So I'll be talking about the traversable wormholes and some applications to quantum cloning. And the subtext will be the surprising simplicity of nearly ADS-2 spacetimes or the gravitational physics of nearly ADS-2 spacetimes. And the talk is based on some work with Douglas Stanford and Chenbin Yang, who's a student at Princeton University.

And it's also based on a recent paper by Gao, Jeffries, and Wall. 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 faraway regions in the same spacetime.

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 null energy condition holds.

So that is to say the integral of T++ along a null base 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 shockwave computation that involves this integral of the null energy condition. And if that integral is positive, then the time delay is 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's a maximum propagation of speed for signals. But today we will talk about some special situations where it makes sense to talk about traversable wormholes.

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-Schwarzschild 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 ideas that this solution is the gravity description or the dual description of a system of an entangled state 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 at the bottom of the transparency where we have some overall energy eigenstates with a factor similar to the thermal factor. Well, the same essentially as the thermal factor. So we have these two corporate quantum systems.

And the idea of Gauss, Jeffries and Wald, 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, 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 those circumstances? What happens when we introduce some interactions? Now, we can couple them in many different ways. However, Gauss, Jeffries and Wald 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 the 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 should think of it as a particular operator of that boundary theory. The operator is 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 boundary, this is a perfectly allowed operator.

So we can add this operator to the interaction as an additional interaction. We here have G as some coupling constant that we can take to be small. And then in the bulk picture, we are just adding this funny operator.

From the point of view of the bulk, it's a weird operator because it's a bilocal operator. It's a non-local interaction, at least from the point of view of the bulk. And such non-local interactions can create a state which has negative energy.

So if we just insert this at these two points, it will create... Let's imagine phi is related to massless connections. It will create shock waves or negative energy along this null race. And it will create negative energy if the sine of G is appropriate.

If we change the sine of G, it will be positive energy. But with the appropriate sine of G, it will create negative energies. 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 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 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. 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, how we describe it, how we describe the back reaction and so on. And it further with the goal. So the eventual goal is to understand the interior of the black hole. So, yeah, we study this phenomenon. This phenomenon happens in black holes in any dimension.

In fact, the guy Jeffries and the wall describe the particular case of ideas three. But as they pointed out, it happens in all dimensions. And we studied in the particular case of Neil geometries, 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.

We'll set up the calculation in such a way that we'll also show that it 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 quantum information in black holes.

So first, I'm going to review some aspects of nearly ideas to gravity. So the idea is to this nearly ideas to gravity is, for example, in well, it arises in any situation where we have an ideas to part of the geometry.

So, for example, if we have near extremal black holes and we and we focus on the low one ashes on the region of the geometry close to the horizon, that geometry is an ideas to space.

And it arises in any number of dimensions so we can have from five dimensions going to ideas to and so on. Now, it's important when we consider we consider perturbations that around ideas to propagate in ideas to to keep the leading effects there by way from ideas to.

That's why we call it nearly ideas to gravity as opposed to ideas 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 in theory and was also studied by American Polchinski. And this is the aggression that you see here in this second term. It contains some scalar field and and and well, of course, the metric, the two dimensional metric.

Now, one would have written an even simpler aggression, 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 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 not lead to a nice gravity theory. So we by adding this this other this other term and we should think of this other term as keeping.

For example, if we if we think of these two dimensions, these two dimensional gravity theories arising from four dimensions, then the field five would be essentially the area of the two sphere. So we can think of this ideas to as arising from a four dimensional space, which is ideas two

times as to so find out would be the area of the extreme of the area of the sphere. And five would be a small deviation away from that external value. And OK, 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 OK to consider this in addition here.

We've written the boundary term that we need to add in order to get the action to be well defined so that the variations of the action with respect to the metric gives us the questions of motion. This is usually the yeah, the usual boundary term that we have in four dimensional gear.

Five is the boundary value of the electron field. So here the questions of motion for five five here appears as a kind of Lagrange multiplier. One thing I didn't mention is that we could imagine adding a kinetic term for five.

And it's fine to add a kinetic term for five. 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 equation of motion for five, we find the metric is is ideas to.

So the question of motion for five fixes the curvature to be minus two. And so the metric is locally ideas 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 five or we integrate out five the field, the delta field five. 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 non trivial dynamics that we'll analyze in a second. So the only dynamical information will be the location of the boundary inside this rigid ideas 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 is moving according to the dynamics given by this action.

So this is the same thing. So we have in general, we have a portion of these two. So this is. Is there a question? Was there a question? OK. Yes. OK, good. So in general, the interior of the space time will be the region that is inside this boundary curve.

So different trajectories for the boundary boundary curve defined somehow different cut out geometry. So we should think of the of the park 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 cut out 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 the circles or circular solution.

And the what we would call the mass of the total energy of the solution is related to the overall size. So the of this curve and that there is actually a family of solutions that are related by ideas to our symmetry so that you can put the circle centered on the center of ideas or we can move it around.

In ideas. But all the cut out geometries that we get in this way all have exactly the same form. And so we should think of all these different solutions as being 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 charge of ideas to black hole or one hole.

So the trajectories of the boundary now, they are denoted by these red lines and they will reach the boundary of ideas at some they will hit the boundary of ideas 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 this 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 if we were driving these from four dimensions, we 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 pendrous diagram of near extremal black hole.

OK, 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 the simplicity is. So we have about we have fields that propagate on a rigid ideas to space.

So if 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 filter approximation.

And then we have some boundaries that move also in a rigid ideas to space following some local dynamical loss. So we can think really of this as some particle that moves in ideas to. And this is somewhat similar. It's actually identical to the UV brain in the Randall syndrome model. 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. And the whole dynamical gravity comes from the dynamics of this brain. So what is dynamics? So, for example, imagine we have let's say the red line here represents the original black hole.

And if and at some point we add some extra energy. So we, for example, insert some operator in the field theory 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. OK, so that's that's what this blue line means. It's 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 dotted line is. 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 of ideas at the earlier than it would have been, would have hit it otherwise.

And that implies that the new position of the horizon is a little bit outside the whole horizon. And this is the way in which we see the growth of the horizon when we send some excitation.

So we send an excitation and normally the horizon sort of moves outwards. And this is what we see in this 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 Gao, Jeffries and wall interaction can do. So what we are doing now is we're inserted 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, its expectation value. So we'll make this approximation and this approximation will be good if G is sufficiently small.

So to make the effect big, we may take a large number of fields in order to amplify the effect. So that was a side remark, but in some approximation, which is a controlled approximation, we can replace this term by simply the expectation value here in the exponent.

And this expectation value, we can think now of this as some kind of effective potential 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 gets kicked inwards.

And so it will take a 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 the other side.

Now, an interesting aspect about this nearly AdS2 dynamics is that when this excitation is sent in, the excitation doesn't feel anything special.

So it can just fall in and go to the other boundary without feeling anything bad, let's say. It doesn't feel any shockwave or anything like that. It just gets peacefully from 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 extra interaction between the two boundaries. Now, here we've said that we've set up this interaction.

And one question one can ask is, what's the most, let's say, economical way? Let me talk to this picture first and mention that. So 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 question one can ask is whether it is possible to also send the 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 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 information or signals propagating through a wormhole. So here the difference relative to a previous protocol is that here we measure the field value on the left side.

And 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 crucial point is that the point of view 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 inwards 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 the function as quantum teleportation.

So as an example of quantum teleportation through the wormhole. Now, one question you can ask is, well, so one interesting feature of this is that the information we send,

we send from the left to the right can involve one field while the information we send through the wormhole can involve another field. It seems to be completely unrelated. But something that shouldn't happen is that we shouldn't be able to send too much information through the wormhole.

OK, so 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 bound for quantum teleportation. Now, so in this whole picture, the question is, what is going to prevent us from sending too much information?

Now, notice that when we send, so I'm going now to describe the physical effect that prevents us from sending too much information. OK, so if we try to send some information, like, for example, we throw in this cut, 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 cut, it will be kicked outwards relative to where it would have been if we hadn't sent in the cut. 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 now, since the force is not so big, so it might be that the cut 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 cut, right?

I hope that's clear. So this is the picture for the physical effect that limits the amount of information that we send in. Now, all of these 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. So I'm just going to present it just to show you that it's 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're putting on the left, and

whatever we look at on the right, in the presence of some interaction, which is given by this double trace operator. So the steps we're going to do to derive the formula is the following.

So first, we'll Fourier transform the signal we want to send. So this will be Fourier transformed. And then we are essentially going to evaluate this correlator on a background with momentum p, 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 correlations is related to the fact that the actual position of these two operators in AdS will depend on the momentum via p-dependent SL2 transformation on b.

So there will be a relative SL2 transformation between the left and the right side. And this effect will be amplified by puss, 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 this two-point function coming from the expectation value of v here in the exponent after considering this new background.

And so this is roughly the structure of the formula. So this p to the 2 delta e to the i p, 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 i p means that the two excitations are on two different sides of the black hole.

Then this factor, e to the minus i g, 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 correlation. So this factor here is 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.

And it turns out that the amount of information we can send is roughly g, if we assume that the square of the operator. So we can smear the operators a little bit over a thermal wavelength and say the square of the operator is one.

And roughly the amount of information bound that we can send is roughly g. Now, that formula takes into account the effects of gravitational back reaction. So these effects that shut off the amount of information we can transfer. But there is the 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 a factor like this from the first order term here. The zeroth 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 i p was somehow the factor that came from the fact that the two operators were at different boundaries that were separated. And this factor is positive. And so it tends to reduce the effect of the e to the i p. So it tends to make the distance shorter.

And this is the effect that will make the wormhole traversable. And so the wormhole really becomes traversable when this whole thing is bigger than one. When the whole term in parentheses is bigger than one, then we can really go from one side to the other. Now, one important point is that we can think of this as just a simple translation or an operate.

I didn't mention, but this momentum p is conjugate to one of the SL2R generators. And so this factor that we get here can be viewed as the action of one of the SL2R generators.

So that means that the signal does not feel anything as it travels from the left 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 they go from one side to the other.

So being teleported through a wormhole in this way, it's a pleasant experience. It's not a traumatic experience. You might feel you are falling into 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 really felt for this reason.

So it's different than the shock waves in higher dimensions, which they have non-trivial transverse dependence. And for that reason, you feel a tidal force. Here, you don't feel any tidal force.

So all this in this nearly ADS-2 spacetime. So now I'll discuss one quantum mechanical model that also has the same, leads to the formula. And this is the so-called, is there a question? Yeah, please.

Say 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. So 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-perturbative effects from this point of view. So this is a completely perturbative effect.

So as we see, well, it involves this simple power of G-Newton. Did that answer the question?

So I guess yesterday, Vladimir gave a talk about the SYK model. So I don't need to tell you all the details. So this model with N-Myrona fermions, discussed by various people. And so it 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 at the bottom of the tensor. So here J is dimension full coupling, which sets the energy scales here in the Hamiltonian.

And if we are interested in inverse temperatures, so times beta J, which is much bigger than one, so beta J is effective coupling of this model. And if 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 analyze it using some large n techniques, which are somewhat similar to the ones that are usually used to define O-N models.

So we define a new variable, which is essentially the expectation value. It's essentially the two-point function. When the questions of motion are obeyed, it's the two-point function of this original Majorana 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 there n appears as a coupling constant or as one over h-bar. So the field G becomes a classical variable in the Russian limit.

So this is similar to what happens in O-N models. And this G function is a function now of two variables. It's 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 doesn't have the locality properties that the bulk gravity plus matter action. But it is somewhat similar in the sense that n appears in the sense that becomes classical in the large n limit. As in other examples of AdSFT.

There is a particular function G that minimizes this action. And this function G is at long distances is SL2R invariant. So this is analogous to the vacuum AdS2 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 one over t minus d prime to the power two delta.

And then we can the model develops an almost reparameterization symmetry where 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 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 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 action is the same as the action for that UB boundary, the gravity description.

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 S2R 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 lower edges, it reduces to an action which involves basically one mode, one mode that is important.

And this mode is 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, we don't know how this model reproduces the full

back matter theory, it does reproduce the gravitational aspects of the gravity in AdS2. So they have this, they are in the same universality class, at least in what respects to this low energy action.

So everything that we said before in the wormhole context depended only on the motion of the UV boundary and the propagation in the AdS2 bulk. And we got the same action for the boundary. And the other modes of G are conformal invariant and they lead to correlators, which are the same as the correlators in AdS2, 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 formula because it was, the formula was

completely determined by these SL2 symmetries and the dynamics of this boundary model. Now, we've discussed these effects in quantum physics and so on. But we can wonder where there is a similar effect in just ordinary classical mechanics.

So in ordinary classical mechanics, in fact, we can ask, is there something similar? So let's try to make a classical analogy. So imagine that we have two classical systems, which would be the analog of the thermofill 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 momenta. So we can picture this as, let's say, two cups of water. So classical water, not quantum water in the thermofill double.

And then let's say we tap one on the left one at some early time. So sometime before time equal to zero when the momenta 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.

So 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 effect wasn't present in classical mechanics, but actually it is also present in classical mechanics. So and the way to say 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.

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 X1 is some other 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 measure p to 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 to right is displaced in the positive direction.

So whole dynamics is complicated and chaotic and so on. But there is this correlation between the two sides. So let's just show this. So here this follows from the following calculations.

So first, 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 particle number one due to a change we did at some early time in particle number two.

Now, due to the interaction Lagrangian that was of this form, the fact that we make this displacement will lead to an extra force in the momentum of 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 we'll change the momentum of particle one at time equal to zero.

And this momentum. So we're changing this momentum. 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 this define this whole product is the final change in particle two due to a change in particle two on the left.

Now we can represent this term on the left as a quantum between X one left and P one left. Right. This is just this quantum bracket here. And we can represent this other one, the second factor again as a quantum bracket.

But this quantum brackets are essentially well, they're they're 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 quantum brackets are equal.

And we'll get something which is the square of something. That means that the sine of the X P two displacement is correlated with the sine of the X two initial displacement that we had in the beginning. OK. And so this shows that we have a similar effect also in classical mechanics. And now another another point is that the magnitude of this whole effect is grows in a chaotic system.

So the magnitude of this quantum bracket. So this quantum bracket is something that will grow in a chaotic system according to an exponential with a given factor, which is the Lyapunov exponent.

So we have sort of chaos fuel growth of this of this quantum 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.

OK, so that was in classical physics. So we have a similar effect. And now we'll discuss some relationships between this and the black hole cloning paradox. So is that suppose you have an old black hole or black hole that is maximally entangled 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 fall into a black hole, scrambling time.

And then the idea is that Bob needs a bit more than M bits of Hawking radiation from the black hole in order to decode the message. So this was something that was pointed out by Hayden and Preskill. So this is another picture. So the idea is Alice sends the message and then there is Hawking radiation coming out that Bob can collect.

And he can decode the message and then he can jump in. Question is, when he jumps in, does he see another copy of the message or not? 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 space time where the message appears to be duplicated. And we want to see whether this happens or not.

So it's again the same picture. So we have the old black hole that is maximally entangled with Bob's computer. But now we imagine 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 Hayden 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 there are nearly at least two 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. This is Alice's message going from right to left.

Then this is just a restatement of what we've discussed before. So Bob here gets some Hawking radiation. For example, a measurement on the Hawking radiation. 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 are 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. So this is a figure of how Alice's message gets to Bob's computer. It gets to Bob's computer through the wormhole.

And there is here no slice where Alice's message is duplicated. And notice also that if we were to extrapolate, so after Bob does this, gets the radiation, if we were to extrapolate backwards the state we get after that,

since this trajectory goes to kick inwards, when we extrapolate backwards, we'll see that we'll have a horizon here, and Alice, by extrapolating this backwards, cannot get the message again. 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 a 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 in such a way that that will not hit her system anymore.

It sends the message deep inside the black hole horizon. And if this happens, then Bob repeats the protocol, and instead of getting the message, he gets, let's say, this empty machinery.

So in this case, Bob did not get the message here 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 machinery, and then evolves the system backwards in time.

And if he 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 transplanting physics to solve the known cloning problem. It's 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 certainly an untrivial thing you need to assume. But once you assume this, you can understand how to solve this cloning paradox. Well, in France, there is a simple picture for the gravitational dynamics of nearly eight to two space times.

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 at least two space times.

And there is also a similar classical dynamics. We discussed some applications to the cloning paradox. And we also discussed how to think, well, we could discuss also how to think about the process of information extraction from black hole. And because we can view this process of Ali 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 thermofield 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.

Four questions. Can you see one? You cannot see the whole room. Can you see the person asking?

I'd like, is it possible to be, 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 qubits or bits per second is it possible to send?

Something about channel capacity, or is it possible to say something more precise? Yeah, so we did not derive a precise formula. By precise formula I mean a factor of two and so on.

But we derived a bound that the information we can send, the number of qubits should be less than g. The coupling g that was appearing in action. So we have some sort of order of magnitude bounds that are consistent with what we expect.

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 some specific fewer states.

Yeah, we studied this with a student here, and we'll probably write a paper soon about this. But it's very similar. So the idea is that if you have a one-sided black hole that has some expectation value for some field, then you can use the...

this information 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 if 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 Lyapunov growth in two glasses of water, water was moving definitely laminar, not turbulent.

Water was moving? You had two glasses. The motion of water was laminar, not turbulent. Do you really need this Lyapunov exponential growth? Yeah, so the exponential growth is useful to isolate this effect from other, let's say,

gravitational interactions and so there are many one over n corrections to the dynamics and having this growth amplifies one of the corrections, which is the one that I've been mainly discussing. Now, the idea of the chaos in the water is the chaotic motion of the water molecules.

It's the microscopic chaos that you have in the physics, not the hydrodynamics. Hydrodynamics might be simple, but below this dynamics you have some complicated chaotic motion of the water molecules. There have been experiments of entanglement of two boxes of atoms at room temperature

done by Eugene Polozin, now he is in his board. So he was making entanglement of two boxes with microscopic number of atoms at room temperature imposed by laser beams, but probably there the chaotic motion of atoms or not was

not really important for him. It looks more like your two glasses of water with luminary. You simply transfer entanglement from one box to another and it was done on the size of one meter that was nature paid around 2000 won or something.

Is it not similar? Well, I mean, 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, the operation you have to do 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 chaos itself simplifies the teleportation protocol. I mean, if you want to say in generality like this, somehow it takes the quantum information

that was in one system, one little piece and it spreads it everywhere in such a way that just by producing this double trace interaction between another subsystem, you can still transfer the quantum information.

Any other question? Just a question of notation. Repeatedly your potential was multiplied by the interaction constant G. But repeatedly after that you have exponents of GB.

Do you intend to have two factor GB? No, that was a typo. Sorry about that. We miss you, we're paying close attention. Any other question?

When the coupling can become very strong, is it possible that trying to make the wormhole will essentially destroy AdS2 in a way similar to fragmentation in other contexts?

Yeah, so here the coupling being strong is the effective size of the gravitational effects and 1 over n corrections. So we could imagine a situation, so here we've always worked in a regime where the coupling is weak in the sense that the 1 over n effects were small.

There was only a 1 over n effect that is amplified by this chaos, but even that amplification was not too large. So in the language of the boundary trajectories, the quantum corrections is that

you have to treat the boundary particles more as quantum mechanical particles. So you have to include their wave functions. So here I describe them as classical trajectories. And the difference is you have to include the wave function and treat them as quantum particles.

And that will modify some aspects of what I've been discussing, certainly. We've analyzed such long times, but we can go to longer times than I discussed here. This effect, if you go to longer times than 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 is a kind of sweet spot for teleportation, which corresponds to times where there is this chaos exponentiation, amplification of the effect, and then after a while the effect shuts off.

And we can almost see it in the formulas. Let me see if I find it. I mean, this is described in our paper, but let me find the formula. So here, if we go to very late times, this factor of g newton e to the t becomes very large.

And so this whole exponential exponent here becomes zero. And we are left purely with a simple phase, so e to the i g, the sum 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 mod of two pi, let's say. Of, roughly speaking, of order one. So we see this turning off of the power to teleport. And then further quantum corrections.

Yeah, this is already some partial summation of one over 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, the phase will oscillate ever more rapidly.

Yeah, yeah, that's right. I see no other corrections, so let's thank Jane for this.