Quantum Gravity & Quantum Information — Closing Session

0:00 Indeed, you know, I was very impressed by this loop quantum gravity program, which seemed to go a very long way, and it was informed by people with loop quantum graphs, which are not there yet, and actually doing the proper gauge fixing to quantize gravity canonically. But I'm not going to do a canonical quantization for gravity, nor is the string figure anything like that. Okay, so let me actually, and I did this in rather, basically a simple-minded way, because that's the only way I can fix it. But let me remind you about general relativity and how Einstein brought general relativity in the first place. And actually, the original papers on this are very illuminating and fun to look at. I mean, for instance, if you imagine, suppose you imagine a bunch of particles moving through space-time, bouncing off of each other, and then going to other places. Okay, this is what Einstein imagined, and each of these is an event. And you can label the events. This is event one, this is event two, this is event three, this is event four, et cetera. Okay? And starting from the principle of equivalence, you know, that the, someone in an accelerated frame, being in an accelerated frame is indistinguishable to first order from being in a gravitational field. Okay? And then also, starting from the principle of general covariance, which we'll describe right now, Einstein's equations. And the principle of general covariance is that your underlying theory, in this case general relativity, should be independent of how you assign coordinates to this event. Or in contemporary language, people say it's independent of how you embed the events in space-time, which means it's diffeomorphism and variance. So if I embed this in some funny other way, so here these things go, here's the first collision, here's the second collision over there, I embed this in some funny other way, using another funny other coordinate system, all right, then here's the same set of events embedded in a different way with a different coordinate system, but they describe the same underlying physics, okay? And what Pottini, Margopula said yesterday, that that's a, so Einstein called it general covariance. It's also called dichromorphism of variance because

2:30 it's invariant under, they fear should be invariant through transformations of space-time. And Pottini yesterday called it background independence. And what Einstein showed was that if you take this principle of general covariance and you demand that your theory depend only on the Lagrangian of the theory depend only on the first order on the metric for space-time, so the metric tells you the distances in the space-time, and that your term for the gravitational action depends only on the second order on the metric, and this implies Einstein's equations. This is the energy momentum tensor for the matter, and these are, this is the Ritchie tensor, the Regine B, Regine Scaler, the famous, infamous cosmological concept, okay. And so the terms on this side depend only on the metric, the terms on this side depend only on the matter, and potentially on the metric. Okay, so that, Einstein's right, this is the principle of general tolerance. Okay, and this is the end of the review of general relativity. Okay, so how do we make this quantum mechanical? Well, as I said, I rather slavishly and simple-mindedly followed this picture. And wherever Einstein talked about... So, you know, this famously tells you... What this says is that in Wheeler's words, if I look at the t mu nu as a source term, then this says that matter tells space-time how to occur. But at the same time, I can look at the curvature terms of the source term for the matter. And then that says that space-time tells matter where to go. Because this includes the kinetic energy of these particles. Okay, so slavishly turning everything into the realm of plot of information, all I really did was, wherever Einstein says, you know, matter. So we deduce the metric by looking at the trajectories that matter follows through space.

5:00 space. Then I simply inserted quantum information wherever matter occurred in that. And of course you can get a picture like this. In fact, quantum computation looks like this. I have input quantum bits, which can always be taken to be in some fiducial state, zero. They interact each other, reporting some quantum logic interactions, operations, sorry, and in the end, I can compare them to some output states, which can also be the same set of fiducial states. And here are some quantum logic operations which tell you how these qubits, or qtrits, or qdits, or qnats, they could you know, two, three, or four continuous variables, if you want, which tell you how they interact with each other at these points. And of course, if I were to choose to embed this in space-time, I could, the computation doesn't care how I embed it in space-time, right? So if I were to actually say that this describes something happening in space-time, then I could equally well say, you know, here are these qubits going right here. I embed the same computation in a different fashion and the amplitude for this computation and the way the quantum information behaves is the same. I mean, it's just a fiction really to embed it in space-time. So the quantum information doesn't care, it doesn't know that it's being embedded in space-time. So if I take this computation embedded in space-time and then attempt to derive a metric from that, which is what I'm going to do now. If I do that, then whatever theory I make will also be generally covariant. And that's the central insight into what I'm going to do here. Having realized this, then because of these rather stringent theories that tell you if you have a generally covariant theory that only has low-level, low-order dependence on what happens with these logic gates and the metric, get Einstein's equations, or in our case, Einstein's equations in a discretized context, the Einstein-Retri equations. So the first, I'm going to try to derive a spacetime, actually we'll see it's a class of spacetimes, for each quantum computation. And then I'll show

7:30 that each of these spacetimes obeys the Einstein-Retri equations. So you can think then of the Einstein-Retri equations as a way of making sense of a quantum computation should you choose to embed that Okay? But that's the key insight, that if I, you know, the quantum computation doesn't depend on how I embed it in space-time, it is background-independent, and therefore it's this theory. Whatever theory I'm going to derive from this has to be generally covariant. So it would actually be rather strange if I didn't get the Einstein-Hertz equations. And that's why I actually felt confident embarking on this approach. And in just a second, And I'll tell you just how you do this to verify. So you have to trust but verify, right? So you have to verify that you actually get the Einstein-Greggie equations. And I'll tell you how you do this quite explicitly. But before I do that, is this okay? I mean, this is, yeah. No, it is not. Because this theory should also have local lawrence invariance. Local, what, sorry? Local lawrence invariance. Ah, yeah, yeah. It does. Right. This is very good. So, in fact, thank you. Thank you. so this theory should the gauge groups of gravity the transformations that don't change the theory so the gauge groups of gravity include the demorphous invariance and local Rett's invariance and I'll show why this is locally Rett's invariance in just a second so that's a very good point, yeah it is locally Rett's invariance but it'll tell you why it was locally Rett's invariance alright any other questions about this program? If you don't buy this, then you might as well, you know, this is a really important point. So by all means, ask questions about it. Yeah? Well, can you set the initial state to be a tensor product of zeros? Oh, well, it doesn't really matter what the initial state is. But, you know, any quantum computation. So if I look at the amplitude for the quantum computation, I can write it as, if you like, I can write it as, transformation so suppose I want to calculate you know the first billion bits of pie right first I take some zeros and I program in the program that calculates the first bit the first I just apply then I apply you know my computation and then I check to see what the answer is so any amplitude for a long computation can be written as just the some fiducial state all zeros and

10:00 in fact I frequently use all singlets rather than all zeros and for instance this is a network project, then some unitary transformation constructed with quantum biology operations ending with all zeros. And who's going to look at the result? Who's going to look at the result? I don't know. Us, right. So actually the observers in the serial force will have to be abandoned in the confrontation. I'm not going to want to get theological here. Well, what does it mean again for it to be generally covariate? Is this a regular grid of or is it some arbitrary conviction? Yeah, so actually, now, right, so a good question is which graded gates do you use? Well, this will well describe the work for any quantum computation. So any quantum computation will correspond to a space-time. But of course, if you want to say which one corresponds to a universe, then some observation is probably necessary. I'm a big fan... But what does it even mean for it to be co-variated and say it's rather regular? It will still be co-variated. You can squish it all around and make it look very irregular. It won't matter. I take some regular array and I squish it all around everywhere. The computation looks fine. It doesn't care that it's regular or irregular. What would be an example of an array that isn't covariate? There are no arrays that are... This is independent of the array. Covariance is independent of the array. The important thing, covariance is a feature of how... The question is, if I choose to embed these events in space time, care how they're met in space-time? And the answer is, unacrobably, no, in terms of both these events. Actually, it was less clear in this case. So that was a great discovery of Einstein that it wasn't true in this case. But it's clearly true in this case. Yeah? Are your lines, which I guess in some sense are between the gates of the internet or time, in this gate that happens? Yeah, yeah. I mean, one wouldn't expect that it necessarily holds you on the planet. Well, so this is a directed graph, and you can actually make cyclic graphs that have cycles in it in quantum computation, but in my case I'm just going to look at the conventional way of thinking of it as a directed acyclic graph. So it's kind of classical in this? No, all aspects are meant to fluctuate, but there is a causal structure to this graph. So the lines can move?

12:30 Well, yeah, look. Time goes like this, right? This is general relativity, so time goes up, and our spatial dimensions are going to go sideways. I love general relativistic notation, you know it. That's what I was asking if those lines are... They're wires. Information passes along them. and at the end of the space time they'll represent some relatively local place for which quantum information passes actually as we'll see in just a second let me actually explain as we'll see now these wires, these should be like-like wires so these should be null null lines of the metric so let me just let me just each of these views is equal to some set of logic operations I'll tell you right now, we'll tell you a lot better of what's going on. So these are the quantum logic gates. And for example, use of L, here's a famous, a famous quantum logic operation. Sorry. It's the square root of swap. So this is just swap. It swaps the inputs. So we'll call it S if you like. when I do it like this, because it swaps the input from one place to another. And you can combine swaps with, for instance, single-cubit gates to do a universal computation. Now, it's not going to matter very much which computation we pick. If we pick an infinite graph, then most of these graphs are going to be able to contain all possible computations of some subgraph. Okay? But here, this is the first part. So is everybody happy with this? This is a gate that takes, there's a superposition of doing nothing and swapping these bits that go in. This is a typical quantum logic gate. And now you see, and this is an important point, that this graph, even though I've only got one computational graph, that a single computational graph, if I put, for instance, a square root of swap here, has different ways of rooting the quantum information through the graph. So, in fact, if, for instance, swap, then this graph is actually can be written as a sum of terms, each of which corresponds to a specific way of rooting the information through the graph.

15:00 Which means that actually, even though it looks like I've drawn one monolithic quantum structure, causal structure, in fact, such a computation is typically the superposition of different causal structures. so the computation is actually a sum of different causal structures that's why I said that's what I meant when I said that my computation will actually rather happily on its own sum over different possible causal structures for the system. So I'm not actually going to bother, as Renata did two days ago, to actually explicitly sum over different causal structures and computation with that for me. Okay? And each, so for instance, each x, each or each u, I can also write it in this form for each u or for instance this x gate can be written as a sum of e to the minus i plus of k e sub k e sub k so I just write this in diagonal form and it's energy eigenfaces and what we're going to do for example this swap gate is famously equal to the projector onto the triplet minus the projector onto the singlet. And so these phases, phi is equal to zero or pi. And so what we're actually going to do is we're going to identify phi, or h bar phi, phi sub k, will be identified with the action associated with a gate, action of gate. And you see that each gate actually is a superposition of terms, a sum of terms, each of which has a definite action, a definite angle of Hilbert's face rotated in that gate. And in the case where I have this 1 plus swap, I have something where nothing happens,

17:30 so there's zero action in the one part. And then this swap part either has nothing happened, or I get a face of pi. Okay? So I'm going to identify in this picture, right? Remember, the picture is that matter tells spacetime how to curve, and then spacetime tells matter where to go. So I have to actually say what I mean by matter. Now, matter is given by, to describe how matter behaves, I need to specify the action of the matter locally. And indeed, you can specify Einstein's equation by specifying the action of this matter as it exists through space-time. And so I'm going to explicitly just identify the action in these quantum mechanical logic gates with a phase of the rotation of the individual energy eigenstates. And this indeed is what action is for some local quantum system. It's just some phase accumulating in filtered space. I'm not trying to do anything fancy here. I'm just doing something that's normal. Okay, so this says that we now know the action of the computational matter fields. This could be, I don't know what you want to call it, quantum computronium. Though actually, I prefer to call these things that are zipping around, you can call them qubits if you like, but they don't have to be qubits, they can be qubits. And I find that whenever I type the word information, because I type it so frequently, it comes out as inframatino. So, the action of these inframatinos, whatever you want to call them, because we don't have to be with it. Okay, so actually, I now actually have half of what I need in order to derive Einstein's equation, or the Einstein-Greggie equation, because I need to, basically, I have all the material I need to get T mu nu, the energy momentum tensor, because the energy momentum tensor is derived directly from the action. But now I have to tell you more about how to get the metric. Okay, now are people happy with this little bit of quantum information and the idea that a quantum computation is actually a sum of different routings of information? This has been, that's true classically as well, right? Well, except that it's not a sum, classically. Classically, you have a graph that's in route information in all sorts of different ways with a circuit.

20:00 So in other words, so upon computation, as a sum of what are called histories, and each history has some structure, here for instance these guys got swapped, here in this case In this logic gate over here, I just took the I part, so these guys go straight through over here. Here they get swamped again, and here they, well, let's say they get swamped again here. So, each and each history has a definite causal structure plus definite phase at each vertex. I'm just going over two cubic gates here I think you could do it with one cubic gates Or you could do it with three cubic gates But people don't normally do that Okay, so actually Now I see this right here This is I summarized just a rather quantum computation over here. I haven't told you much about the metric yet, and I'll now tell you, in particular, about both the lorex invariants, which we'll get to next. But I'm really happy with this. This is pretty conventional quantum computation. I'm just, you know, there's a causal structure. The causal structure, if I put in things like swap, I can put in other things other than swap. There's a sum of different causal structures, and I can decompose it further and have a definite angular rotation of each quantum on the gate. There's nothing fancy about this. I'm just decomposing a quantum computation in a way that will allow me to make contact with Einstein's equations. And what we're going to get is each history each history corresponds to a fixed metric and that metric is called the Einstein ratio equations.

22:30 So that's the sense. So now you see, of course, that I will have fluctuations in the metric. There will be fluctuations in the metric for two reasons. One, because of the fluctuations of the causal structure. And two, because of fluctuations in the action of the matter fields. And that's why this is going to be a... I'm not quantizing gravity, remember. But this will be a quantum theory of gravity because my metric will exhibit quantum fluctuations. These quantum fluctuations will in fact track the behavior of the underlying quantum information. Okay. Everybody happy so far? It seems like everybody's jetty when it's 10 in the morning. I have the advantage that I'm talking so difficult for me to fall asleep. But I could probably do it if I try. Okay, so I'm going to take the stony silence as a yes, everything is fine. but now I need to find a eraser. Okay. So let me get rid of this and see what we want. So now let's get the metric. Add a vertex. so a vertex looks like this I'm going to suppress the phases and the little boxes and stuff like this it has four lines going in sorry, two lines going in and two lines going out and I will identify these with null lines of the metric, whatever metric I have, if this has some if this is in the embedded space this is some vector delta mu mu nu delta mu delta mu is equal to zero. Why do I identify these with null lines? Because remember, action is the angle accumulated. And these lines are the identity. Nothing happens to the quantum information there. Zero action is accumulated. And in general relativity, lines of the problem with zero action accumulated are null lines of the method. Okay? So these are null. Now this feature, which is, in fact, I don't know how long it's been known for, but in half a century at least, this means that... This determines, actually, almost the whole metric

25:00 up to a conformal factor. So if you know four null lines going in at each point, you know the metric at that point, up to an overall length scale. Sorry for people in the back who can't see. Let me put this up like this. And this is called the conformal factor. I'll write this up on the next board. So let me write it up again. In the back, you can't see. And particularly since Ed Hope asked this question, and this is answering his question, I notice that he can't see this board. So if I determine, this is just a standard feature of classical relativity, general relativity. And there is a local Lorentzian structure for this system, and it determines the metric up to, so here's this picture, four in all lines at each point, and point L determines G mu nu of L up to G mu nu goes to omega squared G mu nu. So, the light columns tell you where information can go, and that determines almost all 9 out of 10 components of the metric. And it clearly can't tell you what the conformal factor is, because, you know, the light lines don't care what the conformal factor is. But it determines all of the metric. And it's this determination of the local metric that actually gives you local Lorentz invariance. Oh, not invariance. Lorentz covariance, right. Local Lorentz covariance. but you know there's something like a vacuum state that's invariant in the local world and you still have to show it as such a state oh yeah, I'm not actually, yeah I should say, this is a very incomplete program of research so I'm actually not going to say what the vacuum state of all the space to find mass I'm going to content myself here, if I have time, which I probably do with simply demonstrating that every computation bottom computation corresponds to discretized space-time. And then, of course, you have lots and lots of choices as to which quantum computations you're going to look at. And, yeah, so this question about the local Lorentz and Lorentz, the active, like I said, but the local, so the Lorentz, um, the one of the gauge invariances of gravity is local Lorentz and Lorentz.

27:30 All of these are invariant under local Lorentz transformations. And that will be true here, basically, because we've used this, uh, we've used the fact that these lines are null to define the metric in a way that's consistent with actually local special relativity. Okay? So actually, now you can see we're in pretty good shape. We actually want to, we know, so... We know g mu nu almost holds up the metric up to local omega, which is a function of the different points. and we know the action the action of our computational matter fields and almost we know the action we don't necessarily know the whole energy momentum tensor because the action corresponds to basically the scalar part of the energy momentum tensor and it corresponds to the scalar part of the energy momentum tensor we still don't know about the kinetic energy part of the energy momentum tensor and indeed we're going to derive the kinetic energy part of the momentum tensor by looking at the Einstein-Gregg equations because, famously space-time tells matter where to go and so we know about space-time we're not actually going to figure out what this energy momentum tensor is though of course we know a lot about it because we know that this matter is going along these null lines of the vector from earlier. Where do you get mass from this? Even though this looks like a bunch of light-like particles moving around, their well-known constructions do originally define that, and then investigate by lots of other people since, that show you how you construct things like the Dirac equation by having a bunch of massless spinners going around and bouncing off of each other at points of a lattice. Indeed, if I just take a lattice, put in square roots of swap at every point, you'll find that what you get is like just a gas of fermions moving around. A massive chromosome. So let's go to, now we want to go to in order to make this explicit now, I have to go to regicapulus. We're almost at regicapulus, but we have to derive it. So I should say, now we can derive

30:00 regicapulus. Okay, so what is Regi-Coglose? Regi-Coglose is a very lovely, discretized version of general relativity. There's a discrete version of general relativity, and we need a discrete version of general relativity here because our system is fundamentally discrete, right? It's discrete, okay? Now, the way that Regi works is you divide everything up into triangles, or actually simplices. Excuse me, before I say that. This feature about the null lines is that this actually says that we should embed this into a four-dimensional manifold. That's a natural choice. Of course, three spatial dimensions and one temporal dimension. Because, and this is actually related to Penrose's spin network theory. But why? It's because in four dimensions, this quantum logic gate completely determines the metric of a different formal factor. and fewer than four dimensions that over-determines the metric which is that and more than four dimensions that under-determines the metric so you could say that actually that the four-dimensional the fact that we choose a four-dimensional manifold comes from the fact that we need pairwise interactions between our information that is why we get four dimensions here so there's a natural choice of four dimensions here I'm not going to have it emerge otherwise I'm simply going to choose it to be four dimensions because it's a natural thing okay so back to regi calculus so what you actually do here is you this is not a simplicial or triangular it's a graph a lattice but it's not i've embedded this graph in space time to make it a lattice but it's not a simplicial lattice it's not made up of little triangles well that's fine there's a standard way a standard way to uh uh embed things if i take a graph with a bunch of points in it, there's a standard way to embed it in, to complete it, to make a simplicial lattice. And this is called, let me do a thing. I'm losing my erasers. Oh, there it is, right up there. So we construct, construct, a simplicial

32:30 triangular this is called the Delaunay lattice it's a standard construction and then actually once we've constructed this Delaunay lattice then there's a dual lattice which I'll draw in lines like this like I'll draw a piece of it and this is called the Boronoi lattice and this Delaunay tells you about distances because these are going to be the lines that we're going to assign distances to so D for distances and Boronoi tells you about volumes gives you distances. And B gives you volumes. And there's a very nice and elegant, I should say, construction of regi-typus in terms of these Delaunay and Boronoy lattices by Warner Miller, who used to be Los Alamos and now somewhere in Florida. And I'm just going to follow that. But I should say that my whole strategy and all this stuff has been to take the absolute easiest course that I possibly could at each point. Because this is obviously a wacko thing to do. and quantizing gravity is hard, that's the first thing I noticed. So I chose not to quantize gravity, that makes things much better. And then, at every point along the way, I'm using all standard techniques at every point along the way to try to derive this. And this is a standard technique. Okay, so once I've constructed this lattice, So once the lattice is constructed, now we assign h-lengths by just taking, say, the distance from, say, 0.2 to 3 right here, this distance right here, let's call it delta to 3. I just take the average of the two metrics so the average of the metric well I'll just say GAB one half GAB at two plus GAB at three and then I'll just take this

35:00 delta A delta B so I'm just going to take this vector and I'm going to assign it a length that corresponds to this now I haven't told you what the conformal factors are yet but once I fix the conformal factors then my entire space set will be fixed and I'll tell you what the average length is there everybody happy with this? I agree this is somewhat technical and I would never have done it if I felt that I didn't have to But remember, all I'm doing is I'm explicitly telling you why it is that one of these spacetimes, or each one of these causal histories, actually corresponds to a spacetime. And last night, Fortini was saying, well, why are you doing all this sort of calculus stuff? I mean, you have this beautiful background independent theory. Why do you have to embed it in spacetime? And the reason I have to embed it in spacetime is because it's not clear if I look at something like this, that this actually is something that obeys the Einstein-Reggie equations. So the only reason I'm going through all this stuff, this explicit construction, is just to tell you, to back up my claim, that because this is a generally covariant theory, it obeys the Einstein-Reggie equations. Okay, that's why I'm doing this. So I apologize if it's a little detailed, but that's okay. So we've not yet fixed omega, but when omega is fixed, that the whole geometry is fixed. So the whole discrete geometry, this Buckminster Fuller Geodesic Go geometry is fixed. Geometry. So once I fixed Omega, then this discrete geometry is fixed. And in particular, the gravitational action is equal to the sum over all triangular hinges, H. So to get the curvature here, you actually have to look at hinges. So hinges in the two-dimensional architecture just points. So in the two-dimensional architecture, what you do to get the curvature is you just look at this point, is just given, you look at the deficit angle. So you look at this, theta 1, theta 2, theta 3, theta 4, theta 5, theta 6.

37:30 And epsilon, so the deficit angle, is just equal to 2 pi minus the sum of these theta. Okay? And that's why you do this in two dimensions. And it's slightly trickier to visualize. In three dimensions, that hinges a line. angles around the line and here a hinge is a triangular one of these triangles I know this is not an easy plot to look at but that's okay so the gravitational action is the sum of the hinges is a 1 over 8 height and then I just take the deficit angle for this hinge times an area for the hinge and this is a completely standard construction for the calculus Okay? Now, remember, what is our matter action? Post-matter, or computational matter action. Well, the matter action is equal to h-bar times the sum over the gynecologic base theta sub-l. Right? Remember, that's when we were defining the action. This is a local angle that gets rotated in filtered space. to write this in more familiar kinds of terms. Now I've defined these volumes. I have a volume associated with each logic gate. And you typically write the matter as the integral of the Lagrangian over the volume. So I'm going to write this as this. It's h bar times the sum over L. And here's a kinetic energy term. So this is a kinetic energy term. And then I'm going So I'm going to write this as minus, I'm going to take a one-half here, minus U sub L, where U sub L is just the, well, I shouldn't use U sub L, it's V sub L. This is just the potential, so this is KE, and this is potential energy, and then I'm going to have delta V sub L, where V sub L, this is just the volume at that point. Okay, so I'm just writing down, I'm using the prescription of Einstein, and I'm going to write down the action, because I know what it is, but the action is supposed to be a first-order function of the metric, a zeroth-order function of the metric and a first-order function of the metric.

40:00 And so I'm just writing it down in conventional form to try to relate it to what the matter looks like at each point. And this term right here, this is going to be zero, so it actually doesn't contribute at all. Why is that? for a bunch of things moving along null lines is zero. The trace of the kinetic energy term. If you just think of, you know, things like if you have a, just take the f mu nu for the stress energy tensor for maximum equations, it's traceless. And that's really because, you know, the kinetic energy of something going along a vector like this is proportional to the length of this vector, but the length of all these vectors is zero. and I'll call this this is equal to I for the matter okay I'm running out of time so let me actually just complete this construction I'll complete this I'm sorry but I shouldn't the preface should have gone so much into these details but I want to convince you that you can fill out I've had a claim that you get Einstein-Regi equations then I just want to show you how you do it. So now the Einstein-Rege equations are as follows. You just take the derivative of the gravitational field with respect to the metric and then you add the derivative of the matter field with respect to the metric and set this equal to zero. This is a conventional construction of Einstein equations if I do this in a continuous case and also the Reggie equations. I just take the derivative of these extra and set it to zero. So in other words, I get dG mu nu is equal to minus dG mu nu. So now I put the matter products on the right-hand side and the gravitational products on the left-hand side. it as I come out with the discrete analog to bar of u nu minus 1m g, I guess this should be upper efficiency. And then on this side, I get 8 pi g times t hat u nu minus u g nu

42:30 Oh, I think I should, this would be minus 7 here. Okay? So actually, it takes a very conventional form, these equations. Now, I haven't shown you that they are made because we still have three parameters in our theories. And the step is follows. Step one, the step one is matter will tell space how to curve. So the first step is going to be picture curvature at each point. so that the trace of these equations and the trace of these equations actually just say minus R is equal to this thing right here is the overall energy momentum tensor is equal to A pi G T mu mu and this is a so this I haven't written it down explicitly you can write it down explicitly if I'm almost out of time and this is a non-linear partial difference equation relating omega to theta. So theta are the source terms, the matter terms, and they tell matter how the curve will fix the curvature are at each point. So that's the first step. And then the next step is note that this point right here, remember that T mu new hat is traceless, so T mu new hat doesn't enter into this equation at all. I haven't done anything about T mu new hat. T mu new hat is a kinetic energy term, and I don't know what those are yet because, in fact, before I know what the metric is, I don't know how to assign kinetic energies goes as they zip around. But now I'm going to do that. So two, based on tells matter where to go, we just pick the kinetic energy so that the rest of the remaining terms. The rest of the Iosine-RT2 equation are made.

45:00 And now, actually, so now, what I've shown is basically, I mean, I admit that this is first of all, you have to write down this explicitly, do it and solve it, which is a mess. But the point of the game, I'm just giving a prescription for showing how you can assign the local length scale of your matter. So local light scale gets assigned by matter telling space-time how to occur. Kinetic energy gets assigned by space-time telling matter where to go. I can do this so that these equations are big. And then I'll just put a check there because this just confirms I just said, you know, 20 or 25 minutes in going into gory detail to confirm that general covariance plus assumptions as to the form of the action and its dependence on the metric. Yes, the United States' right here. Okay. All right. Are there any questions about this? I probably shouldn't have gone in so much detail. But, hey, I want to show you the detail in fact consistent, to do last time when I talked about this in August because everybody asked me all these probing questions but luckily because of the early hours of the day I seem to have lulled everybody I'm not sure what okay so I can now these boards right now actually tell you completely how you construct the metric so they're actually just filled up exactly with how you construct the metric of the conformal factor, how you construct the simplicial regi-calculus how you assign the lengths and how then you fix the conformal factor. I don't expect you to remember any of this. It won't be on the exam. And then how you fix the kinetic energy terms so that the Einstein-Mergy equations will work. So all I've done here is perhaps the two great lengths backed up my claim that this system actually, in each specific history, obeys the Einstein-Mergy equations. Okay? Are people happy with this? the preprint should be out there on the preprint server soon as soon as I figure out this is a rather difficult problem to figure out who to reference and how and I've got to collect lots of these things so let me just actually summarize here

47:30 what do you get from this? I mean there's no point in doing this unless you get something from this well actually what you get is something that I mean I've constructed the theory of gravity based on quantum computation, but really all it is is the theory of quantum computation. And quantum computation is simple. It's a very simple discrete theory. There's nothing complex about it. There are certainly deep theorems in quantum computation, but it's basically the simplest possible quantum mechanics. But now, so I get, so this actually provides, in this model now, I can now address lots of these questions that people address You get what happens in, you get a very simple picture of what happens at initial singularities, like the beginning of the universe, and what happens at final singularities, for instance, inside a black hole. and actually what happens is not that much it's rather gentle actually what you find is in fact depending on how you pick your initial omega you could have the curvature effectively go to infinity at this point but nothing really is happening with computation all that happens at an initial singularity is that a bit comes into existence and all that happens at a final bit or bits come into existence and all that happens at a final singularity is that bits go away and this gives you a mechanism that you can either get for black hole evaporation well you can actually have two kinds of black holes you can either get something like the Harworth-Malassane mechanism and I have a paper on the archive you can check out about that show it called almost certain escapes from black holes also have maybe universe production. Those are two possible scenarios. So they're distinct ways that black holes can form and go away. You can get things like the holographic principle. In fact, in this science paper that I

50:00 described, my colleagues Vittorio Bonenio, Lorenzo Bacconi, and I derived a kind of quantum information, a quantum computation version of the holographic principle, which doesn't limit the number of bits that can be contained in space-time. It limits the number of ups that you can perform in a volume of space-time, which actually turns out to be a more stringent requirement than the holographic principle. And actually, these gates, in something like our universe, these gates may, in fact, be quite far apart. They're not on the Planck scale. For instance, in interstellar space, if you look at how accurately quantum matter is measuring space-time, it's only being measured to the accuracy of the microwave background radiation because that's all there is out there so in fact these gates would be very far apart okay so the scale of the discreteness of this quantum mechanics might be quite large and finally you can make various toy models of quantum cosmology the actual qc and grqc and you can and that's what I said you just take some big honking computation and you look at what happens in the early universe the early universe means the first stages of the computation and what happens is the little quantum fluctuations of these logic gates in the early universe then drive primordial curvature fluctuations and depending on what you do actually you typically get for instance if you take these angles to be positive you get things like a period of inflation followed by a normal period coupled to Gaussian curvature fluctuations in the early universe because if these status and these gates are relatively random then you'll just get Gaussian curvature fluctuations when you course create this system. And that actually inflation coupled to Gaussian curvature fluctuations is in fact exactly the model that gets used by people to calculate primordial curvature fluctuations in the inflationary universe. So I hope though we have to actually perform these calculations that we might be able to reconstruct pictures of primordial curvature fluctuations in the universe that we could then compare to an experiment. So there you go. GRQC, a quantum theory of gravity based on quantum computation. Very simple because it's really only about quantum computation and gravity just comes along for the ride. So all this fancy stuff up here was just to show you that gravity comes along for the ride. And because it's very simple

52:30 quantum mechanically, then it allows you evaluate things like what happens with initial final singularities in a straightforward fashion without having to resort to path integrals and all the other stuff that people in quantum gravity do. And I should say that I really love theories of quantum gravity. I've always been interested. I don't think this is necessarily right. I just wanted to construct a theory based on it, which could be compared then to the predictions of conventional quantum gravity. And indeed, it's not clear that this is not conventional quantum gravity. Indeed, this kind of picture that I drew right here looks strikingly similar once one's retrophied it to the kinds of pictures that Renata was describing on Wednesday. So I have no idea if they're actually related or not. Okay, let me stop there. Are there any questions? One person I had was, I mean, you fix the, everything, by law, basically just with these lines. Yeah, yeah, so the, the, the causal structure of the history, I mean, the causal structure of the history, it's everything up there first. And then you, if I, so if I wanted to have, say, a superposition of this thing happening first, or this thing happening first, one gate happening before the other, you said I could mock it up by somehow having other gates and having some, and so, is that? you mean this gate or that gate? The gates don't necessarily have identities, but I could certainly, this, this, my computation is a sum of these different causal structures. So if you want to say, what is the, if you don't want to give a superposition of this causal structure or that causal structure, then it's easy to construct a circuit that will give you that. And indeed that superposition will take place, whatever superposition you're talking about, it will be a piece of almost any circuit. As you mentioned this, if you look first at the simplest possible theory that might be the correct theory of whatever it is you're looking at, that's a pretty good approach, are you pretty confident with the asset, are you confident that this is the simplest approach? that you can take to this idea. Are you sure? Well, I mean, this stuff about recognize

55:00 was maybe quite a little complicated, and I did put in some assumptions, so I was very careful to only put in the same assumptions that Einstein put in. Oh, by the way, this is general to me without a cosmological concept, and a cosmological concept doesn't arise naturally in the system, but you could insert it by hand if you wanted to. So I actually, The initial part, saying we're just going to derive general relativity from quantum computation, I can't really imagine anything similar than that. But I wouldn't surprise me at all if there were alternative approaches to going from the computation to something like Einstein's equations. But I just took the course that was simple. I didn't know if it was as simple as it was the easiest course, because actually each one of these steps is all mapped out and has been investigated many times before. They just haven't been put together. Yeah, so I don't know if it's as simple as it was. and particularly, you know, what I've been looking at for these quantum cosmologies is I either look at large, random compact computations so, you know, something to calculate the first billion digits of pi, right? Or I look at infinite uniform computations that are either like cellular atomic or that have random connections, they don't have the same answers so, and I don't know if those are the correct ones I mean, there's a lot here, but I don't know what's the correct thing to do But, you know, what's here is enough to actually start answering these questions right here. Yeah? Well, I mean, the embedding, so the embedding, all the, the only purpose of this embedding was actually to demonstrate that this is indeed a theory of gravity. Right, so if I embed this in a manifold, and I demand a metric for the manifold, respect the causal structure of the computation. And I get Reggie taught most of the Bayes-Einstein equations, and I answer Reggie equations in a natural way. Of course. Yeah, but this is a fiction in some sense, right? But that's, you know, it's a fiction in the same way that Einstein talks about space-time as being a kind of fiction. You know, space-time is three dimensions, and one spatial dimension is one temporal dimension is a very convenient and, you know, to us human beings, intuitive fiction about how we make sense of events. Okay? So, the underlying theory is completely background to the tangent, which is why it's so easy to do this kind of stuff, right? The stuff that was complicated here came because I embedded it merely to show that my contention that this was, in fact, general relativity is correct.

57:30 So, yeah, so to do these calculations, except for the things that I actually have to talk about, things that have to do with the metric, It's almost all to do with the metrics, so I actually have to relate this to space-times when I talk about this. But the actual dynamics of what it was underneath is just quantum computation, and that's why it's easy. Yeah? When you think over, you think so that you get less information. Yeah? Is there some fundamental reason that you could make that choice? Or is it just to construct general relativity? Yeah, so I did this specifically to construct general relativity, and in fact, I don't have, I followed slavishly, there's a really good, though somewhat dense textbook by Hawking and Ellis, called the Large-scale Structure of Space-Time. So all I did there, this, if you actually read Hawking and Ellis, you would recognize this construction and then this constructing omega as directly taken out of Hawking and Ellis for matter here. So the way that you know the causal structure of a classical space-time, then what you do is first you set up the light post, which determines the metric of your conformal factor. And then, actually, you don't have to pick the conformal factor so Einstein's equations are obeyed. But if you pick the conformal factor in the wrong way, then particles won't follow G and E6. So you then pick the conformal factor so that Einstein's equations are obeyed. So I'm actually just following that prescription here. Now, I didn't have to. If I pick the conformal factor to be something different, then Einstein's equations wouldn't be obeyed. But that's also true in general relativity if you pick the conformal factor wrong that Einstein's equations are obeyed. and it's sort of an arbitrary fashion that, you know, I feel that I have a precedent on my side of this growing. So, I mean, take your favorite model of black hole evaporating, the mechanism for black hole evaporating. Yeah. Presumably you could simulate all of those on a quantum computer. Yeah, yeah, so what happens in a black hole? What happens in a black hole is that some chunk of this computation goes off causally with the rest of the computation. It's easy to visualize what a black hole looks like in such a diagram. They correspond to causal histories, to histories, where there's a causal disconnect. So some feature of this goes off, hits a singularity before the rest can communicate. Indeed, here, this thing right here has a final singularity right here. And once again here, there's a black hole, so far as everybody outside this is concerned.

1:00:00 So if you can simulate all of the mechanisms, how can this predict which mechanism will be? If it's unitary, so if your simulation can simulate all of them, how can it predict what will be the actual mechanism? Oh, well, so if this remained causally disconnected but now just branched out into a whole bunch of other stuff, and didn't end somewhere, then it would be a baby universe. So it just depends on which. You just have to look at the causal structure. Remember that the computation is some of these causal structures. Some of them are black holes, others don't. Actually, all these finite ones, if I make a finite computation, they all have initial singularities and final singularities. I would say better if they're a compact space-time. And how about the Hawking radiation? What would that mean? Yeah, yeah, that's a good question. So I don't actually, so this relates to a first question, but maybe on the left, so I don't have but I'm here. So in order to actually talk about Hawking radiation, I think I would probably have to do something like Neopurium, in this case. But there is, I mean, black hole entropy actually works out quite nicely here. So if I cut off part of this chunk right here so it forms a black hole, and there's a certain number of bits in it at each point. And since you know that actually this thing is going to obey the ordinary gravity with curves, sorry, quantum mechanics on curved spacetime, because, you know, it's quantum mechanical and it has curved spacetime, and it obeys Einstein's equation, then you can actually see what the size of the black hole is quite powerful in this entropy sphere. And you actually find that, in fact, the entropy may be slightly more or less than, sorry, it won't be more, but it might be slightly less than the, you know, hoppy radiation corresponding to some weird-shaped black hole. And when these things go away, Okay, if this were a black hole right here, which it is, this section right here is a black hole so far as this is what we're concerned. As this goes away, then in fact you get something like the Horwitz-Bellis data mechanism, you're destroying information, and a bunch of the information will escape out of that. But in terms of getting thermal locking radiation, I don't know how to get that from that. And do you have like a concept of a genetic perception? Yeah, very simple. So, in the history, it might just have a definite theta in each of these points. The theta is the proper time of proportion to the theta.

1:02:30 So, if I take a path, it maximizes the proper time. That corresponds to a gene testing. And, in fact, if you go to the kind of, if you imagine these are all square swap gates, okay, then, remember, this corresponds to a bunch of particles going around the data in the rapid direction. So, when a bunch of particles go around the data in the rapid direction, all of the geodesics with their wave patterns spreading out around them. So, yeah, so, because Feynman and then of us, a lot of people have to do this very simple picture. In fact, your normal pictures of particles moving around here are rather similar. And different observers? Yeah, so observers, of course, are hard to ask. Where are the observers here, right? So, of course, an observer, to make an observer, you actually have to, like, construct a subroutine that gets information about the rest of the universe. But you can follow some chunk of this computation over time. I can just de facto call that an observer, okay, which may be inertial or non-inertial, but which is getting information about its surroundings due to these quantologic operations. But what would be inertial and non-inertial? Yeah, well, that's a good question. I don't know. I mean, here, everything is interacting, right? Though, actually, if you just take these swap gates, you can just get non-interactive direct particles, or particles that direct only to Pauli-exclusion principle. But whether things interact or not depend on what these views are. So I don't know what an inertial, well, with a direct equation, you get inertial direct particles. They do bounce off of each other, so, I mean, Pauli-exclusion does make them interact in some fashion, so. Yeah, I to lock up on a threshold to a profession. So you're looking for questions over coverage? Yeah. Great. Thank you very much. Final hour of talks. The area I'm coming from is really sort of entanglement theory. And in entanglement theory, we sort of generally try to answer three questions. We want to characterize entanglement. We want to find out whether a state is entangled or not. And then we would like to see how we can manipulate entanglement under a certain constraint. And that gives them all sorts of criteria, what we can do with entanglement, for example,

1:05:00 under local operations and classical communication. And then we also generally would like to quantify entanglement. So either saying, well, generally, how well can we do this state manipulation? How much entanglement resource do we have there? And all this we would like to do ideally for many particles. So that is a rather tricky thing to do. there's quite a lot known now, but for many part of systems, basically the statement is, well, we know very little and it's really very difficult to make a sort of a full methodology characterization. So, therefore, some of us have thought, well, okay, if it's difficult for us to do this, why don't we look at some sort of more natural states in physical systems that tend to appear well in reality and try to learn something about their properties rather than characterizing every completely crazy 17,000 particle state with so and so many special properties. And so of course there are still many questions to answer here and I have been recently with some friends that we're looking at a very particular question. Mainly, look at the following situation. You have here a big lattice. So whatever thing we have. So let's say we're looking at quantum fields. So to make these things treat, well, to make these things a little bit easier to treat, we put them on a lattice. So we just . So this is somehow the lattice here. And initially, I don't specify what these initial points correspond to. In the end, that will be harmonic oscillators. And then you have here, is there actually a pointer? Ah, there. So in the end, we have this whole field. And then we have here a region that we pick out. This is the shaded region. And surely in these systems are interacting with each other, there will be some entanglement between this region and all the rest. And you would like to see what happens when you linearly increase the size of that region.

1:07:30 So the linear dimensions are increased, so like this. So you want to see how does the entanglement actually scale. So that's a relatively simple question. and it was actually, this was a question that had been raised already, well we learned this later, but it had been raised by people like Bombelli and Truditsky and so on in the connections of black hole entropy. And they did a little bit of numerical simulation and they got a a certain result, but they couldn't really prove it. So what I'm going to show you here is how we can use quantum information tools to actually prove these certain statements here. I mean, to get really strict proofs with actually exactly zero amount of the barriers involved. And to do this, we cannot do this in every system. So, what I'm going to do is I will look at harmonic oscillator systems that are coupled harmonically. So, a specific example would be the Klein-Gaulon equation put on a lattice. That would be, actually, that would look roughly like this. So, you would get harmonic oscillators and that would be coupled with my nearest neighbor interactions via strings. Just a very good score. And if you want to get the field again, you do the continuum limit. So, we will always look at the discrete case first and then in the end we might do the continuum we wish. And so you will have a 100-coma operator which is productive in X's and P's. So these are the energies of the harmonic oscillators and this is the coupling between the harmonic oscillators. So, this is of course an infinite dimensional system. So generally this would be rather hard to treat because we already have problems to treat, let's say, spin-1 systems and all the entanglement properties. We don't even know those. Unfortunately, over the last few years, quantum information theorists have been interested quite a bit in these infinite dimensional systems because they are actually, for example, light modes. And they develop the whole big apparatus of, well, entanglement theory in these systems. And I realized that there's a subclass, which is a very important subclass, of which we know almost all the important properties that there are.

1:10:00 And to explain this a little bit, have a look at the following. So instead of characterizing the important state of the density matrices, state vectors, what you really want to do is here, is to characterize it in face-to-face. It's completely equivalent, I don't throw away any information or so. It's just a different representation. So this means I would like to look at the quantum state in terms of maybe Wigner functions or Q functions or what have you. In fact, what we are doing is we are looking at it in terms of the characteristic function, which is just a query transform of the Wigner function. So it's just another representation. And then you can, I mean, you open every quantum optics book and the first pages you see some nice plots of these things, what they look like. And that's an example, so that would be some characteristic function. The Wigner function of that one would actually look the same in phase space. So this would be a single harmonic oscillator. Here would be x direction and here would be momentum direction and it's a Gaussian. And in fact, what I put it here was the vacuum state. And this vacuum state actually has the property that, well, it's a Gaussian distribution, and that's nice, because in that particular case, I don't need to specify, if I want to mathematically describe it, I don't need to specify it in every point in phase space, I just need to give first and second momenta in, so, the variances of x and p and the covariances of x with e. to keep the center of the distribution, and then I know everything. Like a Gaussian in sort of a single variable, in the first and second moments, you know everything about the Gaussian. So that's actually a general feature. In fact, what we want to say is that a state is Gaussian if its face-based distribution is Gaussian. And then we all of a sudden have reduced the program form an infinite dimensional system with infinitely many variables to one where we just have to give the displacements and the second formants. And in terms of those, there's a big entanglement theory. Everything can be computed from those, and it can be computed with you. And the nice thing is that the ground set of Hamiltonians that are quadratic in X and P are Gaussians.

1:12:30 Thermal states of such Hamiltonians are Gaussians. So for all those, I can apply all these tricks from quantum information science. And, in fact, what I'm going to restrict my attention to is I will always look at distributions that are centered about the origin. That's not actually a restriction because displacements are local observable. So you can actually shift individual harmonic oscillators to their origin and thereby make the displacement zero. And that doesn't change, actually, entirely. Yeah, so you said thermostates are also gauges, and how would you be able to distinguish your thermostate from like the back of the back? Oh, it has a bigger variance, it's wider. Oh, meaning that's the only difference? Yeah, you crank up the temperature, it gets wider. Okay, and then the only thing, make this first moment zero, and the only thing that is really left are the second moments, And so the variators and co-variators. So everything is determined in terms of those. And so, of course, not obvious that everything can be computed easily from those. There might be extremely complicated functions. But it turns out that most things can be computed quite easily. For example, if you look at the Hamilton operator, where you have a potential energy that is encoded in the matrix B, then the ground state can be computed very easily. and invert it, that gives you all the covariances in position, and then the square of the magic gives all the covariances in the momentum. So that's quite easy. The way to derive this is you take the big system, you go to normal modes, then you have individual uncoupled harmonic oscillators, you look at the ground state, and then you transform back and you get this result of this. It's very easy. You can also compute all sorts of entanglement measures in terms of these covariance matrices. So, really, I mean, I just give these standard measures like entropy of entanglement, which is good for pure states, and there's some expression in terms of the covariance matrix. This will be useful here and there, but what I really will need is some new entanglement measures that have been derived in quantum information science, namely in situations where you have mixed states also. And these are two of those. There's a logarithmic negativity, which basically is where you take the partial transposition

1:15:00 of a density matrix, and then you compute its singular values, and you add them up. And then you take the logarithm. And there's also the distillable entanglement, which is an incredibly complicated thing to compute, of course, but fortunately, we can give upper and lower bounds, which are in terms of entropy. And this was, the lower bound was proved quite recently. And it was sort of conjectured for a while, but then Devetak and Winter proved that this is really a restricted overall. So all these measurement measures will feature somewhere in my calculation, in fact. And the reason is that, of course, the first one will show up here and there when I talk about the ground state of the whole system at temperature t equals zero. But I also want to make statements, for example, at finite temperature. And that could also be possible. And that's, these are sort of questions that actually could not really be asked in the first place without having sort of a more detailed understanding of entanglement altogether. Because we didn't know what mixed data entanglement was. Right. So, that transparency did exactly what, let's say, these people did. They took a lattice, they picked out a region here like these red oscillators and then the rest of the world is blue oscillators and the world is very small, it's 30 by 30 lattice points. And you just compute numerically the entanglement between the red region and the blue region and you vary the size of that region and that's the size is here, so the linear dimension and here's the entanglement and you see it's a straight line. So, well, okay, so what you see is that the entanglement does not scale with the volume but with the surface area, actually. Well, that's, of course, only numerics, and in fact, when you look more carefully, then you see that there are all sorts of wobbles here going up and down, and it also, of course, depends on the strength of the coupling between neighboring oscillators, so when you go to the continuum limit, you have to increase that strength, and what then happens is that actually this is not going to be a straight line for very small numbers but there will be big sort of oscillations and models and you don't actually know whether this is going to be true in the complete the continuum limit so you really need to make a proper proof of these sort of things okay so first thought that one could have is one well once one has a look whether there's something like a schmidt decomposition or something like that

1:17:30 there is something like this, also for these Gaussian states. So what you can do is, you can go to the blue oscillators and you rotate your bases a little bit so that the Gaussian character is preserved, so you look at it in a different way. And you do the same thing in the red region. And the effect of this will be in the ground state is that almost everything is disentangled now, except, well, a few oscillators. Because, namely, each of these red ones is entangled with exactly one of the blue ones. So that is already a little bit of a progress in complexity here, because we can disregard a lot of these blue oscillators, they're totally irrelevant. Now the problem is that this is still a number of entangled states that is proportional to the volume. So that's not good. So one has to look at it a little bit more detail, and the first thing we have done is We simplified the system. So initially, you have a system like this where you have sort of nearest neighbor coupling by springs. And this is going to be a little bit tricky to solve. I'll show you how to do this later, but we want to simplify the situation, and we simplify the situation by introducing extra springs, which couple next nearest neighbors. And on the level of this potential energy matrix, this means just squaring the matrix. It doesn't look like a simplification, but the great thing, of course, is that the ground state looks a little bit easier now, because now we take the ground state, it was 3.0, it was the square root of this matrix, and now it's just, well, the matrix and it's inverse, so that's easier. In fact, now the ground state has sort of a very nice band structure. Well, that's the trick. And when you look at this now, then you see that the normal form simplifies enormously. Now you have a situation where when you look at the right basis, so you rotate the basis in the blue region, in the red region, what will happen is it will only have entanglement alone between oscillators, well, between a number of oscillators, linear exactly, and a number of oscillators on the surface. And now you already sort of start to believe, aha, well, there should be something like the entanglement is proportional to the surface. constructing this is a real pain in fact you don't ever do this you just need to know by specific arguments

1:20:00 that there is such a transformation and then what you only need to do is you need to find the pair with the largest amount of entanglement which corresponds to making some eigenvalue estimates so that's very easy well, matrix analysis you open the partia and form a Johnson you look long enough and you find the tricks that you need There's still a little bit of work to do, of course, but in this situation, for this special type of interaction, you will find that there is a given area. And the neat thing here is, you can do the same for thermal states. The same trick works. And this is very, very important when you want to analyze thermal states. Because what will happen is that if you just compute the entropy of this region here, it would be proportional to the volume. and it's very bad. And actually the entropy here is also proportional to the value. However, we know that most of them, most of this entropy is really due to the sort of interaction with the environment and not between different oscillators. So we can just throw away those oscillators and just consider again these ones on the surface and then we do more complicated eigenvalue estimates and we have the theorem again. So that's the way that works. Now, for general because you cannot find this symbol to a normal form, but you can make statements about how the correlations between different oscillators sort of contribute. Because the farther you are away from the surface, you can imagine this one will be less and less correlated to these ones out there. You just have to get a handle on how to compute this. And this is awfully complicated for using entropy. This is really a big pain. So, but we have this new antagonist measure. everything becomes, using that, everything becomes very easy. And logarithmic negativity is always, is an upper bound to the entropy. So that's a nice thing. So we find an upper bound to the entropy. Well, the point is, you have to gather courage, basically, and make some statements about these queries. And first we did this by actual calculations. Later on, we discovered that some people have done very general theorems about what happens What happens to matrix elements that couple oscillators that are a certain distance away? Well, distance means take this oscillator here at position 3, 2,

1:22:30 take another one here at position 6, 5, and the distance between them is just the smallest number of steps on the lattice that they have before. And in that number of steps, these off-diagonal elements, these coupling elements, fall off exponentially. at least when you take sort of the nearest neighborhood directions. So this can be formulated in a much more general way. Basically, you give me a matrix, I can tell you how the off-diagonal elements of the square root will behave. And what these off-diagonal elements are, are just correlations between positions and momenta of distant oscillators. That's it. Now, how does this? This has nothing to do with entanglement in the first place. that the log-negativity is bounded by these elements from above. And in a very simple way, you just have to sum up all the elements that couple red oscillators with blue oscillators. And you just sum them, sum all of them. And that's an extremely easy task, actually. It's just a counting argument. So basically, you have here the red region, and then you look for all the oscillators that are exactly one step away from the surface. They give one order, one certain contribution, that's the linear. Then you take the next ones, they are two steps away. Well, that gives a square contribution. And all of them, the number of oscillators that are one step away, two steps away, three steps away, they're all essentially proportional to the surface. Well, you do that, obviously, that sums up to something finite in the optimization. So that's actually a very easy thing. And you can do this really, you can even do this when the correlations don't fall off exponentially, but when they fall off like a power law. So that actually shows you that this theorem does not really have necessarily anything to do with sort of phase transitions or existence of phase transitions or so on. It's a completely separate phenomenon. Is it through all power laws or just for sufficiently power laws? Well, they have to be sufficiently slow so that the sum is added up, adds up to However, what you see is when you look at the system, so you try to make a potential matrix that has a sufficiently slow fall-off in the correlations. If the fall-off becomes too slow, this matrix that is not positive anymore and doesn't correspond to a physical system. So, in fact, for all physical matrices, it's fine.

1:25:00 Okay, so now you can ask the question, where does the entanglement sit? And so this we did just numerically, so we can say, well, let's take this red region here and then cut out just some of these oscillators. And then look again at the entanglement, and what you will find is it's the same, essentially. All the entanglement sits in a thin layer on the surface here, because the layer is the length, or it's the correlation length, basically, here and here. That's where all the entanglement is sitting. The inner part is totally uncorrelated. Okay. And the last thing I want to say is you can do the same thing. Actually, you could think this is quantum mechanical, so it's entanglement and all these things, but this is really a statement about correlations. So you can actually do the same thing in classical systems. You give me a bunch of harmonic oscillators, classical ones, so in all the calculations I take the hats off, basically, and do them again. And you can ask for how do the classical correlations between, let's say, the interior and the exterior scale, and they scale in the same way. They are proportional to the surface area. And in fact, what comes out is that the calculations even are the same. So every classical system which corresponds to a certain potential matrix, potential energy, has the same behavior as the quantum system where you square the connection matrix. So nearest neighbor interaction with springs in the classical system corresponds to the nearest and next nearest neighbors in the quantum system. And you see, after a few steps of calculations, it's all the same. Because in classical, when you have zero temperature, then you don't have any correlations. Well, I mean, okay, so what I say is, yeah, I compare thermal stakes. Right. Yeah, so. Right. And basically, that's it. So basically, what, it's a little bit of an advertisement also that methods from quantum information theory and entanglement theory that have been developed in recent years, So, it can actually allow you to actually prove, strictly, statements that have been around for, I don't know, 20 years in this case, but have not really been proven. Yeah? They have been sort of conjectured numerically, and everybody believed it in a way. In fact, someone I get statements like, well, this has been proven, hasn't it?

1:27:30 But it's always sort of numerically. Here, you can actually answer those questions, and you can actually make them more general, because here I answered a question about mixed-state entanglement as well, so thermal fields in this case. And this was impossible to be asked before because simply there were not these tools of entanglement theory around. And so basically, the general statement here, so if you have a system that is interacting with its environment, generally the tendency will be that the correlations between the inside and the outside, be it quantum or classical, will scale with the surface area rather than the volume. That's a fairly generic feature. It's actually also true for many spin systems, but not all. But you can also make similar proofs for some spin systems. Okay, that's it. We have time for one or two questions, if there are any. So you've never said anything about general relativity or an autograph? Well, I mean, this, actually this statement has nothing to do with general relativity and it doesn't even have anything to do with quantum mechanics, in a way. Because I could get a similar statement, of course, saying, well, we talk about classical correlations and quantum correlations now. I could get the same statement in classical physics. So this is, I mean, in a way, it's a very universal phenomenon. Is it, um, it's working against five dimensions? All right, okay, I mean, the plots here, of course, I cannot draw so well, I mean, it also takes an unbelievable amount of time to do this on a computer. Everything was drawn two-dimensional, the proofs are d-dimensional, so any spatial dimension you want, you can do. And you can also use for extremely large classes of potential matrices. So you give me a setting of springs, how they couple to neighbors, I can give you a proof. Let's thank Margie again, and there'll be time for, since these talks are fairly short, there'll be time for more questions, you know, after we can have coffee after the set of talks. Next speaker is John Smolin, who's, the title of his talk is... I don't remember. Something about locking correlations in blackboards. Something like that. Is that part of the screen?

1:30:00 Oh, the little picture of Chuck. Oh, it's alright, I can talk loud. Oh, okay. I will stand quietly. I've done valuable years from it. So what I'm going to talk about is the black oil information loss paradox, which since we've heard so much about it, I'll belabor some more, but really my point is that we can save unitarity, which is much beloved by quantum information theorists and probably as much. But we can save it without strongly modifying general relativity, which is good because I don't know anything about general relativity. But basically, the resolution lies in one new idea from quantum information and in just understanding information itself correctly. You get a lot of different answers about why people think there is a black hole information loss problem. I want to argue that the information isn't lost and that really nobody said it was. So why do people think information is lost in a black hole? There's kind of two main reasons. One is the Hawking radiation is thorough. Okay, so everybody says this, and I'm banding around, but it's hard to get a straight answer out of people exactly what they mean. So to me, what this means is one of two possible things. That when you have a black hole, and after a while, a photon comes out, you can write down the density matrix corresponding to this emission, and people say, okay, this is rho, and it's thermal. To oversimplify, it's just some density matrix that's maybe

1:32:30 diagonal and has an entry okay but what happens if two photons come out which will happen sooner or later well you get rho square and after a while you know n photons come out maybe the black hole is shrunk and you have all sorts of light coming out and I mean the precise statement of it being thermal would be that the whole thing is row to the end. But really all you know, I think, is that if I trace out n minus 1 degrees of freedom of the sort of big row that comes out of this thing, that the small thing left is the same row from before. And this is because each row that comes out isn't perfectly how you interpret the semi-classical Hawking calculation, it's really, there could be a small correction. And for a cosmologically sized black hole, there'll be a very small correction. The sort of unit-based argument is that correction should be something like the size of the mass of the photon, and I wrote this too small, but the photon is small, divided by the mass of the black hole. And you say, okay, this is a very, very, very small correction by any standard. I think the mass of the photon is zero, which is... Well, it's energy. I'm being imprecise, but okay. The amount of mass the black hole loses is what I really should put there. So this is a very small correction. However, if these corrections add up, when I go to the nth power, when nth becomes at left, well, the corrections, you know, if I have something that has, well, let's say, Fidelity to the thermal state is 1 minus epsilon for this teeny little epsilon. I take this to the n power, so it's 1 minus n epsilon. I get to multiply by n. What's n? n is how many photons come out of the black hole when it's lost, say, 90% of its mass. Well, how many photons is that? Well, that's 90% of the mass divided by the mass of the photon. So all these things cancel out, and you get a correction here of order 1. and if you buy that it means that the thermal radiation that's come out of this

1:35:00 can very easily look thermal piecewise but the whole thing looks nothing at all like a thermal distribution and indeed could have very low entropy or anything at all that you want so you only have to accept that the original locking semi-classical calculation has really any correction at all of a very reasonable size to say okay the entropy that comes out is all accounted for and it's just the same as encyclopedia getting chucked into the black hole and evaporating out. If I light that encyclopedia on fire, everybody says, oh, what comes out is thermal, this nice temperature of the fire. But nobody says it's non-unitary. So what is the source of this correction in the original Hawking calculation? I'm not familiar with it. Well, you get all sorts of different answers, depending on which general optimist you ask. But the main thing is that it's considered like the black hole is unchanging, and really it shrinks a little. So it's like Really, any reasonable sort should get some sort of direction. And that's almost, you know, you don't even need a quantum gravity to say that. So, okay, I sort of feel like I've dispensed with this. And you get arguments about this, but at least it's plausible. I've done too much theory lately, and worse, too much interpretations of quantum mechanics. So this actually suggests an experimental program, if anybody has any funding. The experimental program is the following. You create some zero temperature flat space and then you put a lot of mass into it in a pure state and you create a black hole and you wait around to evaporate 90% of the way and you collect all the radiation and then you do a measurement with rank 1 VOVMs on this. and that gives you a tomographic way of finding out what the state is and whether it's thermal or not and of course you have to do this many times you have to do an ensemble of these things so you need to make probably something like well you have to do this many many times and there's a number of times you probably have to do it in something like this order which would be a difficult experimental program but plausible so basically if there's any funding at all I need all Also, the black hole of rights, people would probably go to try to liberate the black holes. Oh, that's right. I'm exploiting the black holes. Well, all right. There's nothing much I can do about that.

1:37:30 I mean, they're not cute, right? They're only hairy animals. There's a really dope constituency. So, when I was talking, I think to John Preston, but I forget exactly who said this at this point. Well, the other reason, or any other reason, we don't like unitarity for the black hole, and we say the information is lost, is because of causality. It's because there's an event horizon, and there's a lot of information into it. It better not come out. And the only time that it can come out without violating causality is when the black hole has evaporated down to this tiny little sort of plank-sized thing. And then you say, well, okay, then quantum gravity for sure is going to dominate. like is that no information comes out at all until that point. And then the argument goes, okay, I had a big black hole evaporated for a while into a little black hole, and a field of radiation around it, and it seems unreasonable to say, well, there's no information in this field, so it's information that I don't want to find yet, and it's kind of roughly zero, and then the information in here is all the into it. So this is much, much greater than zero, maybe a bad scalar constant. But there's a whole lot of information in here, and this is independent of how big the black hole started out, that eventually you get down to this kind of small core, it needs a lot of information. So the information density goes to infinity as the size of the initial black hole. People don't like that, because it violates the holographic principle in it, and also just seems unreasonable. So this is where this new result from quantum information theory comes in which basically says or the point of it is that you can have a lot of stuff having come out and very little entropy left in the black hole and yet the stuff that comes out won't have any information in it about what was dropped into the black hole. And this is counterintuitive because it violates sort of a fundamental tenet of classical information. Classically, if I have two people and they have a classical random variable, x for one of them and y for the other one, I can calculate the mutual information that they have and say i of x and y. Okay, and then suppose somebody says a small amount of information, say 1 bit from x to y.

1:40:00 Well, classically, I can say, well, what's the new information after I've done this? Well, it's always just the old information plus 1. So if I send you a bit, your mutual information with me increases by one bit or less. If I send you a bit, you already knew it would increase by one bit. Well, okay, that's kind of fundamental classically, and indeed, mutual information is defined, so this will happen. But quantum mechanically... I wonder if I've already seen too much information. So quantum mechanically what you have to talk about is that I have a density matrix shared between two people, and the mutual information I want to define is the thing called the accessible information. The accessible information is the maximum over all measurements that have a tensor product structure, so the two people that share the density matrix do these measurements separately, which lead to classical variables x and y of the mutual information between x and y. So you have a quantum state. You do the best measurement and then see how much mutual information. And this thing violates this sort of arbitrarily badly. If I have a density matrix where the mutual information is some I zero and then I row A, B and I send one bit from A to B the mutual information I final minus I zero can be it can be infinity. You can make it arbitrarily high. It's a function of the dimension in Hilbert's space you're talking about. So this is It's sort of surprising, but it's actually very simple to see how this could happen.

1:42:30 So, I'll write down an example of this, which is a little less impressive, but gives you the basic idea. Now, suppose Alice and Bob share a quantum density matrix like the following. Bob has one of these four polarization states. So there's just four non-orthogonal states that come in two orthogonal basis pairs. So Bob has this quantum state. And what Alice has is the classical information of which one of the four states Bob has. Very simple system. Well, you can calculate the mutual information between Alice and Bob. And it turns out that for the optimal measurement, this equals exactly one half of a bit. We think of this as Bob doesn't know the basis, so he picks one at random and measures it and he gets the answer correctly half the time when he gets the right basis. So it comes out to half a bit. Now, if Alice, who knows the basis, just tells Bob, then the mutual information, well, Bob knows which basis to measure it, so he gets the bit correctly every time. And he also knows the bit that Alex sent to me, which is the basis. So that's two bits. The bit that he measured and the bit that got sent. So we went from half a bit to two bits by Alex mailing him only one bit, which violates the classical thing already, although only by a tiny amount. Now you can generalize this and say, well, instead of four states and two dimensions, I can put and if I do that I can just adjust this using exactly the same plan there's still only two bases but now there's two the end states and it goes from half of the log of the dimension and the final information is the log of the dimension plus one which was the bit that was sent and so here is an arbitrary increase in information by sending one bit and went from some big number to some number twice as big, so the information we call this unlocking of classical correlation or classical information

1:45:00 it went from one number to another number anonymously largely and then it turns out that you use a result from what is it called this is a result for Winter and Aiden who else is the helpers? Peter Shaw Peter Shaw, of course alright, so it's a result from Peter Shaw and if you use that result, you can actually have a system where you start out with mutual information that's constant it starts out at zero and then you send some number of bits which is something like blah, blah of the dimension of the space, and then the final mutual information is log of the dimension of the space. So I start out with essentially no information. This constant could be something like, you know, just three. I start out with three bits of information, I send a small amount of information, and I end up with an arbitrarily large amount of information. So let's find the picture of the small black hole again, and say, okay, well, let's put inside of it this tiny amount of information, log log of n. And what's out here is this big Hilbert space of dimension n. And I can write down a state where the information in here is this tiny little thing, and the information which has escaped is only three bits instead of this huge amount. And so I can avoid violating causality, at least very much. Three bits may be leaked out, but at least I didn't have dimension of log n bits leaking out. And this sort of rescues unitarianism without violating causality. You can wait around until the end, and the information leaks out. Now, there's problems with this. This log log n has the unfortunate property, perhaps, that it depends on n. So I can still say that, well, the Planck scale should be some fixed size, and if I keep making my black hole bigger, it should have more and more information in it, and eventually I won't be able to, even this small core won't be enough to save causality. That's a little disturbing, but there's sort of two escapes for that. One is, we don't know if this locking is optimal. Maybe you can make this grow even slower, although probably not a constant. And the other is to say, well, how big a black hole do I really have to worry about serving unitarity over.

1:47:30 This was a beautiful idea from, I think it was Carol Bardetsky, over dinner one day. He says, well, okay, let's take all the mass of the universe, figure out the dimension of that Hilbert space. Pretend it went into a black hole. And they say, well, how big a core do we need? So, that core is, you know, something like log, log, the dimension of the universe. But two logs, it got pretty small, pretty quickly. And you say, okay, well, I'm just going to call that, you know, Planck scale. It defines it. So this is just my second experimental proposal, which is, count the mass very accurately of the equations in the upper space that corresponds to. And that tells you the Planck scale. We also already know the Planck scale from other experimental evidence, and see if the number is the same. The thing is, the Planck scale is numbers. And the mass of the universe is a very, very big number. So, okay, they're reciprocally related by this relation. Wouldn't that be nice? So, that's just a modest experiment. But also, the number of degrees of freedom in the universe of its maxiometrically depends on the Planck scale. So, you've already put the Planck scale in there in order to calculate how many dimensions are over the space. Oh, that may be true, yeah. There's got to be some way to define this consistently. Anyway, so that's really the idea. What I don't want to suggest is that this solves everything, because it doesn't... I sort of feel like it does, but I'm going to generalize this. its biggest flaw is that the states that do this locking are of a very particular form and there's no mechanism proposed for why when a black hole evaporates it should make exactly these states other than that if it did it would preserve military I think that's a good mechanism a mechanism so much as a reason but really my main point is that tell you for why unitarity fails, like the Hawking radiation is thermal or the information can't leak out until the end, they don't really tell me that unitarity has failed.

1:50:00 All they told me is that we don't know exactly what black holes are actually doing when they evaporate. But even in the semi-classical limit, I see no reason to just accept that my beloved unitarity has failed and I have to modify quantum mechanics and fix it because there is really no evidence that it has. So you showed that the maximum accessible classical information has this property. But somebody might say, like I'm going to say right now, that what you should really be looking at is the bottom of my mutual information, trace row of A, plus trace row of B, trace row of log row of A, and then in that case, I don't think you get this property That is true. But I don't think it's the right quantity to look at. I definitely don't claim to be able to lock information of that type, but really what you're talking about when you say causality isn't violated is not that somehow quantum information or even regular information leaks out. What you're saying is can I predict the outcome of somebody's experiment before it happens, right? That's what causality means. And if I start saying things like experiment, then I'm really talking classical information and I don't know I don't know what it means to predict things in this in superposition because then you end up predicting nothing except that everything happens in the many worlds so so you have to tie it down to an experiment at some point or at least I prefer to tie it down to something that you get it can get a grip on at some point so I'm confused because you propose a tomographic experiment the tomographic experiment would actually to give you the answer from the outside, so I would seem to be that a... Well, really all I want to do in the tomographic experiment is calculate the entropy that comes out and see if it's zero or pi, to see how good the Hawking centiglassical calculation is. But that's really facetious, because I'm really starting out by assuming that everything is unitary, so I know what the answer could be before I do that. He's counting on nobody doing that experiment. Well, but if somebody does, I mean, it would be a very direct evidence of whether things are unitary or not. But, of course, it's basically impossible to do that experiment, so you have to drop back to theory. But even that experiment, it's tied to, you know, outcomes of detectors.

1:52:30 Everything's macroscopic by the time it's done. I have some detectors, the tomography is I have a whole bunch of things, and I wait for a detector to click, it clicks and I write it down. Then I do the whole experiment again with magnetic black holes. Experiments are always classical. I have one question about these states. So I guess they are, I mean, you said they are very particular kind of states that have this property. So they're probably like a set of zero measure or something like that. So how would the black hole achieve that? I mean, for whatever initial condition you end up in one of these states that's not a unitary process then I don't think they are a set of measure 0 I don't know, I mean, I'm asking I'm not exactly sure of the answer to that but I think they're not because their whole construction is you take a random unitary matrix and apply it to your state and that kind of encrypts it and a random unitary matrix you can choose ahead of time you just choose from a set of random things where the total number in the set is log D Well, let's say, I wouldn't say measure zero, I mean, there are fewer, I mean, most states will not have this property, let's put it that way, is that the, no, I just said the measure, the set of measure one of these states will have this property, you start with a random indicator of nature, so you can code your state that way, that states can be achieved by that, or by definition, the set of measure one, so, right, okay, okay, okay, fine, because I have to think about the counting of this, but I do think it works out OK. Then let's thank John again, and our next speaker. And we'll mic up our next speaker to be Charlie Bennett. Thank you.

1:55:00 I think the only thing that I don't know is... I think the only thing that I don't know is... Well, I think that's not that. I think that's true. That's true. You know, information... I won't wait, but it's not the answer. I think you can get your conduit's on it. He says, I can get your conduit's on it. Charlie claims his title is the same as it was on the program, but that's actually not true. The title of his talk is Black Holes Simulated Time Travel and Cross-Horizon Romance. I'm trying to take seriously my responsibility as the last speaker to keep people from getting bogged down in excessively serious thoughts on their way home. In fact, I had a boring title until last night that I was inspired by talking to Harry Berman and Andreas Winter. But they are not responsible for this idea here at all, but it's all mine. I mean, the bad aspects of it, the science is there, I guess. So this is all about some pretty suspicious ideas. We're going to talk about simulated time travel, classical and quantum. This teleporting out of a black hole by what I like to call the imperative bell measurement, or IBM, including the Horowitz and Valdesena proposal.

1:57:30 And then my main interest here is this new stuff because you've heard about this already and this maybe you've heard a little bit about it. So I'll try to zip along here. Originally I thought of this in connection with a conference in honor of the 50th anniversary of the Monte Carlo Method in Statistical Mechanics that was introduced in the famous paper of Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller, and this was a three-day conference partly in honor of Marshall Rosenbluth, who died a little while after that. But the basic idea of this method is that you want to explore the equilibrium configurations of something like a gas or a liquid, hard-sphere liquid, by making trial moves. And the trial moves usually fail, so you have to try another one, and you keep trying until you get one to succeed. So that involves a lot of rejection. And all this failure and rejection can be bruising to the ego. And so in what might be called the Woody Allen version of the Monte Carlo method, of this testing until the end. And of course, that makes the failure more likely, but at least you only fail once. So this is sort of this type of idea of repeating the rest of it. So now we're going to use this in connection with simulated timecraft. And this is work that I did with Ben Chewbacher, more like a play that works. Also, it isn't even new, because David Deutsch looked at some of her ideas back in 1991. But it is fun. We can't deny it that. So imagine that we have some physical time machine, such as a wormhole. So the question Ben and I asked each other is, what would it mean to interact with your younger self? And so we say, well, let's travel back in time, then you can interact with another you, and then you can come out in some state.

2:00:00 And how do you work out the logic of this interaction? Now, this is an idea that people have explored both classically for a long time and quantitatively. So here's how the Woody Allen Monte Carlo method can be used to simulate time travel without any exotic wormhole or physical movement. So we start off with a person in some state, and then we, at this point, we guess a trial future state for yourself, that is, for the other version of yourself, from an alphabet of possible states. You make two copies of them. One you put here, and the other you put there. So we've got this guest trials, we make two copies, an original and a back-off. And then, now we have the wherewithal to do this interaction. We have our initial state, we have the guest state of this other person. We interact, we get a result from the final state. And at this point, we then make a test of whether the guest was correct, and if it is, we keep the outcome. And so we'd have, by virtue of this post-selection, a faithful simulation of having experienced time travel without any special equipment. And so you can work it out as the states are bits. In this case, if the bit is zero and we have a control dot here, the output is actually probabilistic. The success problem is 100%, but the output can be either 0 or 1. So this is sort of the other side of the grandfather paradox, which Ben describes in this kind of story that Time Traveler, on his way to work one morning, meets a stranger who looks a lot like him, and they smile, then he goes to his office and in the course of his duties meets the same stranger and smiles back. if he met the same guy on the way to work, in another possible history, the stranger punches him, and so later in the day, when he sees the stranger again, punches him back. So this is a case of a single, an initial condition on a space-like surface which gives rise to multiple classical futures.

2:02:30 Now, the other thing, of course, that can happen is that there's no future. And this is the grandfather paradox. And so that's manifested in this diagram by saying that the success probability is zero. So you never get a prediction at all. Now, if you do this, this is a practically pretty well-known idea in the world of philosophers and science fiction writers who think about time travel. But if you quantize this, the thing to do here is to put in an entangled state and put a bell measurement here, and so then we do the post-selection in the same way. We, again, get a simulated time travel, but we're a quantum system. And the main difference here is that we don't get any randomness here. Pure input states are mapped onto pure output states. Here's an example. The multiple futures case that I gave with this experiment involving Ben Schumacher. it has a success probability of a half, and the output state is this pure superposition. And this is, in fact, represents a, oh, here's a case where it always succeeds. You have a swap. So it's pretty clear why it always succeeds, because this, it's being, you're doing a bell measurement on the, on the maximum integral state, and the other state's just going through. But in the grandfather paradox case, again, we get zero success probability. And as I said, in this general situation where we put it in an arbitrary superposition, we get the grandfather paradox only on the suppose of measure zero. In all other cases, we get a function which is a projection. That is a function that is linear, but not unitary. It is dependent on the initial statement. So this is my answer to these questions. It's a time travel. And when I gave this talk, this was shortly after the Clinton impeachment hearing. So this is a quote of the time. It doesn't involve phoning that the older and younger versions are not independent.

2:05:00 And what does the output state of the post-selection avails and the answer that is shut up. So, well here's how we've gotten revived, and I think you've heard about this from Danny Magonisman. Let's say we're having this state at the event horizon, and some of the radiation falls in. required this positive energy, so the Hawking radiation, and then we have ordinary matter going into the black hole. The Horowitz and Valdezena proposal was essentially to have a bell measurement here, but impose a boundary condition, so that the result of the special bell measurement was always the same standard state that you drop in. And therefore, there's no need to communicate it out of the hole. So, and that was the proposal, and the flaw in it that was pointed out by the Gottesman of Preskill, is that if you have an interaction here on the way in, then it becomes a non-unitary mapping. And now, this possible resolution that occurred to me was to perform the IBM not at the singularity but on the vent horizon, thereby preempting the interaction, and so we'll have this sort of thing. Now, of course, this is sort of the old hat for this conference. The main objection is event horizon is not locally distinguished from other places, so there's no reason to expect unusual physics. And my answer to that, which I now realize comes from my religious background as a point of information theory, is that the difference at event horizon is not a physical difference, but a legal difference. As soon as you cross event horizon, I'm no longer responsible for what happens to you or what you think. But I'm going to get into that some more later. And this is what's known as black hole complementarity, which people like myself like almost as much as we like in church at a larger number space. And the guy even made a picture to illustrate it. So imagine that Alice buys a luxury house from Bob. The British television is largely involved with buying and selling houses. So imagine that Alice buys this nice house from Bob.

2:07:30 And actually, she doesn't know Bob. he's a friend of a friend but she reluctantly allows him to continue living in the house for another week after the ownership uh has passed to alice but what happens is this really nice house begins to to look not so nice after a little while and uh the uh just because bob doesn't care about it anymore uh now that he no longer owns it okay well this is getting very controversial I don't want to talk about this black hole information paradox. So let's go to a more productive topic. Romance across event horizon. Now, the idea is, suppose that Alice has fallen into a black hole, and Bob, although he's not willing or able to join her, still wishes to flirt with her. Now, how does he do this? And when I say how, I mean both the techniques involved and the impairs. So, well, of course we have this nice metaphor for the black hole, which is just a hydrodynamic metaphor. And I actually, even before this conference several years earlier, I made a picture of one outdoors. This is a beaver pond with a leak in the dam, so there's a natural vortex of the water running out. And I think if you look carefully here, well, let's put Alice near the hole. She's fallen in. And Bob is out here. Here you can see what I think is an evangelize in the creek. I mean, it's not quite the same thing as a hydraulic jump, but it's one of these hydrodynamic phenomena where there's a rather sharp transition here to here. But in any case, imagine this as a black hole with an event horizon. Alice is inside and Bob is outside. So, this non-local box is a device that has been considered a great deal in this conference. It's a distributed two-input, two-output device that maps the inputs onto the outputs in such a way that the XOR of the outputs equals the AND of the inputs,

2:10:00 but otherwise the outputs are random. Now, these have been widely discussed in this program, but actually their main importance, as everyone knows, is that they are very popular Valentine's Day gigs because they enable people to flirt rather politely and discreetly. By each privately inputting into their input, the desire or lack of desire for the other person and then exchanging the outputs, Alice and Bob can answer the perennial question, does she love me, does he love me, with minimum hurt feelings. And this means in particular, I mean, there's no way why some hurt feelings, for example, if Alice puts in, yes, I love Bob, and she gets the answer, no, then it is clearly because he doesn't love her. But in some other circumstances, for example, Alice puts in no, and the output tells her the answer is no. She doesn't learn whether Bob loves her or not. And this is mainly a good thing because it spares him the embarrassment. So this is really some idea. Something that Alice doesn't learn reduces Bob's embarrassment. But this is the kind of thing, etiquette is all about. Well now the good news for trans-horizon romance is that non-local boxes have no communication ability in either direction. Therefore we can postulate them extending across a horizon without blaming anybody's religious passions. So let us, for the rest of this program, assume that there of local boxes across the vendor eyes. We're going to ask them how would they be used in the flirtation between Alice on the inside and Bob on the outside. Well, of course I should say there's a bad news. The bad news is that this follow-up communication is only possible in one direction. So Alice cannot talk back to Bob. So in order to compare their outputs and get the joint result of the consultation, Only Alice learns that, or could learn that.

2:12:30 Bob is forever ignorant. So, Bob never learns whether Alice loves her or not. However, he can send a message in to where Alice is, and she can learn the results. And this message could be a single bit, or it could be something a little bit more chivalrous than that. But let's just explore the case where it's a single bit. So here we have the cross-horizon flirting using a non-local box. Now, as I said, Alice above words, nothing. So we're not going to think about his results. We'll only think about the results from Alice's viewpoint, which, however, includes her thoughts about his feelings about her right feet. This is a subtle thing. We're only going to look from Alice's viewpoint. So let's assume that Alice gets this letter that Bob drops in, and he says, my non-local box gave the output zero, thought you'd like to know, because my forever box. Okay, so let's imagine the possible inputs that Alice might have had, and outputs, and there's three of them, only possible because one of them is excluded by the fact that Bob got a zero, meaning that the and, that the and was false. Like, if one of them was excluded for reasons, you will soon see. So Alice thinks if her input was a one, meaning she loves him, and her output was a one, meaning that the x order of that and this is one, she says he loves me. I only wish I could tell him that I love him too. Forever. So this is very poignant. Now, suppose Alice didn't love him, and she got out the answer zero. In other words, zero X or zero is zero, and so that the date is off. Well, that's pretty clear it's off because her input was that she didn't want it. So this is what she thinks to herself then. I didn't even have to open his note. I already know that the XOR of the outputs would be zero. But now let's look at the third possibility. And Alice gets, and puts in a one,

2:15:00 and she, that is she loves him, and when she gets back a zero, which when she XORed with that, that tells her that the answer is no. And so she says to herself, Alas, my love for him is unrequited. At least he will never know how I felt. So, but this, as I indicated before, this is not the only kind of letter Bob can send. Let's imagine some other kinds of letter. Suppose Alice gets a letter like this, and she says, Never mind the damned discreet non-blooded boxes. This is no correct for discretion, and I love you, I'm missing you forever by. Well, this is really very nice, because Alice learns that Bob loves him, even if she didn't love Bob. In fact, she learns that he loved her so much that he didn't mind getting his feelings hurt when she finds out that his feelings are unacquainted. So, short of joining her in the black hole, this is the most, the most chivalrous thing you can do. But then And then there are also really nasty bobs out there who send letters like this, and in this case, you can see Alice learns of Bob's lack of interest, even if she's also twerks. This is just plain mean and is adding insult to misfortune. I want to leave you finally with a single ethical question a really important ethical question which is is it wrong to offend someone or each of them after they've fallen into a black hole thank you so are there any scientific or ethical questions for Charlie Don't you think that it would be more simple risk for Bob to send some, you know, quality entertainment programming or something? Instead of just being blown up so that people are occupied longer. Well, it depends on what kind of person Alice is. Yeah, she likes that sort of thing. Sports results measure. Stock market. That's right. Dear Alice, since you fell in the black hole, the Red Sox actually won the World Series.

2:17:30 okay well let's thank charlie again and thanks and if people want to ask more questions there's still coffee available Thank you. Yeah, it's actually, in fact, there are, I haven't spoken secretly to people out here in the world before I woke up like I could actually talk about a lot of times, I'd talk about a lot of times. That's how the confusion started, because they always have a question, and then the What kind of plan is an issue between each other. It's part of the state. It must be an initial condition. The beach is unknown. Whatever it is. It's an initial condition. The beach is exactly the state of it. There he is. And then... I don't know if I understand that either, but the love is a ton. Basically, things are causally separated by the metabolizers. They're not information if you've ever crossed the metabolizers.

2:20:00 Well, I don't exactly... I can't understand it either. But if you accept that as the important feature of the metabolizers, It's possible when the black hole is kind of like scale and there's no more memorizing anyway. Okay. And that's why the desire is there. Okay. See them sticking around a little? Can we come out to dinner with us later? Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Thank you. Thank you.