you have causally related region on the other side. So this suggests to compare this picture this suggests that just like the size of the Einstein-Rosen bridge connecting the two causally disconnected but entangled region gives you the entanglement entropy it is natural to expect that the area of this gives you the entanglement. So this led Ryu and Takayanagi to propose that for this kind of pure state the entanglement entropy is given by the area of this surface let me call sigma associated to A and this has various features that sigma of A should end on the boundary should be anchored by the boundary of A and it should be it should extrematize with respect to the metric in the bulk. And this has been checked in various examples this proposal has been checked for example I mentioned in CFT2 we have this formula by Cardi and Karaburese that I mentioned the other day where if you have so it's 1 plus 1 dimensional conformal field series so if you consider boundary with length L then the conformal sorry the entanglement entropy is given by this plus constant so this is what I mentioned the other day and you can show that if you consider a pure ADS so this is for the vacuum state so if you consider pure ADS3 and then if you consider minimum surface ending for the region with length L then they evaluate the area and divide by 4G Newton and translate that into conformal field series language this is exactly reproduced this is a hyperbolic space so the metric actually diverges near the boundary so you have to introduce some regularization which is related to the fact that on this side you also have to introduce ultraviolet regularization which affects this constant piece so this ambiguity also matches up in that sense so this leads us to new kind of idea called entanglement at which which is a slightly different notion compared to the causal wedge that we have been talking about so I have defined in the couple of lectures ago the notion of causal wedge which is that if you take the region A so I draw it over here if you take the space like sub space of the space like section of conformal field series you can define domain of defendants and then you can consider path and future of that domain of defendants consider you intersection then you have a causal wedge associated to region A you can also define entanglement wedge associated to this region which is naturally related to the Ryuta-Kayanagi surface in some cases causal wedge and entanglement wedge are closely related but in other cases they are not necessarily related and in fact generally speaking entanglement wedge is bigger than causal wedge so I have to first define what entanglement wedge means the entanglement wedge is the following notion so you have suppose you have a state which asymptote to entreat a space towards the boundary so you have some solution to gravitational equation satisfying the ADS boundary condition and consider some space like section of this and then again as usual I consider some section some space like section A for example so then you can consider Ryuta-Kayanagi surface associated to it which I denoted by sigma of A so then you have some region surrounded by A and the Ryuta-Kayanagi surface so let me denote this by this character small A and then entanglement wedge is defined as the domain of dependence not of big A but small A so entanglement domain is entanglement wedge is a region in the bark so I cannot draw it here so let me draw it on this diagram so suppose this is small A and you press this dimension along the boundary so suppose this is small A then what you do is just like before you consider the past and the future of this A and then consider the intersection of this so this is entanglement wedge I didn't speak it correctly so that's what I meant by the domain of dependence is that you pick a point A so this region is defined in such a way that if you pick any point inside of this and then extend this point along any time-like path it would have to necessarily intersect this small A so that's the definition of the domain of dependence so for example if you pick a point here you can extend this indefinitely in past and future in a time-like direction without intersecting small A so you see that this point is outside of this domain of dependence whereas this point is inside of this domain of dependence so this is the definition so this is entanglement wedge and you can actually show that entanglement wedge in general is bigger than the causal wedge so for example so this can be most easily be seen when you consider the case where the boundary region A has multiple components you can certainly consider a case when the A has a multiple component all these definitions does not prevent you from doing that so suppose you consider a case when you have the region A has multiple components so suppose the region A is a union of these fellows so and then here I'm just considering the space-like section of these wedges so if you have this situation like that the Ryutakayanagi surface would go like that the definition of Ryutakayanagi surface is that it's a minimum surface connecting the boundary of the region but here you have two regions so you have four boundaries in this particular case when the boundary is one-dimensional but in more general case you have two boundaries these are connected but suppose A is slightly bigger so suppose A is slightly bigger and suppose A goes like this and then goes like this so this is one segment of A one segment of A and this is the other segment of A so in this case if you are interested in Ryutakayanagi surface you will be choosing this type of boundary so then from this definition of entanglement wedge entanglement wedge would choose this whereas for the same configuration the causal wedge would be smaller because if you follow the definition of causal wedge it will have to choose the other region like that so this is the causal wedge so if I draw it over here it's going to overlap this region but then these regions in the middle are not causally connected to either of these so if you consider domain of dependence of A here and A here and if you pick a point here this point is specially separated from these points so you cannot send a signal from here to the causal domain of dependence of A here so this point is not included in the causal wedge causal wedge would be like that on the other hand this should be properly regarded as entanglement wedge from this definition so they overlap this is included in here if this is bigger than they could overlap but for example you can actually come up with a situation where they don't touch with each other so there is one more condition I should have mentioned which is that the surface has to be homologous to A so this is a case when in pure state often you don't have to worry about it but when you have a black hole for example then you have a horizon so I should mention that in Ryutaka Yanagi formula there is a condition I have time so I have about 40 minutes left I guess so in Ryutaka Yanagi formula we have a condition called homology constraint that is that the Ryutaka Yanagi surface has to be homologous to the boundary domain A so suppose you have a black hole and you have a horizon so if A is small well you can choose like this so then this would be homologous to A but suppose A becomes bigger and then suppose A covers this region then you can have two types of minimum surfaces you can have a minimum surface which goes like that or you can have minimum surface which actually goes like this so you have these two possibilities so it turns out that homology constraint requires this condition and in fact this gives you the correct result because what you can see is that as this becomes bigger and bigger eventually what minimum surface chooses is that this would grow so then this is going to pinch off and then it's going to become like that so this will be the combination of this cycle and this cycle is still homologous to this boundary A and eventually this disappears so that means that when A covers the entire boundary A then the real Takayanagi surface in this case would be the horizon itself and that's the correct answer because the entanglement entropy for the entire region is a Beckenstein hooking entropy as we calculated over here and so this give rise to the correct answer and in fact you can also see that from this type of picture that when A is small then it goes like that but then when A grows you start growing toward the boundary and then as A wraps the entire space you need to seek out the minimum surface which is homologous to this and the minimum surface homologous to the boundary is this neck of the Beckenstein-Rosen bridge so this fits with this kind of picture here but the fact that this existence of this entanglement which suggests that reconstruction idea can also be possibly generalized to the domain bigger than the causal domain you can ask well can you actually reconstruct stating entanglement wage rather than the causal wage so in the remaining half an hour I would like to actually give an argument that you can actually reconstruct local operator in principle in this larger space of entanglement wage rather than the causal wage but in order to do that I have to tell you a little bit about various entropy inequalities that you can derive for entanglement entropy and fortunately I know that last week Nima Alcani-Hamet discussed entropy inequality in particular the strong sub-aditivity and as I watched his video he was deriving that from the positivity and monotonicity of relative entropy do you remember that? Excellent so I don't have to go through the derivation of this but I will be utilizing that so entropy inequalities so I think that Nima was talking about the strong sub-aditivity and if you remember my first lecture then strong sub-aditivity is related to the property of mutual information so this is a mutual information so mutual information is an information quantity and I will tell you how much you learn about A when you learn about B or more precisely how much uncertainty about A is reduced by learning about B so that's a mutual information and gaining information should also always give you non-negative amount of information for the other quantity so that means that this has to be positive or non-negative and that is actually called sub-aditivity and the other condition is that if you learn something more then you will be less uncertain that would be the sort of slogan and so this can be summarized in this type of formula and then this is actually strong sub-aditivity and then almost a year after Ryu and Takayanagi proposed that formula it was pointed out by Matt Hendrick and Takayanagi that Ryu Takayanagi formula actually satisfies this strong sub-aditivity so that was sort of a very nice confirmation of consistency of the Ryu Takayanagi proposal and as Nima was showing that deriving this in general quantum system is very hard but deriving this for a Ryu Takayanagi surface is actually relatively simple so let me just explain that in one line so suppose you have region A, B, C and suppose these are entanglement entropy for A, B, C, etc. so then Ryu and Takayanagi says so what strong sub-aditivity says strong sub-aditivity says that SAB plus SBC is greater than or equal to SB plus SABC okay so let me draw these four terms pictorially here so SAB would be Ryu Takayanagi surface like this so namely the area of this surface divided by 4 Newton SBC would be this so this would be the right-hand side in contrast the left-hand side excuse me in contrast the right-hand side here is SB which is this and SAB which is this do you see that some of red curves some of the lengths of the red-carbon reds is smaller than the some of the white curves and can you see why? well it is easy to see because this red curve here connecting a boundary of B homologous to the one here so evidently this is longer than this because of the triangle inequality all the facts that this is actually stuck at the intersection here so this gives you additional constraint so this is actually the minimum surface and this is a surface with one additional constraint so this should be in general bigger so reducing this over here should reduce the area similarly moving this up over here should reduce the area so that means that this is this sum of this should be less than this one so this is a very simple derivation of strong sub-additivity which was actually more generally difficult to derive in the general circumstance so this is a Reuter-Geynani formula is supposed to be applicable actually I should have mentioned it so this formula was originally proposed for a static state so more generally a state which is time-symmetric and a reflection around the space-like surface and then subsequently Fubine, Rangamani and Takayanani proposed more general formula for time-dependent case and those are called HRT formula so you can ask well is this strong sub-additivity true for HRT formula and so for Reuter-Geynani formula so this for Reuter-Geynani formula strong sub-additivity can be proven in this way but for more general HRT formula actually the same kind of argument works provided that the space-time satisfies the null curvature condition which is to say that for any k which is null that means that k square is 0 this has to be non-negative so if you think that geometry satisfies Einstein equation with cosmological constant with matter-energy momentum tensor then the matter-energy momentum tensor should satisfy the null positivity condition null energy condition excuse me so if that is satisfied then you can actually derive strong sub-additivity condition this was shown by Alan Wall 1, 2, 1, 1, 3, 4, 9, 4 so this is entropy this is a strong sub-additivity condition and you can ask are there any other inequality you can derive for Reuter-Geynani formula and in fact understanding entropy inequality in general is very interesting question it is known that for general Shannon entropy that is a classical entropy that I talked at the very beginning on my first lecture it has been proven that there are infinitely many independent entropy inequality for general number of regions and it was actually very recently proven just 10 years ago so even though this is about statement about classical Shannon entropy but it was just proven 2007 my math that there are infinitely many entropy inequality for Shannon entropy for Neumann entropy for general quantum system it is not known although there are numerical evidence suggesting that these are also infinite now recently Nimbau Sefa Nezami Bogdan Stoika and James Sully and Michael Water and myself proved that for Reuter-Geynani surface actually entropy inequality is finite the independent entropy inequality is finite and can be classified by finite algorithm for every given number of domains and we were able to actually classify entropy inequality for small number of regions explicitly and enumerate them so I actually was planning to talk a little bit about that but in view of time I will skip this subject but I just wanted to mention that the totality of entropy inequality satisfied by Reuter-Geynani formula is in principle knowable and there is a finite algorithm to classify them and in fact there are inequality generalizing this type of strong sub-adjectivity inequality for more number of regions and those have various interesting interpretation and are characterizing entanglement property of kind of state which belongs to the cold subspace that is the kind of state which can be described by smooth geometry in the bulk but given the time we move on to another entropy inequality which as pointed out by Nima is closely related to strong sub-adjectivity which is an inequality involving relative entropy so this is actually a very important notion in information theory and its significance has been sort of motivated in Nima's lecture so I don't have to go through that but let me just remind you that relative entropy is defined for given two density matrices relative entropy is a quantity which somehow tells you how close these two are so it's like a distance except that it's not quite symmetric in rho and sigma so relative entropy is defined as trace of rho log rho minus trace of rho log sigma so this is a relative entropy and it has various important property I think that one of the fundamental property is that this is actually non-negative and is equal to 0 if and only if rho is equal to sigma so this is actually a very important fact that if relative entropy vanishes then that means that relative matrices has to be identical I'm going to use that so please remember this and another useful thing is that for quantum field theory the entanglement entropy requires UV regularization whereas in this combination they are cancelled so it's actually a UV finite quantity so it's more mathematically well defined and it has various inequality this is one of the inequality is that if the region A contains region B then the relative entropy between the entanglement density matrices associated to rho A and rho B also increases so this is known as monotonicity of relative entropy okay now we can calculate this kind of relative entropy for holographic cases and for example you can consider the following object so suppose let's say sigma is a grand state and then rho is some excited state but still in the cold subspace in the way that I define so the nice thing about grand state is that if I choose A to be ball-like region so you have D minus one-dimensional boundary which is a sphere and then on the sphere you can have D minus one-dimensional ball so if I choose A as a ball-like region then I can actually find an explicit expression for log of sigma A so minus of log of sigma A is called modular Hamiltonian and in particular when A is like ball-like which is whose boundary is D minus two-dimensional sphere then explicit form of HA is known it's actually one of the conformal generator of the boundary and the way you constructed is first that you can by using conformal transformation you can always map this region into hemisphere by rescaling it and if you have a hemisphere where pictorially it's half of the space so suppose this is a space-like section of conformal fin theory it's half of the space then there is a Rindler Hamiltonian which is actually a boost Hamiltonian associated to this half of the space for which this is a fixed point so this is exactly the conformal transformation which fixes the boundary and then you do the conformal transformation of this generator back into this region and this gives you the modular Hamiltonian so with this therefore what this is showing is that therefore this S of in this case is given by suppose this is region A then this gives you the first term is minus of Ryutakayanagi formula the Ryutakayanagi entropy so this is entanglement entropy entanglement entropy for this for the state A, rho of A so this is entropy for rho of A and the second term is just expectation value of minus of rho of sigma for rho so namely this is an expectation value of the modular Hamiltonian evaluated for rho so this looks like a free energy this looks like a free energy right the Hamiltonian minus an entropy so this looks like a free energy and in fact this is actually a very useful analogy so you can think of a relative entropy as free energy for the modular Hamiltonian now as I said relative entropy is non-negative and vanishes when they coincide so that means that by continuity if you actually consider a small variation of rho around sigma so suppose you consider infinitesimal variations like that then relative entropy between them well when del rho vanishes it should vanish but then actually del rho vanishes if and only if del rho is 0 so that means that it should start quadratically in del rho by continuity because this is positive non-negative and vanishes if and only if del rho is 0 it cannot be linear because it is linear then it should be negative on one side or the other so it should be quadratic it should have some kind of quadratic expression plus some higher order right and in fact in quantum information theory this is known as quantum fissure information the fissure information is sort of natural notion of metric in the space of probability distribution and in the quantum version this is a metric in the space of density matrices so this is known as fissure information so this starts with a quadratic part so that means that of course this is 0 this will be 0 in the linear order in order del rho right and it was shown by various people I have reference here but I guess you have started citing them it gets complicated it was shown that if you take sigma to be pure ads geometry and if you consider rho to be infinitesimal fluctuation about that geometry then this is actually equivalent to linearized Einstein equation in that case so namely if you characterize rho state rho as some fluctuation of the geometry around the pure ads this give rise to actually constraint on the geometry that it has to subject to the Einstein equation so it's very interesting that you can actually derive in some sense Einstein equation from this type of information theoretical consideration yes yes it's a non-local thing but in this kind of linearized situation it reduced to a local equation because fluctuation at different point can be treated separately okay and then subsequently Nima Rashikeli and Jennifer Lin Bogdan Stoika and Mark Van Rang and myself for example have shown that we are interested in extending this to higher order we are able to show that actually in general the notion of positivity here is related to positivity of certain kind of energy associated to the entanglement wedge but again I only have like 10 minutes so I am not able to go into that but I do want to mention one more thing which is sort of one of the main things that I wanted to convey which is that this type of entanglement entropy and equality actually give you some conceptual reason of why this reconstruction of bulk works so let me try to end this lecture by explaining the basic idea so again the idea come from this this relation by the way this is known as the first law of entanglement in analogy with the first law of thermodynamics so now let me actually try to explain why this is related to the entanglement wedge reconstruction concept so first of all again consider this situation where you have this region A and then you have the energy surface associated A and then you have this bulk region which I denote by small A okay so if I compute so suppose rho is some state in the cold subspace and then consider entanglement density matrix associated to that and so it is computed by CFT which is restricted to the region A of rho A log of rho A and this is given by 1 over 4G Newton times area of sigma A minus and it was shown by Forkner-Lukovitz-Maldesena Forkner-Lukovitz-Maldesena in 7, 2, 8, 9, 2 that the correction to this can be regarded as entanglement entropy calculated for the bulk quantum field theory so the leading part if you expand it in Einstein gravity with small fluctuation by loop expansion the leading term the classical piece which is given by this but then there are and they pointed out that the can be calculated by thinking that in the bulk you have in the small fluctuation so then you can divide that quantum field theory into region small A and its complement and then you can consider for that quantum field theory and compute the entanglement entropy this is actually order 1 as opposed to classical this is order H bar if you like so order H bar collection can be calculated as entanglement entropy in the bulk okay now this is a classical piece and then this is a quantum piece and in some sense you can think of this as the expectation value of some operator because this is something that you can compute for given geometry so you can imagine that there is actually an operator both are UV, this is UV divergent and this is UV divergent there is an order H bar UV divergent also possibly contained in here so you can imagine that there is an operator in the quantum theory of gravity which whose expectation value gives you for row A gives you area of sigma A because evaluating operator in semi-classical case is just evaluating the so this can be calculated in terms of the metric in the bulk so you can certainly consider that there is an operator whose expectation value gives you area so that means that for simplicity let's also divide by 4G newton so that means that S of row A can be evaluated as trace of A hat row A plus S of row A so namely that left hand side is given by boundary CFT density matrix and the right hand side is given by the bulk formula like that this is an operator in the bulk Hilbert space because you are evaluating it so because in the bulk you have geometry and fluctuation around it and then small a because this is specifying the bulk quantum state for which you oh yeah so I should have written like that yes thank you very much this holds only for cold subspace that's important so now let's apply the first row of entanglement for this formula so if I apply first row if I apply the first row what's going to happen well first row states that 0 is equal to S of row plus row row and let me see yes but S is this quantity S is this quantity so that means that this is equal to trace of H of row del row minus del of S row but now we know what row S is so that means that if I calculate the del of S row A this is equal to trace of A hat del row A plus trace of H of row A del row A because S of row is given by trace of H row delta row and this is true for any density matrix so in particular if I apply this over here here you have S of row A but that's according to the first row of entanglement should be given by trace of H row del row on the other hand this is equal according to this formula is equal to trace of H A delta row A so this is interesting because this means that you can actually compare this with this and this should hold for any variation so that means that operatorially there must be a relation between this operator and this operator so actually I'm going into 0 minutes I guess I started a little bit late maybe 5 more minutes yeah okay so I'll try to come to logical conclusion in 5 minutes so this means that if I consider projection of modular Hamiltonian for the cold subspace note that this is acting on the conformal cell-huber space so if you consider projection of this to the cold subspace then this is given by A hat plus the modular Hamiltonian for this region A so there is such a relation you can derive so this is actually a very important relation because you can now that you know the relation between modular Hamiltonian you can plug that back into relative entropy and derive the following inequality so this is algebraic you can just do some algebra to prove it and let me just state the final expression after going through the algebra so if I use this relation and by remembering that relative entropy is given by expectation value of modular Hamiltonian and entropy itself you can derive the formula that relative entropy between entanglement density matrix on the boundary is given by relative entropy of the corresponding density matrix in the bulk so what is this telling me what this is telling me is the following so suppose you have two states rho and sigma so these both of these belongs to the cold subspace so these are both some state in the cold subspace and then you consider region A and region A and then you consider you takayanagi surface and then you can consider domain A what this is telling you is that if you consider density matrix rho of A and sigma of A and rho of A and sigma of A in the bulk then these two are the same if and only if these two are the same and vice versa so namely that whatever the change that occur over here you can detect on the boundary and similarly whatever the change you make on the boundary you can detect in the bulk so that means that for example if you excite this thing by applying some local operator on this side that can be detected by the boundary so that means that you can actually simulate that effect by doing some operation on the boundary this is actually the basic idea of reconstruction in fact it was proven in a very nice paper by Shidon and Daniel Harrow and Alan Wall that this equality is equivalent from this you can prove that for any operator in cold subspace acting on the cold subspace there is some operator on the CFT side such that for any state in the cold subspace you can reconstruct so namely that operator acting on this cold subspace can be reconstructed by the operator acting on the boundary so this give you sort of a prior reconstruction of why local excitation in the entanglement domain can be reconstructed from conformal few-series operator on the boundary so this may be a good place to stop so thank you very much for your attention