Entanglement and Complexity: Gravity and Quantum Mechanics

As you go, the

entanglement

s on smaller and smaller scales develop and eventually when you get to the top where the real, honest degrees of freedom of the system live, not some mathematical aids that you have introduced to describe it, but the degrees of real freedom in which the

entanglement

structure has been constructed using this crazy Fineman diagram that is not really a fan diagram. I just drew it, but it has some similarities to a fan diagram and represents the entanglement structure of the

quantum

state of a A lot of them could be atoms, they don't actually spin, they could be atoms in a line, this was a breakthrough, this was a breakthrough because it allowed us to navigate through the enormous

complexity

of Hilbert space for certain systems here's another example in this example the real degrees of freedom the red dots are again arranged in a circle a periodic lattice this is something condensed metap physicists are always interested in a periodic lattice how do you fill it with a Tensor Lattice to describe the state of the limit here is the type of Tensor Network that you would draw again it is a tool to calculate the limit to calculate the wave function of the little elbows that live in the limit the tensor network is purely auxiliary massive geometry for geometry, I mean, well, it has a kind of geometry look, if you move your eyes a little bit you might see some kind of geometry there, but it's not real, real is on the edge. is the Tensor Network that you draw to describe a single state, not even approximately, there are many tensor networks that can describe the same

quantum

state, but for any limit theory quantum state there is a Minimum Tensor Network, the Minimum Tensor Network means the most small. thing of this type that you can use to construct the limit state the

complexity

of the state an important word the complexity of the state is simply a measure of the smallest tensor Network that can describe that state describes how complicated it is to navigate your Through the space of Hilbert to find that state, the smallest tensor network and that's an important concept, he also defines a type of cross-grain geometry, a lattice geometry almost well.

Here is a question that tensor networks can answer. I've chosen a particular tensor network, uh. sorry, one question in particular, you might be interested in splitting the system into two parts, the system now consists of the limit. Remember that you might be interested in describing and dividing into a and b and asking how much entanglement there is between side a and side B. There is a technical question that is not very important if you have a precise idea of ​​what it means, but it means some measure of the correlation between the lower half and the upper half.

What Swingle and others discovered was that there was a simple answer to this from the tensioner. Network, that is, you draw the minimum path through the Network tensor that divides the system and minimum means the smallest number of links that it intersects, that smallest number of links that it intersects is a direct measure of the amount of entanglement between A and B that was very interesting, but was it something important? Well, if you're interested in entanglement in condensed matter systems, yeah, if you're interested in anything else, probably not, um, yeah, but another question also came up that also yeah, sorry, I'm not here. tell them I'm not here this happened once this happened once once I had a class in the middle of class the phone rang like this and someone on the other end picked up the phone there was a phone on the wall in one of the in one of the classrooms I pick it up and the guy says uh I'll have a pepperoni pizza and two beers come on this is not a pepperoni pizza place he says come on Ray I know your voice I know your voice send me a pepperoni. pizza with two beers I know I'm sorry, this isn't Ray, who do you think you're talking to?

It's Domino's Pizza, right, no, it's not Domino's Pizza. We argued for a while and finally, you know, I hung up on him. This turned out. It was one of the students in the class who had set this up, never mind, okay, here's another question you might ask yourself: is there a way to break it, given that two parts of a system are intertwined? Can you break the entanglement? Can you do something to destroy the entanglement the answer is yes you can, you have to feed energy to the system, but can you destroy the entanglement, can you untangle A and B and the question is what would the Tensor Network look like for the state in which A and B had their intertwinements. destroyed and that's what it looks like, it disconnects the tensor network into two pieces so that they are no longer connected and the message here is that the connectivity of the tensor network is connected to the entanglement of the boundary degrees of freedom, the actual degrees of freedom, because?

Am I spending so much time on this because it's deeply connected to something that has really revolutionized our understanding of a lot of things? It's called CFT advertising. It is a duality. It is a duality between gravitation on the one hand and anti-space and conformal field theories. That's a kind of quantum field theory, it was discovered in 1998 by Juan Mesa and it's just to look at how important it is. It is, by far, the most cited article in all of theoretical physics, of all time. more, I don't know 15,000 quotes, it's very important and I'm going to try to tell you a little bit about how it fits into this Tensor Network story.

This is an image of anti-Deiter space, of course not. It's an image. a simpler drawing but it is an image of the geometry of anti-anti-space now the anti-space is a space-time it is not just a space it is a space-time with a t direction and a space direction This is an image of space cut through anti-Deiter space in an instant of time, what does good mean? What it means is that this is an image of a geometry. You must imagine that each angel or each demon is exactly the same size as any other demon, but due to the curvature of space and warping. of space when you draw it like this it makes the demons near the limit seem very small and the angels and demons near the centa seem big the phenomenon is the same mathematical phenomenon that makes Greenland seem bigger than Africa Greenland is not bigger than Africa but because of the curvature of the Earth when you plot it on a plane, Greenland appears much larger.

The same phenomenon. The mathematical phenomenon distorts all curved spaces when you try to place them in the plane and this is what anti-de space. Space would look good, antispace is a space-time, it has a time axis, it has a space axis up here that we put all the angels and demons on. I couldn't draw them. I can't draw them. so I left them blank and most of the space the SpaceTime looks like a soup can which I have repeated repeatedly of course it has no bottom or top it just goes on forever but it looks like this this is anti - Deiter space What he discovered badly is that there were two descriptions of the same thing, a duality and the duality had to do with quantum field theory, where quantum fields live at the edge of space-time on the label of the can. of soup and something else, there's something else which is

gravity

, but we'll get to that, okay, what is this drawing here?

This is not the poorly drawn Campbell Soup can label. This is a financial diagram for field theory on the label and it is supposed to be representing a Fineman diagram starting a particle enters here a particle enters here and leaves here and here a scattering process why is it so complicated It's complicated because very strongly coupled quantum field theory is one of these quantum field theories, which is very difficult because the Finan diagrams are so complicated that no one can calculate it. What Mala discovered is that it is completely equivalent to a very simple Fineman diagram, except that the lines of the Fineman diagram go through the soup, through the dough, through the interior. from can number one, but they are very simple, easy to calculate and those lines represent particles, what type of particles, gravitons, gravitons or mass gravitation is somehow equivalent to quantum field theory and the limit, this is what which Mal discovered how long ago in 1998 and has revolutionized theoretical physics tremendously.

Well, this again is a space image of anti-space and let me ask you a question. It's not the kind of question you'd probably ask, but I'm going to ask it anyway. remember that there is a limit Quantum field theory that lives on the limit and assuming again that we were interested in the entropy of entanglement between two halves of the field Theory field theory on the top and field theory on the bottom the void how much entanglement is there? Well, a few years ago, in another very brilliant discovery, two physicists named Ryu and Takagi discovered that the mathematical answer to this is to draw the shortest geodesic that cuts A and B into two parts and then just the length of that geodesic sounds familiar.

It should because I just said something pretty similar a few minutes ago, but that's what they discovered Next, another physicist called Ran Ramdon, who spoke here about two weeks ago, asked what would happen if you did something that destroyed the entanglement between A and B which What it does is break the anti-de space into two pieces, this should look familiar. I just said something like that about tensor networks and condensed matter physics systems. There again is the Tensor Network which describes a system that lives on the limit, but is described by this type of Findan diagram tensor network discovered by swingle and others, if you are interested in the entanglement between the upper half and the lower half , simply find the shortest path through the network and count the number of links much like calculating a geodesic length and if you want to destroy the entanglement, break the Tensor Network into two parts which is the message here.

I just gave you some very simple examples of something, but the message is that somehow anti-Deiter space thought of as real spacetime. A cosmological space-time. Most of it is in some way the tensor network to represent the state of a boundary. Quantum field theory. This was great news. I mean, this is no small thing. Most of it is being described by quantum writing. The limit. by a strongly coupled quantum field theory, but bulk spacetime, the soup in the can, is also a real spacetime in its own right, anti-cosmology, spacetime is the soup in the can, not the limit label, the only difference between the limit label and the soup can is the limit label.

Theory has no

gravity

, it's just common garden variety. Quantum field theory. The soup in the can, the spacetime inside, has real gravity and quantum gravity, so this obviously caught a lot of attention from physicists. Okay, let me go ahead and tell you that studying the ground state of anti-space and the ground state of condensed matter physics systems is very interesting, but it's not all there is: you can heat systems, you can put energy into them, you can make them. vibrate. you can force them to do things, and when you do, you can ask what quantum states look like when represented as tensor networks in particular, assuming you put in enough energy to heat up the degrees of freedom described in the limit of the condensed matter system.

How do you describe the resulting state as a tensor network? The answer is, first of all, excited states are much more complex than the ground state. They must require a larger tensor network. That's practically a theorem. The most complex state is in the sense. Fineman's sense of this enormous amount of complexity that Fineman talked about, excited states are more complex and therefore require a larger Tensor Network. The answer. I'll tell you the answer. You cut a piece of your original Tensioner Network and connect it. it's a larger tensor network that has more structure than the original one that described the ground state, the more complex the state, the bigger this um bag of uh goodies is, okay and you just plug it in.

I couldn't draw it plugged in. but that's how it works, but there's something very interesting about it because of the nature of quantum complexity and because it can potentially be so large. What happens to a quantum state? An excited quantum state is one in which its complexity begins to increase over time as time evolves. States are getting bigger And getting bigger, that's not too surprising. You take just a random starting point for a bunch of degrees of freedom and you let them go and as time goes on they become harder and harder to describe, especially if the system is chaotic, that hardness of description.

It is stated in terms of these tensor networks by saying that as time passes, the tensor network needed to describe the quantum state grows and grows in a certain kind of linear way. This is not something easy to see, but it is something that we have learned to understand that tensor networks grow over time, in other words, the tensor network describes a state and an instant of time as time evolves, the tensor network of a complex system grows to a certain point, okay,Keep that in mind and let's get back to anti-Deiter. space let's go back to this quantum gravity uh duality of the label Supan but now let's do something let's add heat add energy throw energy into the soup can and what happens the energy gravitates towards the center the anti-of space has a gravity field that attracts things towards the center the energy is attracted to the center and if there is enough a black hole is formed so black holes enter it we throw energy into anti-space we make black holes we have learned a lot about these black holes, for example, we have learned that the thermodynamics of quantum field theory the theory on the label is essentially exactly the same as the thermodynamics of black holes, the Hawking Beckenstein thermodynamics of a black hole, this back and forth interaction between field theory and the gravity. has played a central role in our understanding of many things so far, okay, but what's inside the black hole black holes have their interior, not just their exterior, but you can fall into a black hole.

If you fall into a black hole, you will discover that there is space behind it. black hole in some particular description space looks like this here is the outside of the black hole here is the black hole seen from the outside here is its Horizon but behind the Horizon there is a big bag of I call it the bag of complexity but it is just a big bag of space and oops, let's not go there yet that bag of space when you solve Einstein's equations there is no complexity just Einstein's equations that bag type structure grows it gets longer and longer very familiar yes I heard what I said five minutes ago on tensor networks.

The tensor network of a thermally excited complex system looks like a bag of complexity that grows over time, so it seems that these ideas connect antispatial tensor networks. Duality between quantum gravity, on the one hand, and condensed gravity. systems of matter or quantum field theory, on the other, seem to involve not only simple things but also things as complicated as black holes, well, here is the black hole again and here is its interior, imagine someone falls inside, why they can't get out and the reason. They cannot get out, it is because the interior grows too quickly.

The growth of the Interior drags down everything that has fallen and prevents anything from coming out. That is one way of thinking about the fact that you cannot escape from a black hole. let's go to a completely different problem, we'll come back to this of course, it's the problem that Alice and Bob have Alice and Bob are space cadets, they are in love, but the captain of the ship tells them that Alice for one reason or another. they have to go out on a separate small ship and bomb on another separate small ship, then they have to do some exploring and they get separated by accident and they get so far apart that they end up 10 billion light years away from each other, how are they going to get back? together again they no longer have 10 billion light years they have used a good part of their 10 billion of their 12 billion light years 12 billion years they are not going to get back together when they get back together so they have the idea what would happen if we created a pair of black holes so Alice says let's create a pair of black holes and B says what the hell is that going to do I don't know I don't know what's inside maybe maybe if we make a pair of black holes we can jump around and meet up and do those kinds of things that we like to do and okay, there are two versions of the experiment, the two versions of the experiment, one is this: Alice and Bob are the same.

They are equipped with a bunch of particles now each one now the particles may be particles that they found there wherever they are they are particles that never had anything to do with each other in the past completely disconnected Eh, very far from each other from each other and they have never had nothing to do with each other technically they are untangled they have not become entangled they are completely independent and what are they going to do with those particles are they going to take those particles jam each other gravity and create two black holes this is not going to do anything very interesting it will create two black holes very separated from each other the green here indicates 10 billion light years each of those black holes will have an interior it will look like this bag of complexity the bag of complexity will grow but they will be completely independent of each other and there will be nothing interesting between them , so they will lose, they can't win that way.

Here's the other experiment before Alice and Bob split up, they had a bunch of particles in their lab and they entangled them, they created pairs of entangled particles, they're called Bell Bell pairs for John Bell pairs of particles that got entangled like that first one. diagram that I drew you at the beginning and for each pair of particles that Alice has one of them and Bob has the other, they take them and leave and leave their 10 billion light years each one has half of the entangled particles, so that the two states of the particle system are highly entangled, they have entanglements between them. that prevents them from separating them no, it does not cost any energy to separate entangled particles the particles are now very distant but they are entangled the meaning of that is that when black holes are created the black holes will be quantum entangled The states of the black holes will be highly entangled, be it whatever the meaning of the entanglement.

Can we get any idea what the quantum state of entangled black holes looks like from tensor networks? And yes, we can, here are a couple of tensor networks. I've omitted the interior here. the way I drew it uh two tensor networks for two completely disconnected systems of limiting degrees of freedom or two completely disconnected systems are not intertwined here is one here is the other here is what it looks like let me compare that to what the Network would look like tensor if these and these could be very far away from each other, it doesn't matter, they could be 10 billion light years from each other, but what the Tensor Network would look like if the two halves were tightly intertwined here is the best I can do to draw it would mean that the Tensor Network would have an edge for each endpoint here it would be connected to an endpoint here there would be some kind of bridge in the Tensor Network mathematically connecting them to each other, which suggests an answer to the question of how you will see two entangled black holes the answer is now familiar to people who study this kind of thing here it is here there are two black holes in very, very distant layers of space they are so far apart, I have drawn them as completely independent and if the black holes they are intertwined, so they form what is called an Einstein Rosen bridge behind the Horizon, entering through one where you exit, you can't actually exit for reasons that will become clear in a minute. but if you could get out, you would enter through one and exit through the other.

Sometimes called a wormhole. If 5 years ago someone said they would be interested in wormholes, I would say you were crazy because there is no such thing, but now we know. There is a virtual certainty that if two black holes are created with a high degree of entanglement between them, they will have what is called an Einstein Rosen bridge between them. This type of geometry was discovered in 1935 by Einstein Rosen and P. Einstein and Rosen, excuse me, here is another image of the same thing and here I have imagined that these are the 10 years, the 10 billion light years that separate Alice and Bob, of course, I haven't drawn these 10 billion, but they are very far away and the distance through the wormhole can be very short, so Alice and Bob, who have created this highly entangled black hole, now say, ah, let's jump in and do our thing inside the black hole that we won't be able to get out of again for some reason, but you know, 5 minutes of happiness is worth a lifetime of agony. and so on so let's do it, let's do it right there's a little problem and the problem goes something like this here is our pair of entangled black holes starts in some quantum state but that quantum state evolves towards increasing complexity. getting more complicated as time goes on, what does that look like, well I would love to do this, it's so much fun, look, look closely, this is what happens, complexity makes Einstein Rosen's bridge grow.

This phenomenon can also be seen in tensor. networks exactly the same phenomenon let me do it again oh boy I love this okay that's why it's so hard for Bob and Alice to find each other in the center before they can meet. The thing grows, it's also the reason why you can't escape and get out. of a black hole now they can do something they have the greatest technology in the world it has been extremely powerful the most powerful quantum computers in the world they can put their black holes into quantum computers and prevent this from happening the answer is believed to be yes it does not violate any principles of quantum

mechanics

to prevent it, but it is very, very difficult whether it is possible or not.

It's not clear in principle, but Alice and Bob can't easily stop the system from growing in the middle, so they get frustrated because they can't. do it, but um, but still, in principle there is a connection between their entangled black holes that does not go through ordinary space, not the long path but a short path, that is called the Einstein Rosen bridge and the equivalence of the Einstein bridges. Einstein Rosen with interlacing. Yes, I didn't tell you the acronym that goes with this, the acronym for this equivalence between entanglement and spatial connectivity is called E for Einstein Puente Rosen and E which stands for Einstein Podolski Rosen himself Einstein himself Rosen Podolski was a newcomer here the funny thing is that Einstein Rosen's discovery of bridges and Einstein's discovery of entanglement were the same year 1935.

I don't think he had any idea that they were connected in any way, but you know it was Einstein, so they didn't call. He Einstein for nothing is fine, so there we are E equals epr and that is a genuinely new and unexpected principle that is really something very surprising. There seems to be a deep connection between quantum

mechanics

and general relativity, something much deeper than we had any idea. 10 years ago, another thing we discovered that I explained is how increasing the complexity of a quantum state causes the geometry to expand. This is very interesting, the space behind the black hole did not exist before the black hole formed. and then once a space forms inside the black hole and grows, what that growth of space has to do with other forms of growth of space, like in cosmology, we don't know, but it is some kind of appearance of space out of the tangle that uh that's just beginning it's just beginning to be explored okay, this is good for anything, does it help us solve any problems?

How much time do I have? Yes, I still have a few minutes. Okay, can you help us solve some physics problems? Apart from the physics problems of quantum gravity, does it help us solve anything else? You know, that's usually the case. Newton invented calculus to solve gravity problems. Calculus was a very powerful tool and solved all kinds of problems. I think we are starting to discover them thanks to these. dualities between gravity on the one hand and other types of systems on the other hand that these ideas are starting to play hopefully, like I said, I don't like to sell things, but I think it's happening that these ideas connection between gravity on the one hand hand Quantum field theory condensed matter systems tensor networks are beginning to solve problems that were generally considered too difficult to solve, what kind of problems?

Well, there are non-equilibrium statistical mechanics problems. What is called left side here is non-equilibrium statistics. mechanical phenomena like revolt, which is another word for chaos and thermalization, on the one hand, on the other hand, believe it or not, computer science problems are also not being solved but are related to the problems of gravity and the problems of black holes. I'll just show you a couple of examples. I have listed a few names here. I want to read the names because these are the names of the people, many of them are around here, in fact, most of them are around here.

This is a sitp. project along with no, it's not the only place where it's being done, but it has a very healthy representation at the sitp sind Institute for Theoretical Physics, okay, so I'm just going to read the names of Hayden, which is Patrick Hayden, who may or may not. Sitting here I don't know, it's a professor in our department John Preskill, who is a student of Hayden, no, that's not true, me and Sakino, Stanford and Shanker, this is not Stanford, this is not um, this is not Stanford nor Stanford University, this is Douglas. Stanford, who was a student here for several years and then did his Mark Stanford Shanker and Mala KV, whose name probably those of you who know condensed metaphysics will know swing very well and there is another name here that is a name.

That sounds familiar to many of us Monica SCH Smith, who is an experimental physicist. I put this here because Brian Swingle, who was a post here, and Monica, who was a teacher here, have come up with an experiment to try to do to test some of these ideas. Lab test. The test isis taking place on the other side of the column. Connections are being made here to very theoretical questions in computer science. Quantum computing science questions. One of the discoveries is very recent by people here as well. Amed Al uh shiang Daniel Harlo, who was also a Here and Now Gone student, discovered that these tensonic networks that are apparently equivalent to anti-Deiter space are also quantum error-correcting codes.

This is very important because if you want to build a quantum computer, error-correcting code is absolutely essential. mathematical error correction codes are not things that are built in the laboratory maybe you build them in the laboratory and then complexity theory Complexity theory is an important mathematical topic that emerged from computer science, its heroes were Alan Turing and Alonzo Church and his quantum version um Charlie Bennett and dot dot dot dot to the present and I'll tell you a little bit, a little bit about how complexity theory is connected to black holes, okay, first I'm going to study.

First I'm going to tell you about the coding problem. The problem with coding is if you have a complicated system, say in thermal equilibrium and you perturb it, how do you transmit the perturbation information? propagate through the system how long does it take to get lost in all degrees of freedom? how can it? how long can it be traced?, etc., encryption is the process by which information is disseminated through a system to eventually be lost. in many degrees of freedom it is also called chaos, it is also called thermalization or elbows, a system of spins and we will only assume that the Hamiltonian is a sum of pair terms each particle interacts with all the other particles, it may or may not interact with all the others particles, but at most it interacts in the Hamiltonian, say with another particle, so pairwise interactions, an example again would be a spin in which each particle interacts with its neighbor.

Well, with its two neighbors on either side, if it were higher dimensions, each particle could interact with three, four or five, that's fine, it doesn't matter for systems like this, it's pretty clear how long it takes for information to diffuse through the system if you disturb one of these and it is in thermal equilibrium it will kick the next it will kick the next it will kick the next and the signal will propagate through the system and what is called ballistically with some type of speed the speed is called butterfly speed for the effect butterfly uh and uh it will take a time that is linear in the distance or linear that the size of the system that is fighting for a system like this, there are other systems where these terms in the Hamiltonian connect each pair of particles, each pair of degrees of freedom Are you connected.

An example that could be something like this would be a large, very complex molecule where everything is linked together enough that everyone couples with everyone else, but again with pairwise interactions this is a large complex molecule this is a spin chain alright the large complex molecule thermalizes or encodes much faster than the spin network and I'm going to give you a model of how it's encoded this is how it works imagine In this room here, someone is infected, well let's not be so Grim , someone has some black ink on their hand, a nasty kind of black ink, but not enough to kill you.

Only Pierre has a black ink in his hand. We're going to play the next game every 10 seconds, we blindfold everyone every 10 seconds, we spin around like crazy and everyone finds a partner and shakes their hand and then we do it again every 10 seconds, they spin around and shake hands. the hand again how many? people after the end times steps will have been infected with the black ink that Pierre originally had on his hand. Well, the answer is clearly growing exponentially, by the way, this is the same math of the way that, um uh, the contagion spread after one step.

Pierre and his partner will be affected the partner he shook hands with after two steps each of them will infect two other people the infection will grow exponentially but this assumes that anyone can interact with any other person if we had a long chain of people with each can only interact with its neighbor, it would spread much more slowly, so after the end time or after T time steps, the number of infected people is 2 to the power T. Well, how long does it take to affect everyone in the room if the number of people in the If there is space, then we take the logarithm of both sides, the time it takes to infect everyone, that is called tar.

It is standard terminology. Forget this factor 1 over Here just the logarithm of n log to base 2 of n. Now, which one is open? getting right what is the speed at which people interact, obviously the temperature, if the temperature was zero everyone would be too cold to move, as the temperature goes up people become more energized as the temperature goes up a lot, they move like crazy, so here's a rate the rate is the temperature the time it takes for everyone to get infected is one more than the temperature times the logarithm of the number of degrees of freedom and the number of degrees of freedom for a system like this is proportional to its entropy s is entropy now if you don't follow all this it is not very important the point is that it is logarithmic in the size of the system with a coefficient that is the temperature this is called fast coding this is called fast coding and this is what is expected from this type of large molecule, this is called slow coding, there was a conjecture, the conjecture was originally due to a combination of Hayden and myself and uh and uh But and the conjecture had two parts, it went as follows, no physical system can encode faster than the contagion model than the handshake model no physical system can encode faster than this the fastest the shortest time it can take to completely encode a system of s degrees of freedom It's one about T*log s that was a guess, okay?

The second part of the conjecture is that black holes move as fast as black holes when something falls on them, they erase their information at that rate, okay, these were conjectures, they are not new conjectures, I don't know, five, six , seven years. old, I don't remember, but very recently a couple of really notable articles have demonstrated these conjectures. Conjecture 2 was in fact, in a sense, the first to be proven. Shanker, our own Steve Shanker and Douglas Stanford, showed that black holes are fast encoders and that they encode on such a time scale, okay, and then number one, which is an absolute limit, now the coefficient here I haven't written the coefficient numerical, there is a numerical coefficient and they discovered that no physical system can code faster. that, they started from an assumption, you know the assumptions that went into it, but the assumptions are very plausible and therefore this quickly confused conjecture has now been proven.

I've only got a couple of minutes to tell you how it was proven, uh, part of It's how it was proven how they proved that black holes are fast encoders, so I'll jump in now for those who don't know general relativity, you just have to close your eyes and Be patient with those who do recognize them as Penrose. diagrams that images of black holes I will not explain them I will not explain them at all this is a black hole in anti-is space the yellow region here is behind the Horizon the white regions out here are in front of the Horizon and some of you know this image, this image here is an antida space with a black hole or actually a pair of entangled black holes connected through a wormhole in the center and this is the region behind the horizon, the yellow region here.

It's more or less now I'm paraphrasing, I'm paraphrasing what Shanker and uh Stanford did, but it goes something like this, they said disturb the system a little bit, oh, by the way, the limits of anti-de space are these vertical lines here, this it's the limit in spacetime these are the limits this is the mass these are the singularities for whatever it's worth they said imagine going back deep into the past and disturbing the black hole a small disturbance a small butterfly disturbance a butterfly flaps its wings at the black hole and makes a little perturbation here that little perturbation will evolve as time goes on two ways to think about it if you think about it from the point of view of the limit Quantum field theory is just chaos oh I mean cha By chaos I mean chaos the butterfly flaps its wings that affects some neighboring molecules the neighboring molecules affect other neighboring molecules affect other neighboring molecules and whatever state of the system it would have made, it moves away exponentially and goes to Some other state has to do with why a small butterfly can flap its wings and affect the climate 2000 years from now.

Well, that was the image at the posterior limit, but there was a much easier image to deal with and the one you used on yourself. the global geometry and in the global geometry what they found is this small perturbation that shifts towards the blue and becomes a powerful shock wave that passes through the interior geometry. The powerful shock wave was much easier to calculate than trying to track the chaotic behavior of the boundary. The theory here is again, I'm paraphrasing what they found, what a combination of some of us found is that the encoding phenomenon, the phenomenon that the system has spread its information over everything, that time scale is exactly the same as the time scale. for the shock wave to reach a plon and a distance from the horizon, it goes in and out and this is easy to calculate, just a little general relativity will allow you to calculate how long it takes for the shock wave to arrive within a plun and a Horizon distance and the answer was here, is here essentially exactly the same as the conjectured limit on encoding time.

What they provided also was a numerical coefficient of 1 over 2 pi, which is very, very accurate, this was done by maldosa Shanker and Stanford several different papers and also something very similar by uh by um by kayv by Alexi kayv so this essentially showed that black holes are fast encoders. They were also able to prove that nothing can code faster than that, okay I think. um, yeah, let me finish sort of. No, I don't have time to talk about the complexity, uh, which is a fascinating topic in itself, but I think I'm done, so I'll leave it. here but some of these ideas are testable they are testable in the laboratory why because quantum mechanical systems you can build large systems of quantum mechanical atoms you can make them interact quite strongly remember what I said in the first place that now It is believed that there are dualities that connect strongly in large systems that interact with gravity problems and you cannot test gravity, which is too difficult to test, but you can use gravity to calculate and then test in the laboratory how quickly they occur. various types of coding phenomena.

Monica and Brian designed a Brian's experiment was based heavily on the work of Mala Shener and Stanford and believe that they can actually do the experiment effectively on what is a system of cold atoms that represents a kind of large molecule in the one where everyone interacts with everyone else, disturbs him and asks what's going on with him so some of these things can be checked. This is very interesting. I do not think they do. I don't think they expect to find any big surprises. Of course, it would be very surprising if they violated the limit. and I found that the encoding happened faster, that probably won't happen, but I think I'll leave it at that.

I won't try to get into complexity, which is my own interest in this topic, but complexity theory is starting to play a central role. role in understanding the growth of the interior of black holes, so I hope that answered the question of what happened in this secret meeting, but of course this is not what I did, what happened is that we talked about recorders video, okay? that will be all for today piece of quantum complexity of a state this is Fineman's observation where there are enormous numbers of states in Hilbert space and the time it takes to explore them all can be exponentially long uh the increase in complexity techn technical complexity technical complexity from the gate whatever the increasing technology of the technical complexity of the quantum state of anti-space black holes that appear to be dual, which means another description of the growing space behind the horizon of the black hole, so That's something I don't have time to describe, but what we think about now is that complexity is actually equal to the increasing action of a region of spacetime and that's being explored now, but that's a topic. for a um uh for another time because it's just not the time you have two two photons thatThey are entangled, yes, and they separate, you can measure the spin of one photon and predict the spin of the other, what does that kind of entanglement have to do with heat?

It can be connected with the same type of entanglement, it is exactly the same type of entanglement if what the entanglement represents is a degree of correlation that is beyond ordinary classical correlation, um, okay, pretend this is a red pen , okay and pretend it's a black pen, this is a black pen. pen Pretend this is a red pen if I put them behind my back and shuffled them around and then, you know, I gave one to Bob but I told him not to look at it and uh and uh one, uh, to Art here he said not to look at it. and then go Bob goes to Alpha centuri Sorry Bob, no, you don't have to go to Ala Cent, go home, but I have, I have one of the P, yes, Bob is going home. he goes home and they live very far from each other and then Bob looks at his pen and says it's red, he instantly knows that Arts is black, okay, that's the classic correlation, that's the classic correlation.

Quantum correlation or entanglement is a very strong form of this, he says. is that Bob before, and Bob could decide at the end of the game if he's going to measure the Z component of the spin, in which case if this is down, this will be up, but he can say no, I'm going to measure the X component of the spin. , in which case if this is this way this is this way this this way then it is a very very strong form of correlation that goes beyond anything that classical correlation can do, it is called entanglement and it is in its own right a fascinating topic, but we are talking about exactly the same type of entanglement, so if Alice and Bob have two black holes and Alice decides to do an experiment on her black hole, if they are entangled, she will immediately know something about Bob's black hole.

There are many experiments she can do on Bob's black hole to determine the properties of Bob's black hole and she will instantly discover the properties of Bob's black hole if they are entangled. It's the same kind of intertwining except on a huge scale where we're talking. about many, many, many degrees of freedom, yes, or it happens, that's right, nothing can go from one outside to the other outside, however, it seems that what is possible and does not violate anything we know, that two people can Jump inside and find yourself in the center. but they can't recover that information, so there's no tangles or tangles.

Einstein Rosen bridges don't allow signals from outside one system to another, but they do allow it, or at least the Einstein Rosen bridges do allow this gathering in the center, so that up here, yes, you made this equivalence between bridges and interlacing. it's space just full of worms BS, it's just worms Pro, yeah that's the kind of message, however, you know, if you just have two particles that are entangled, let's call it a wormhole, the wormhole is highly character quantum mechanical, probably too quantum mechanical to say that there is any kind of classical geometry is only when there is enough concentrated entanglement in, you know, very, very dense collections of entangled things, which means black holes, which manifests as spatial connectivity of a kind of classical geometry, uh, yeah, within the framework of the tens network as well. understand the classical relationship between the mass of a black hole and its area oh well, yes, um within the tensor network, tensor networks are descriptions of quantum states, they are not actually descriptions of dynamical energy, it is an aspect of the dynamic, so I think the tension It is fair to say that the network does not describe that connection, but the CFT version of the ads has limits.

Quantum field theory. You heat it up, you put some power into it and, um and uh, you ask the question you can ask is what is it? thermodynamics what is the connection between mass and entropy, for example, and you discover that that connection is exactly the same as the connection between mass and entropy of a black hole, so the tensor network is a kind of static instantaneous description of energy and Hamiltonians are all about Evolution, so I think the answer is that you can't read from there, surely you can read the connection between entanglement entropy and area, yes, yes, that can be done, yes , also.

Can we use Duality in the other direction? Take a simple quantum field theory and learn something about a highly quantum gravity fluctuation situation. Yes, I think we have been learning. I've been learning things like that entangled black holes have bridges between them, that's pretty surprising but simple. field theories oh simple field theories no I think it's true no I don't think it's true I know it's true that if a field theory is going to be complex enough to have black holes and do what gravity requires , it should be chaotic and in fact it should be a fast Scrambler, that means it won't be.

I mean, it could be a simple field theory with a simple GA built in, but it means it's coupled enough. I mean dualities, generally, you can use. in both directions yes, yes, yes, okay, no, that's more or less true here. The duality of the very complex fuel theory, at least some questions are dual questions or simple gravity questions, but then you can get into regions where both are complicated and you can force them. Go to regions where both are complicated and, you know, half of the phase diagram of the two-dimensional model is neither high nor low temperature and it's work to get there from both sides.

I had a crazy idea the other day and, uh. I get a lot of them, yeah, so forgive me because I'm Rusty on most of the things I'm going to talk about and I'm also going to throw out a lot of words that I probably don't fully understand the meaning of um. Okay, but you get a word I don't understand and we stop. Okay, so let's assume that a photon is emitted by what it is emitted. A photon is emitted, yes, and then it is absorbed by another MH particle and then the two particles can become entangled. from that point you can entangle particles by interactions between them, yes, so you see the frame of the photon, you know, you don't see time, you see emission and absorption at the same time, which is more or less the same idea as if the photon I just went through a wormhole, so it's over, let's not go there.

I'd love to talk to you about this later, but I can see it's going to be a more complicated problem than I can answer right now, so I think. It's too complicated a question and I'm going to call it, sorry, I guess we're on the right track, but let me ask you a question that's been bothering me for quite some time. The study of black holes and the study of advertisements tend. towards the study of something that is not really dynamic, but studies like the question you asked meanwhile, when we study cosmology, we study well DS, an exponentially expanding universe and there are no limits to it, so if this cannot have any implications Yes, that is an absolutely central question, but let me point out that the interior of the black hole is growing.

The black holes on either side are static and do nothing. The inner geometry between them is growing and it is growing in a pattern that is not too much. Unlike a cosmological geometry which looks more or less like a cosmological geometry, the blue lines simply represent the growth of the Interior, so I think we have learned something about the growth of space, but you are absolutely right that we are always using this crutch of having Limit field theory and the real world don't have a limit like that, so I think the answer is this is the next question and I don't think we have a good idea how to do it.

Okay, then we should invite the lady. Well, thank you so much for more, visit us at stanford.edu.