 Okay, I'm sorry I cannot be there in person, but hopefully this will work well. So I'll be talking about the traversal wormholes and some applications to quantum cloning. And the subtext will be the surprising simplicity of nearly ADS-2 space times or the gravitational physics of nearly ADS-2 space times. And the talk is based on some work with Douglas Stanford and Shembin Yang who's a student at Princeton University and it's also based on a recent paper by Gao, Jeffries and Walden. So we all know that general relativity has wormhole-like solutions and the simplest version is the maximally extended Schwarzschild solution and that has two asymptotic regions and there is a closely related solution where we can have two asymptotic regions could represent really two far away regions in the same space time. It's not exactly the same solution but it's closely related. Now all such wormholes in general relativity are not traversable so even if you take into account quantum mechanical corrections you find that you cannot send signals from one side to the other and the fact that you cannot send signals is related to the fact that the integrated Nile energy condition holds. So that is to say the integral of T++ along Nile rays is positive and there's been some recent papers proving this in flat space using entanglement inequalities and well the same ideas probably work in this case. So it's a property of GR that if you send a signal let's say from the left side then the time delay is computed by a kind of shock wave computation that involves this integral of the Nile energy condition and if that integral is positive then the time delay always positive it's always a delay so the signal sort of gets deeper inside the interior and cannot get outside so it ends up falling into the singularity and well this is in general good because otherwise general relativity would lead to violations of the principle in which it is based so the idea that there is a maximum propagation of speed for signals but today we will talk about some special situations where it makes sense to talk about reversible one-folds and this do not violate any of the above principles but they will tell us some interesting things about black holes. So before talking about that we'll have to discuss the Kruskal-Schwarzschal-AdS black hole so that is the maximally extended solution that describes an ADS black hole and that solution has two asymptotic regions these two solutions are asymptotically ADS and because we are in ADS we can think about also the dual quantum system or quantum field theory and we can view the idea is that this solution is the gravity description or the dual description of a system of an entangled state in that in those quantum systems so the idea is that you have two non-interacting quantum systems and you have an entangled state a very special entangled state which is the thermal field it has a form here the bottom of the transparency where we have some overall energy eigenstates with a factor similar to the thermal factor well the same essentially the thermal factor so we have these two quantum systems and the idea of Gauss, Jafferys and Wolff well let's connect the two systems by putting a direct interaction between the two systems so we can imagine these two quantum systems as being systems of spins that we have in the lab and if we have two separate systems of spins we can nobody forbids us from coupling them to each other and then the systems of spins have a dual gravity description which is given by this space-time geometry the space-time geometry of the Schwarzschild ADS solution and so the question is what happens under that those circumstances what happens when we introduce some interactions now we can couple them in many different ways however Gauss, Jafferys and Wolff proposed to couple them using a simple interaction Hamiltonian that involves two operators so one on the left side and one on the right side so the left side will be an operator in the system of spins or quantum system that is dual to a field that propagates in the bulk of this space-time and so we are imagining that I'm going to denote by the same letter the operator in the boundary theory and the operator in the bulk so here we should think of phi left at zero as in the bulk we think of it as a field operator very close to the boundary and in the boundary theory we just think of it as a particular operator of that boundary theory the operator of this dual to the field expectation value near the boundary of ADS and here similarly we have the right one which is close to the right boundary so in the in the boundary this is a perfectly allowed operator so we can add this operator to the interaction as an interaction as an additional interaction we here have G's some coupling constant that we can take to be small and then in the bulk picture we are just adding this funny operator it's a bi-log from the point of view of the bulk is a weird operator because it's a bi-local operator it's a non-local interaction and this from the point of view of the bulk and such non-local interactions can create a state which has a negative inertia so if we just insert this at these two points it will create that let's imagine we phi is related to let's say massless interactions it will create shock shock waves or negative inertia along along this null rays and it will create negative inertia if the sine of G is appropriate if we change the sine of G it will be positive inertia but with the appropriate sine of G it will create negative inertia and then a signal that comes from the left side can have a time advance as opposed to a time delay and can emerge on the right and this okay so this doesn't violate any principle because we of course have introduced an interaction between the two sides so the fact that the signal can go from the left to the right is not what is important what is interesting and what we will try to discuss is how the signal goes so what's in what is interesting is how the signal goes from the left because the signal by going from the left to the right is exploring the interior of the black hole as it's exploring the geometric connection that we have between the two sides so the idea is to study this process in a little more detail and try to understand how it happens how we can give a mathematical well how we describe it how we describe the back reaction and so on and it further with the hole so the eventual goal is to understand the interior of the black hole so yeah we'll study this phenomenon this phenomenon happens in black holes in any dimension in fact the God Jaffer is involved describe the particular case of ideas 3 but as they pointed out it happens in all dimensions and we'll study it in the particular case of near geometrics where this effect is particularly simple so we will show that the in addition we will show that this s y k quantum mechanical theory displays the same phenomenon will set up the calculation in such a way that we will also show that happens in s y k so in the quantum mechanical system and then we'll show that we can use this to analyze some aspects of cloning of quantum information in black holes so first I'm going to review some aspects of nearly 80s to gravity so the idea is to this nearly 80s to gravity is for example in well it arises in any situation where we have an 80s to part of the geometry so for example if we have near extremal black holes and we and and and we focus on the low energy is on the region of the geometry close to the horizon that geometry is an 80s to space and it arises in any number of dimensions so we can have from five dimensions going to 80s to and so on now it's important when we consider when we consider perturbations that around 80s to propagate in 80s to to keep the leading effects there by way from 80s to that's why we call it the nearly 80s to gravity as opposed to 80s to gravity so it is to gravity on itself only makes sense for the ground states but not for any excitation now this this there is a simple aggression that takes into account the leading gravitational effects which is the so-called keep title-born theory and was also studied by almerian polchinski and this is the laceration that you see here in this second term it contains some scalar field and and well of course the metric the two-dimensional metric now what one would have written it in even simpler lacration which is here the first term which would be just simply the Einstein action in two dimensions but that in two dimensions is completely topological so this term has some effect which is the it contributes to the ground state entropy but it has no other effect and if we only have this first term then the Einstein the Einstein equations would imply that the stress tensor is actually zero the stress tensor of the matter theory zero and so it's not would not lead to a nice gravity theory so we by adding this this other this other term and which we think of this other term as keeping for example if we yeah if we think of this two-dimension this two-dimensional gravity theory as a rising from four conditions then the field phi would be essentially the area of the two sphere so we can think of this ads to as arising from a four-dimensional space which is ads to times s2 so phi not would be the area of the the extreme out of the area of the sphere and phi would be a small deviation away from that extremal value and okay this is more or less they already said now we can also consider a full theory that contains also a matter action so the matter will contain a coupling from the metric and some matter fields in principle we could also have a coupling to the to this five field but to lead in order it's okay to consider this in addition here we've written the boundary term that we need to add in order to get the the action to be well defined so that the variations of the action with respect to the metric gives that's the patience of motion this is usual the yeah the usual boundary term that we have in four-dimensional year 5b is the boundary value of the electron field so here the questions of motion for phi phi here appears as a kind of Lagrange multiplier one thing I didn't mention is that we could imagine adding a kinetic term for phi and it's fine to add a kinetic term for phi but we can do a field redefinition that removes the kinetic term so this is the most general action so now if we if we look at the question of motion for phi we find that metric is is a DS to so the question of motion for five fixes the curvature to be minus two and so the metric is locally a DS to so the metric does not have any fluctuations it's completely rigid rigid geometry so we find that the only dynamical information will be the location of the boundary and we have after we we impose the question of motion for phi or we integrate out five the field the dilaton field phi then we lose this bulk term in the action and we're left purely with this boundary term so all the gravitational effects come from this boundary term and this will this will have some retrieval dynamics that will analyze in a second so the only dynamical information will be the location of the boundary inside this rigid a DS to geometry and this will be the boundary action is just given by the extrinsic curvature and one important point about this action that will be necessary for us is that it's a local action along the boundary so it's an action defined locally along the boundary so we can think of it as a particle that lives at the boundary and it's moving according to dynamics given by this action so this is the same thing so we have in general we have a portion so this is is there a question was there a question okay yes okay good so in general the interior of the spacetime will be the region that is inside this boundary curve right so different trajectories for the boundary boundary curve define somehow different cutout geometry so we should think of the of the bulk space as the space which is within this red curve sort of like taking some dough and cutting it with a cookie cutter so we have a cutout geometry which depends on the boundary curve and whose action is given by that action in terms of extrinsic curvature so the minimum of the action is just a circle so circular solution and the what we would call the ADM mass of the total energy of the solution is related to the overall size so the of this curve and the there is actually a family of solutions that are related by ideas to our geometry so you can put the circle centered on the center of ideas or we can move it around in in ideas but all the cutout geometries that we get in this way all have exactly the same form and so we should think of all these different solutions has been totally equivalent to each other so basically physically we have only one solution so this is the same picture so we have the trajectory of the circular boundary here in Euclidean space and as usual we can cut here at this moment of time reflection symmetry and go to the Lorentz solution that will give us a short ADS to black hole or worm hole so the trajectories of the boundary now they are denoted by these red lines and they will reach the boundary at some they will keep the boundary of ideas at some at some time so the proper time along this boundary is infinite so it takes an infinite proper time to get to the boundary of ideas here so that's the diagram of these black holes we should think of the region that prolongs here as a kind of probably some kind of singularity in the full geometry so we were deriving this from four dimensions would have a singularity here can you hear can you see my the mouse on the screen yeah so here in this upper part we'll see the singularity and that's the usual fendros diagram of near extremal black hole okay so one message that I want to convey in the next few slides is that the whole gravitational dynamics becomes very simple and so I'll try to explain what this implicit is so we have about we have fields that propagate on a rigid ADS to space so we have the matter fields that propagate on a fixed space time there are no interaction no gravitational interactions between these fields in the bulk so bulk observer will not see any gravitational interactions only interactions that we could have in the in the filter approximation and then we have some boundaries that move also in a rigid ADS to space following some local dynamical loss so we can think really of this is some particle that moves in ADS to and this is somewhat similar well it's actually identical to the UV brain in the Randall's syndrome model right so you you cut the space and now you have some dynamical particle moving here dynamical brain and that gives rise to some dynamical gravity now but and the whole dynamical gravity comes from the dynamics of this brain so what is that dynamics so for example imagine we have let's say the red line here represents the the original black hole and if and at some point we add some extra energy so we for example insert some operator in the filtering or boundary conditions for some bulk field and what that will do is it will create a bulk excitation that will propagate into the interior okay so that's that's what this blue line means is some bulk excitation now if we had not sent this bulk excitation then this red line would have hit the boundary here at the top corner here and the horizon of the black hole would have been where this red line red dotted line is or orange dot liners however because we sent in this excitation the boundary here gets a little kick outwards and this kick is essentially determined by momentum conservation at this vertex so we can think of the dynamics here as coming from some dynamics that conserves momentum in two dimensions locally conserves momentum in two dimensions so we send some energy inwards some energy and momentum inwards there will be some well we'll have to conserve momentum and the boundary is kicked outwards and because it's kicked outwards it will hit the boundary phadias earlier than it would have been would have hidden otherwise okay and that implies that the new position of the horizon is a little bit outside the old horizon right and this is the way in which we see the growth of the horizon when we send some excitation so we sent an excitation and normally the horizon sort of moves outwards and this is what we see in this diagram now we have something similar happening if there was a bulk excitation before and we and it hits the boundary it's also kicked outwards let's see what the this God Jeffers and wall interaction can do so what we are doing now is we are inserting in the path integral we're inserting something that involves this field field values at the two boundaries so at these two points and so we can approximate this this term in the path integral in terms of its expectation by its expectation value so we'll make this approximation and this approximation will be good if G is efficiently small so to make the effect big would make taking large number of fields and nor to amplify the fact so that's what's a side remark but in some approximation which is a controlled approximation we can replace this term in the by simply the expectation value here in the exponent and this expectation value we can think now of this as some kind of effective between the two boundaries it's a potential that turns on only for an instant of time you can view it as an impulsive force so it's a force between the two boundaries and the interesting aspect is that this force can be attractive so if you choose the sign of G appropriately then the force can be attractive and can pull the boundaries inwards and so it means that after the force acts the particle get gets kicked inwards and so it will take longer time now to reach the boundary of ideas at the horizon of the black hole now has moved inwards so it's a piece of the geometry that we could not explore before but after we turn on this interaction we can now explore so that means that after we turn on this interaction we can send if we send in some excitation it can reach other side now an interesting aspect about this nearly ideas to dynamics is that when this excitation is sent in the excitation doesn't feel any anything special so it can just fall in and go to the other boundary and without feeling any anything bad let's say doesn't be nearly shockwave or anything like that just gets peacefully one boundary to the other and well of course there is no contradiction with the wormholes being not traversable in general because we had put this this extra interaction between the two boundaries now here we've said that we've we've set up this interaction and one question one can ask is what's the most let's say economical way of let me this picture first and mentioned that so we here we can send the cat or we can send some quantum information between the left and the right side and the question is but of course there is no contradiction because also we need to send some quantum information to set up this double trace interaction we originally had now one one question one can ask is whether it is possible to also send the thing from quantum information via the interior by sending only classical information between one side and the other and this also can be done and this is a particular instance of quantum teleportation so it's a particular well this is how that quantum teleportation occurs in this situation so quantum teleportation of course is a general phenomenon that can happen in general and in this setup the picture for quantum teleportation is transmission of well of information or signals propagating through a warm call so here the difference relative to the previous protocol is that here we measure the field value on the left side and after so that might create some excitations here on the left side when we measure the field value and then after we know the field value we transfer the results to the right observer and the right observer acts with a unitary operator which is given by the classical measured field value that was measured by the left observer and then the quantum operator here on the right-hand side and here the the crucial point is that the point of view on of the right observer we essentially get the same picture so we get again we can think of there will be a force so we can take the expectation value of this operator now here in the exponent and again we'll get the key in words for the trajectory and we get the same physics on the right that we had before so we can also view we can have a small variant of action as a quantum teleportation so as an example of quantum teleportation through the warm hole now one question you can ask is well so well one one interesting feature of this is that the the information we send we send from the left to the right can involve one field while the information we send through the warm hole can involve another field it seems to be completely unrelated but something that should shouldn't happen is that we shouldn't be able to send too much information through the warm hole okay so if we here measure a few field values and we send some information that we can let's say quantify in terms of some number of qubits we shouldn't in terms of some number of bits let's say these are the classical bits that we transfer from the left to the right we shouldn't be able to send the number of qubits which is bigger than half the number of classical bits that we send from the left to the right that's the usual bounds for quantum teleportation now the so what is so in this whole picture the question is what is going to prevent us from sending to formation now notice that when we send in send so I'm going now to describe the physical effect that prevents us from sending too much information okay so if we try to send some information like for example we throw in this cat the trajectory of the boundary will be kicked a little bit outwards right so it will be kicked outwards and because it will be kicked outwards the distance now so after we send in the cat it will be kicked outwards relative to where it would have been if we hadn't sent in the cat and because it will be more distant the correlations between the boundary values of the field are going to be weaker because these fields are going to be further apart from each other and so the expectation value of this operator becomes smaller and then the attractive force that you had between these two particles will become a little smaller and so so now since it's the force is not so big so it might be that the cat doesn't doesn't make it out so it might be that because the force is weaker we will follow this dotted line trajectory as opposed to the blue line trajectory which would have been the force in the case that we did not send the cat right I hope that's clear so this is the picture for the the physical effect that limits the amount of information that we send in now all of these have been have been pictures but there is a precise formula that you can write down that describes all of this and the precise formula is simply a calculation that follows the steps given in the pictures but so I'm just going to present it just to show you that there is not just only the pictures but there is an actual formula so what we're interested in calculating is the two-point function between the excitation we put in on the left so this is and whatever we look at on the right in the presence of some interaction which is by this double trace operator so the steps we're going to do to to derive the formula is the following so first we'll Fourier transform the signal we want to say to send so this will be Fourier transform and then we are essentially going to evaluate this this correlator on a background with momentum p with moment with this momentum which is the momentum of the Fourier component and this effect that I was mentioning in the previous transparency of decreasing the correlation is related to the fact that will the actual position of these two operators in ADS will depend on the momentum via a p-dependent SL2 transformation on V so this there will be a relative SL transformation between the left and the right side and this this effect will be amplified by a boost or let's say chaos so they will have an amplification factor which is e to the t where it is the time difference between the time at which we send in the signal and the time where the double trace interaction is acting and then so we will get an extra phase in the this two-point function coming from the expectation value of v here in the exponent after considering this new new background and so this is roughly the the structure of the formula so this this p to the 2 delta e to the IP this would be just the two point function in the absence of any gravitational effect that's just the usual Fourier transform of the two-point function for the two-sided black hole the e to the IP is that means that the two excitations are on two different sides of the black hole then this factor of e to the minus IG comes from the expectation value of this term and here this whole factor comes from the expectation value of this term where we acted by this p-dependent SL2 transformation that as we had said decreases the the correlation so this factor here is bigger than bigger than zero so here I'm thinking of p as being bigger than zero well it is bigger than zero in the calculation and this effect is amplified by a factor of e to the t okay and it turns out that the amount of information we can send this is roughly g if we assume that the square of the operator so we can smear the operators a little bit over thermal wavelength and say the square the square of the operators one and roughly the amount of information bound that can send this roughly g now that that formula takes into account the facts of back gravitational back reaction so this effects that shut off the amount of information we can transfer but there is the simply more simplified limit where we look at this term so here I reinstated the g newton factor which I had suppressed in the previous transparency and then we can imagine that g newton is small and expand to first order in g newton so expand this term to first order in g newton and then we get the factor like this from the first order term here the zero third order term in the g newton expansion cancels this e to the minus i g and then the first term gives us this and this has well this whole term in parentheses we can think of this as some kind of gravitational time advance that e to the ip was somehow the factor that came from the fact that the two operators were at different different boundaries that were separated and this factor is positive and so it tends to reduce the effect of the e to the ip so it then tends to make the distance shorter and this is the effect that will make the one hole traversable and so the one hole really becomes traversable when this whole thing is bigger than one the whole term in the parentheses is bigger than one then we can really go from one side to the other now one one important point is that we can think of this as just a simple translation or an operate some one of the yeah I didn't mention but this momentum p is conjugate one of the s2r generators and so this factor that we get here can be viewed as the action of one of the s2r generators so that means that the signal does not feel anything as it travels from the left to the to the right side or said in a different way if you had a composite object that contains many particles they're all translated by symmetry and so they don't feel anything when you go from one side to the other so being teleported through one hole in this way it's a pleasant experience it's not a traumatic experience you might feel you're falling to a black hole but then you get rescued on the other side so this is also saying that shock waves in two dimensions are not not really felt for this reason so it's different than the shock waves in higher dimensions which they have no trivial transfers dependence and for that reason you feel a tidal force here you don't feel any tidal force so this in this nearly 80s to space time so now I'll discuss one quantum mechanical model that also has the same leads to the formula and that's this is the so-called is there a question yeah so again well that fragmentation is something that involves the four-dimensional structure so the fact that there is a two sphere and the two sphere can split into smaller spheres and so on but here we are considering essentially the gravitational physics of the single center solution so we're not taking into account that any effect like this that those would be non-perterrative effects from this point of view so this is a completely perterrative effect so as we see is well it involves well with this simple power of the Newton did that answer the question so I guess yesterday Vladimir gave a talk about the S-Y-K model so I don't tell you all the details so this this model with n-myrona fermions discussed by various people and so it has a has the advantage of being a simple quantum mechanical model with a finite number of degrees of freedom and at lower nashes so where lower nashes is defined in this way here the bottom of the message so here J is dimensionful coupling which sets the energy scales here in the Hamiltonian and if we are interested in inverse temperatures so times bet J which is much bigger than one so beta J is effective coupling of this model and we're interested in times which are relatively big compared to one but still small compared to n so the second inequality makes sure that the one over n expansion is still valid so this is a simple limit that we can study and in this limit we can model and we can analyze it using some large n techniques which are somewhat similar to the ones that are usually used to define over n models so we define a new variable which is essentially the expectation value is essentially the two-point function it is when the questions of motion are obeys the two-point function of the this original mariana fermion variable and then we can integrate out the fermions and get an action essentially in terms of g and this action has the feature that it's just linear in n so the n appears as a coupling constant or as one over h bar so the field g becomes a classical variable in the large n limit so it's similar to what happens in on models and this g function is a function now of two variables these two times and this action we should think of it as somewhat analogous to the bulk gravity plus matter action I mean it's not quite the doesn't have the locality properties that the bulk gravity plus matter action but it's somewhat similar in the sense that n appears well in the sense that becomes classical in the large n limit as in other examples of adsfd there is a particular function g that minimizes this action and this function g is at the long distances is a s l twaring variant so this is analogous to the vacuum ads to geometry and then there are a set of low action fluctuations around the solution and they are parameterized by a function a single variable which is sometimes called the reparameterization mode so we have the conformal so this g here now denotes the conformal solution which we can think of it as being 1 over t minus d prime to the power to that and and then we can the model develops an almost reparameterization symmetry where you can you can make those transformations that look like a reparameterization and these transformations have an action which is relatively small so smaller than the action of any other fluctuation and so the this these fluctuations have an action which is given in terms of function f in terms of a schwarzian action and so one can deduce this schwarzian action as being the simplest action that is consistent with with this to our symmetry and the locality of the well in terms of locality so it should be a local function of f now this this action is the same as the action for that u b boundary in the gravity description so in the gravity description i didn't say exactly what the action was i only said that it was a local action and it was with the s to our symmetry that implied for example the momentum conservation at the vertices that i discussed and here we have the same action so it's basically because it's determined from the same entries and the same principles so in other words we have some microscopic model in terms of majorana fermions and some interactions and at low furnaces it reduces to an action which involves basically one mode one mode that is important and this mode as the trajectory of a particle on the boundary of ads as we discussed before so it's given explicitly by this action so while this model but we don't know how this model reproduces the full bulk matter theory it does reproduce the gravitational aspects of the gravity in ads too so they have this they're in the same universality class at least in what respects to this low energy action so everything that we said before in the one whole context depended only on the motion of the u b boundary and the propagation in ads to bulk and we got the same action for the boundary and the other modes of g are conformal variant and they lead to correlators which are the same as the correlators in ads too at least at the free level and so we get the same precise formula for the two point function so we had some formula for the correlator of the fields here in the presence of some interaction and in the syk model where we could consider fermions for example and again some interaction among the fermions or if one is worried about an interaction that changes fermion number in one side one could put interactions which contain products well further products of fermions but we get exactly the same the same formula because it was the formula was completely determined by this sl2 symmetries and uh and the dynamics of this boundary mode now we we've discussed the this effect in quantum physics and so on but we can wonder where there is a similar effect in just ordinary classical mechanics so in in ordinary classical mechanics in fact that we can ask is there something similar so let's try to make a classical analogy so imagine that um we have uh two classical systems which would be the analog of the thermo field double so we start them at time equal to zero they have all the molecules all the particles let's say have the same positions but opposite momentum so we can picture this as let's say two cups of water so classical water not quantum water um in the thermo field double and then uh let's say we tap one on the the left one at some early time so some time before time equal to zero when the momentous and positions are the same and then at t equal to zero we let them touch each other and they transfer some vibrations and at the time t on the right the question is whether we feel a bump on the right cup or not right um so this is the this is the kind of effect that we saw with the wormholes and the kind of effect that we saw in the syk model and so we can ask whether this effect is present in classical mechanics or not i initially thought that this this effect is not present in classical mechanics but actually it is also present in classical mechanics so um so and the way to see it is the following so let's imagine a classical system and let's say just for the sake of the argument that at some early time we perturb one of the positions on the left side um and this is the position of particle number two let's say on the left side and at time equal to zero we cup some other degree of freedom let's say particle number one so x one is a matter coordinate we put this term in the Lagrangian or more precisely we put this interaction term in the Lagrangian and at time t on the right we uh measure p two right so the momentum of the second particle so here the index two is the same as the one we perturb on the right and it turns out that we find that it is displaced in a manner which is correlated with initial displacement so if we displace this in the positive direction then p two right is displaced in the positive direction so whole dynamics is complicated and chaotic and so on but there are there is this correlation um between the two sides so let's just show this so here this follows from the following calculations so first uh we we said that what we did was to displace particle number two on the left side right that will cause a displacement of particle number one at time equal to zero right so it's the amount of displacement is given by this derivative the change in in particle number one due to a change we did at some early time in particle number two now due to the interaction Lagrangian um that was of this form um this uh this this the fact that we make this displacement will lead to an extra force in uh the momentum of uh so this force initially so this was not doing anything because x one left and x one right were zero at time equal to zero but because we did a small displacement it will now this term will not be zero and will give a little impulsive force on the right system so it will change the momentum of particle one at time equal to zero and uh this momentum um so we're changing this momentum um and um and then the momentum due to this change in the momentum the momentum of particle two on the right will also change right so that's uh this is the final this whole product is the final change in particle two due to a change in uh in particle two on the left now we can represent this term on the left as a Poisson bracket between x one left and p one left right this is just this Poisson bracket here and we can represent this other one the second factor again as a Poisson bracket but um this Poisson brackets are essentially well they are they are going to be equal because the two systems are identical and the left system is sort of the time reversal of the right system and so that's that's why these two Poisson brackets are equal and we'll get something which is the square of something that means that the sign of the x p two displacement is correlated with the sign of the x two initial displacement that we had in the beginning okay and so this shows that we have a similar effect also classical mechanics and now another another point is that the magnitude of this whole effect is grows in a chaotic system so the the magnitude of this Poisson bracket so this Poisson bracket is something that will grow in a chaotic system according to um an exponential with a given factor which is the the apunov exponent so we have a sort of chaos fuel growth of this of this Poisson brackets and of this effect of course this growth saturates when the trajectories are not near each other and we cannot use this simple derivative formula anymore and that's when the effect saturates and it's not growing anymore okay so that was in classical physics so we have a similar effect um and now we'll discuss some relationships between this and the black hole cloning paradox so it is that the suppose you have an old black hole or black hole that is maximally entangled in a different system and Bob is someone who has access to the second system and to an infinitely powerful quantum computer but not to the black hole then Alice sends an m-bit message and waits for a relatively short time for the message to effectively fall into a back hole scrambled in time and then the idea is that Bob needs a bit more than m bits of hocking radiation from the black hole in order to decode the message so this was something that was pointed out by Haydn and Preskell so this is another picture so the idea is Alice sends the message and then there is hocking radiation coming out that Bob can collect and he can decode the message and then he can jump in the question is when he jumps in that does he see another copy of the message or not okay so this is the cloning paradox because the message seems to be in two places at the same time this is what we would like to analyze now so the cloning paradox is the fact that the message appears to be in the there is a spatial section of the spacetime where the message appears to be duplicated okay and we want to see whether this happens or not so it's again same picture so we have the old black hole that is maximally entangled with Bob's computer but now we're marching in a situation where Bob with his infinitely powerful quantum computer produces a second black hole that is maximally entangled with the first black hole this is hard to do and it's exponentially complicated as Harlow and Haydn have shown but we are going to assume that Bob can do this with his infinitely powerful quantum computer so now we have two black holes that are maximally entangled and so we have the black hole which Bob has access to and then the other black hole that Alice has access to and in addition we'll say that they are nearly 80s to black holes to apply the previous discussion so we have the left black hole which is part of Bob's computer and then we have Alice's black hole so Alice sends a message okay this is the Alice's message going from right to left then we this is just a restatement of what we've discussed before so Bob Bob here takes get some Hawking radiation okay that's important for example a measurement on the Hawking radiation once this results of the measurement to some early time here in his quantum computer and then the trajectory of the signal so if the message he does is just he measures the expectation value of some field as we discussed before and here we act with the unitary we were acting before in the teleportation protocol then the trajectory of this boundary will be attracted to the right and then Bob will be able to catch Alice's message right so this is a state figure of how Alice's message gets to Bob's computer it gets to Bob's computer through the wormhole okay and there is here no slides where Alice's message is duplicated okay and notice also that if we were to extrapolate so after Bob does this gets the radiation if we were going to if we were to extrapolate backwards the state we get after that since this trajectory got a kick inwards when we extrapolate backwards we'll have we'll see that we'll have a horizon here and Alice by extrapolating this backwards cannot get the message again okay so in some sense the message got to a vision of space time that was accessible by Bob but not by Alice so somehow we can say that the message left Alice's system and went to Bob's system so before the transfer Alice had the message but Bob doesn't and after the transfer Bob has the message but Alice does not have it now let's do something a little more complicated so now Alice sends the message with the machine so she sends a message and it has a machine such that after a while it sends the message right and for the sake of the argument let's say that it sends the message such in such a way that that will not hit her system anymore okay it sends the message deep inside the black hole horizon and if this happens then Bob can get repeats the protocol and instead of getting the message he gets let's say this empty machinery okay so in this case Bob did not get the message here on the on the left side now who has the message in this situation well something that Bob can do is Bob can extract this machinery and then he has infinite power on this system so he extracts the machine machinery and then evolves the system backwards in time and evolves the system backwards in time then he will eventually get Alice's message by evolving the system backwards in time so in this situation still Bob can recover the message okay so the point is that the process of extracting the message puts it out of reach from Alice the message is never duplicated in the bulk and there is no need to involve unknown transplankian physics to solve the non-cloning problem it's under understandable from standard rules of gravity on the wormhole geometry of course there is one thing you need to assume which is you need to assume something like ER equal to EPR or the fact that if you have the perfectly entangled wormhole you get the geometric connection so that's suddenly a non-trivial thing you need to assume but once you assume this you can understand how to solve this governing paradox so it's not well in terms there is a simple picture for the gravitational dynamics of nearly eight of two spacetimes and traversability has a simple operation and nothing special is felt by the traveler who's teleported from the left to the right and we have the same description in quantum mechanics like in syk and gravity in nearly eight of these two spacetimes and there is also a similar classical in order classical dynamics we said we discussed some applications to the cloning paradox and we also discussed how to think well we we could discuss also how to think about the process of information extraction from black hole and because we can view this process of Alice sending the message as extracting information from a black hole and so we can see that somehow the information we can view it as traveling through a wormhole now there are many questions so one is where there are other ways to extract information from a black hole in order to extract the simple information we send in do we need to go through the process of making the thermofill double and so on or there is a simpler way for example and perhaps the most important question is what is this telling us about the interior and extract better lessons about the interior okay thank you very much time for questions can you see Juan you cannot see the whole room can you see that person asking I'd like is it possible is there more known about the information carrying capacity of this wormhole construction you alluded to somewhat briefly at some point but is it possible for instance how many how many qubits or bits per second is it possible to sound and something about channel capacity to say something more precise yeah so we did not derive a precise formula by a precise formula I mean a part of two and so on but we derive we derive the bound that the the information we can send the number of qubits should be less than g that the coupling g that was appearing in action so we have some sort of order of magnitude bounds that well are consistent with what we expect right have you thought about setting up a similar experiment for a one-sided black hole yes there are similar effects that occur for one-sided black holes so black holes that form from from some specific fewer states and yeah we studied this with a student here in here and we'll probably write the papers one about this but it's very similar so the idea is that if you have one-sided black hole that has some expectation value for some field then you can use the disinformation about the expectation value to somehow slow in a way what happens is that you slow the formation of the black hole so you can extract some of the things that would have been behind the horizon they are now not behind the horizon or said in a different way you can modify the trajectory of this boundary particle so that it goes to the boundary of ads more slowly and so you can see more of the space time I also understand that almeri was thinking about similar ideas do you really need exponential level of growth in two glasses of water water was moving definitely laminar not turbulent water was moving you you had two glasses that yes the motion of water was laminar not how not turbulent do you really need this exponential growth yeah so the the exponential growth is useful to isolate this effect from other let's say gravitational interactions and so there are many one-of-one corrections to the dynamics and having this growth amplifies one of the one of the corrections which is the one that I've been mainly discussing now the the idea of the of the chaos in the water is the chaotic motion of the water molecules is the microscopic chaos that we have in the physics not not the hydrodynamics so hydrodynamics might be simple but below these dynamics you have some complicated chaotic motion of the water molecules there had been experiments of anti-colonel of two boxes of atoms at room temperature done by Eugene Tulsey now he is in his board so he was making an entanglement of two boxes this macroscopic number of atoms at room temperature imposed by laser beams but probably there's a chaotic motion of atoms or not was not really important for him it looks like more like your two glasses of water is laminar you simply transfer entanglement from one box to another and it was done on the size of one meter there was nature plate around 2001 or something is it not similar well I mean for well that you can certainly have quantum teleportation without chaos or anything so suddenly quantum teleportation exists without any appeal to chaotic dynamics so here the chaotic dynamics was to give you a relatively simpler protocol for transferring information so in general if you do quantum teleportation in a complicated system it's the the operation you have to do on the on one of the systems is complicated here the idea is that the operation you have to do is essentially the same as ordinary time evolution so it's chaos itself simplifies the teleportation protocol I mean if you want to say it in generality like this somehow it takes the quantum information that was in one that's a system one little piece and it spreads it everywhere in such a way just by producing that this double trace interaction between another subsystem you can still transfer the quantum information any other just a question of notation repeatedly your potential was multiplied by the interaction constant g but repeatedly after that you had exponential of gv do you intend to have two factor g no I that was the type sorry about that we mean you were paying close attention any other questions yes when the coupling can become very strong is it possible that trying to make the wormhole will essentially destroy ads to in a way similar to fragmentation in other context yeah so um well here the coupling being strong is the defective size of the gravitational effects and one of our n corrections right so we could imagine a situation so here we've always worked on in a regime where the coupling is weak in the sense that um the one over n effects were small there was only a one over n effect that is amplified by these chaos but even that amplification was not too large so in the language of the boundary trajectories you the the quantum corrections is that the you have to treat the boundary particles more as quantum mechanical particles so they there are more they are you have to include their way function so here describe them as classical trajectories right and the difference is you have to include the wave function and you know treat them as quantum particles um and that uh well we'll modify some aspects of what I've uh I've been discussing certainly and we've been analyzing such long times but um we can go to longer times and we that I discussed here and the effect this effect if you go to longer times and the ones I've discussed here but within the approximations we made it gets smaller so you can transfer less if you wait longer so there's a kind of sweet spot for teleportation which is sometimes where that corresponds to times where there is this chaos explanation amplification of the effect and then after a while the the effect shuts off and we we can almost see it in the formulas let me see if I find it I mean this is described in our paper but it's let me find the formula so here um if we go to very late times the this factor of g newton e to the t is becomes very large and so this whole exponential exponent here becomes zero right and we are left purely with a simple phase so e to the i g there's some phase and this phase implies that we can send some amount of information but not very large not proportional to g but proportional to g model to pi let's say of roughly speaking of order one so we see this turning off of the power to teleport and then uh further quantum corrections uh yeah this this is already some partial resumption of uh one of our n corrections now you're asking about more drastic corrections like non-perturbative corrections and so on yeah those probably will make life more complicated we haven't analyzed those but already for g large oh this is the phase the phase will oscillate ever more rapidly yeah yeah i see no other corrections so let's thank you