Quantum & Topological Aspects of Entanglement & Nonlocality

0:00 So I'm going to talk about entangles in two modes, at least two modes. there's pharma entanglement and there's topological entanglement I imagine that I understand topological entanglement a little better than I understand pharma entanglement because topological entanglement is just a lot more phenomenon of linking or nodding so everybody's going to experience on an ordinary and on the other hand pharma entanglement is exemplified state over here in a tensor product, which could represent separated particles which could be observed from two different locations, one from one location and the other from the other location. And once one of them is observed, then the other one's state is determined. If one is observed as zero, the other one is observed as one, and vice versa. And so Mathematics can represent non-locality, quantum mechanics, and entanglement. It's interesting, actually, how little mathematics is necessary to represent it. And how, when you do represent it that way, you don't include any spatial information about where those guys are that's outside of this bit of mathematics, which is part of the reason why it seems so convincing mathematically to talk about non-locality. And on the other hand, they're confusing, of where those particles might be. The other thing about quantum entanglement, which I'm going to jump into for a moment, is the way people normally speak of it, they don't speak about the one locality when they're analyzing this. They speak about the property of the element of the tensor product. And there's a good reason for that. But you will notice that I'm sitting here with this element in a tensor product which is not itself a tensor product of elements from the individual two-dimensional spaces of which it is composed. It's not a tensor product of some linear combination of zero and one with some other linear combination of zero and one, as you could check by multiplying something out

2:30 and seeing what happened. So often people say a state is entangled if it is not written as a tensor product of So that's what entangled means in linear algebra and of course that does have something to do with nonlocality and I thought I would remind you of that in a moment but on the other hand, here's topological entanglement on the other hand, which is certainly a nonlocal property of things and nonlocal in the spatial sense but not obviously related to this other thing, maybe it's just a pun. Another favorite of mine in terms of linking is this guy, which will come up in a moment. Might as well call it B for Boromian rings. That one is a tripartite relation topologically. You can't remove any part of it without it decomposing into something simpler. remove any one of the rings, the other two pull apart, right? So it's holistic in that sense. You can't get rid of any part of it without removing its whole property. And, of course, that's the same thing that's true here with the linked rings. If you remove one of them, there isn't any linking. And so there is an analogy here, which was pointed out by Erwin a while back in paper, the analogy being that when you observe, you project into one of the parts of the state, and project out one of the coordinates. And then, of course, it's no longer entangled. It's no longer not a tensor product. It's much simpler. And here, if you were to cut a link out, clip it, and remove it, then it becomes simple. And it's more interesting to look at the Boromian rings than it is to look at the simple link in that regard because when you clip one of the components of the Boromian rings, it really does simplify the other two slide apart. Here's part of the analogy that one would like to pursue. Now, as I said, I thought I would step back into something slightly more complicated, just to talk about why it is that people say entanglement and then say not a tensor product.

5:00 It really has to do with the dev inequalities, and what I have encapsulated on this slide for my own understanding is an example of doing modality qualities. I have here four operators, four little unitary operators with eigenvalue plus or minus one, and a state. I'm interested in observing this concatenation about this combination of concatenations of operators I want the expectation value of it now this was this is part of that clever evolution of things that came from Bell but if you if you if you compute this as a quantum expectation then you find that you get this quantity here 2 to the minus 4 times a plus b squared minus 4 times a minus bc divided by the square root of 2 and on the other hand if you think of these as classical operators randomly giving you lighting values of plus or minus 1, and to ask yourself, well, what will this do? You calculate that it's going to be plus or minus 2, and then its expectation is less than or equal to 2, if it were just classical, if this was just classical, not quantum operators. And then you can ask yourself, well, what is the relationship between the classical expectation and the quantum expectation? So, for example, if you were to take this state here, the one that was on the other side, you would find that the delta of the quantum expectation is 6 root 2, 6 over root 2, which is bigger than 2. And so that tells you that whatever quantum theory is telling you, it's not telling you the same thing as the possible observation. That's the Bell inequality. The Bell inequality would be that it should be less than or equal to 2. That's the way people speak. And the Bell inequality is violated by this state. So that's an example of the difference between Bonner and classical. And then when do you get a violation? That's the reason for talking about this a little bit. You get a violation when this is greater than 2 times the square root of 2. Why? Because this is divided by square root of 2, so I'm just saying it should be greater than 2, and I put the square root of 2 over on the right. And then I manipulate the algebra a little bit and find that that says root 2 minus 1 over 2 should be less than this little determinant minus a plus b squared.

7:30 So then you see that in order for this to work, for the bounding quality to be violated, you need this determinant to be non-zero. Now, that determinant, let's just do a little exercise. I think I have a slide later on about that. But we can do it on the next slide. is actually telling you when that state is entangled. You see, what I'm claiming is that that, you see, where's that determinant here? It's still sitting around, AB minus BC. AB, the end terms minus the middle terms, prompt, yes? So let's do a little exercise and find out when a state would be entangled. I think we have a little room at the bottom of this slide. We'll just use it. Suppose I take x0 plus y times 1, and I take its tensor product with z0 plus w1, and write that out. So I get xz, 0, 0, plus xw, 0, 1, plus yz, 1, 0, plus yw, 1, 1. And you see that x times z times y times w is the same as x times w times y times z. So that if a state happens to be un-entangled, then the determinant vanishes. So if it's entangled, if the determinant doesn't vanish, then it will be entangled. And then what did we see in the other bit here? We saw that in order for it to be, in order for it to violate Gauss inequality, in order for it to look non-mothal, you would need the determinant to not vanish. Now that doesn't say that the determinant of non-vanishing tells you that it will violate those inequalities. It won't. Not for these operators fixed. You can get some things in the interface where the state is entangled but it's not violating these operators. But on the other hand, you need this in order for it to have a chance of violating those inequalities. And so that's the basic elementary linear algebra story in a nutshell. And I found it useful to kind of garner that into a nutshell for myself

10:00 because there's lots and lots of talk about Bell's inequalities. And if you do a little bit more exercises than I actually did, then you have that whole story in a small space or at least part of it. So now let's go back to our thinking about entanglement, where entanglement just means that there's a tensor product that isn't decomposed. And think about it from a point of view of links a bit. So I already said this. Here's the Boromian rings. And if you were caught one of them, it falls apart. You could caught any one of them, and it falls apart. Now, there's a quantum state that has the same property, this one. If I observe in the first variable, it'll give me 0, 0, 0, 1, 1. And if I observe in the second variable, it'll give me 0, 0, 1, 1. variables are all online. So it's just like the Boromian rings except that you see immediately that there's something a little funny here because the mathematics of this is often simple. And when you start to think about the structure of a gadget like the Boromian rings, it's not so simple. For example, how do you even prove, if you're a topologist, how do you prove that the Boromian rings is actually linked? There aren't any linking numbers there. You have to and go a little deeper. You could do it by the method that Keith was discussing last time, you could set up a circuit and show that its current was non-participal. But you have to do something other than linking. So there's another thing, and this was Erwin's observation too, that makes the analogy interesting and complex. And that's that you can change the basis and you get a different effect. It's worth looking at this. I mean, I'm taking a very simple basis change like this, okay? And so these are going to those sums and differences, and that means that the original one is the sum of the new elements and the other one is the difference. And you put them in, in one of the core. You could put them in all three, but I'm only putting them into one, and rewrite. And then ask yourself, well, what happens if I observe in the first coordinate now? Well, so I collected out the first coordinate, and you see that if I observe in the first coordinate,

12:30 the resulting states are entangled. Not are entangled. So the analogy would be a link where when you cut a component, it was still entangled. It didn't fall apart. And then what happened in the second coordinate? it two because I removed it from the middle, but this one is not entangled anymore, the zero factors out, and this one the one factors out. So that one is unentangled when you make the operation, and the last one on the right is also unentangled. So in this case you have to draw a picture of a link which was itself entangled, here's one, but when you cut one it was still entangled, it is, and if you cut two, then it falls apart, and it does, and if you cut three, then it falls apart, and it does. This goes a step farther than Aravind, who didn't write down pictures for his example when he went beyond the point of changing basis, but this kind of thing certainly happens. And furthermore, you can have examples, and I forgot to call one out of my notebook, so we'll probably have to reconstruct it later. You could have examples where after you make the basis change in something like this, when you observe in one way, it's entangled, but when you observe the other way, it isn't. So all sorts of possibilities can occur. There is, associating to any given quantum state in a sense, patterns of entanglement, which depend on what basis you chose. And so you can ask, what happens if I go through all different possible choices of basis? I would like to catalog all the different possible patterns of entanglement that can occur. So this is an interesting problem which Sam and I, Sam is my collaborator there I mentioned on the first slide, are starting to try to investigate. The thing is, you would like to say that two, but I'm not going to go into the details of that part, but it's a very interesting question and worth stating. You would say that two states are locally equivalent. if there's a unitary transformation in one coordinate which takes one to the other. This will not disturb anything. All the entanglement patterning will be the same under those. But if you change basis, then you can get something different happening. So there's a finer structure in there than just that.

15:00 So you want to think about what happens when you change basis, and you want to think about what that does involved. So there's some very interesting structure, and it's worth trying to understand exactly what happens, even in the case of the bromine rings, where it turns out, in fact, that everything happens. All of the different, if you analyze this, every different possible pattern and entanglement actually occurs in that GHC state when you start switching things around. Almost everything. So that means that there's a lot of relationship with of topology, or else there's no relationship with topology. How do you want to think of cutting the cake? I'm absolutely atorn to think that entanglement, as I'm afraid of the answer, is racist. What are we doing, isn't it? Yeah, that's a little worrying, isn't it? But after all, choosing a basis is choosing a way to observe. So I take it as being meaningful. Yeah, I see. Yeah, I think that's a very helpful answer. Yeah. I don't mean basis, I really mean I've chosen a particular direction in which I'm going to do the observation. The light observed intrudes into this. I've got to be a serious, serious way. Yeah, it says that you have this state and whether or not it ends up being entangled or not after you observe it, It depends on how you observe it. This is not an unfamiliar situation on quantum mechanics. It's something that depends on how you observe it. In this case, it's spatial. Observe it spatially one way, it will come out one way. If you observe it spatially another way, it will come out another way. So this also occurs in conventional quantum patterns. Well, you know, I mean, if you go ordinarily and think, well, if I sell to observe momentum, I'll get my momentum information, but I won't get the position information. But this isn't complementarity in that sense. It's more like I'm going to measure the Z component of angular momentum rather than the X component. Nevertheless, all those components, including the magnitude, are all constants of the motion, which is nice.

17:30 but they don't commute. You can't measure them all at once. So you do have to set up to decide which one you're going to do. Now, from the point of view of knots, it suggests generalizing knots in what I think actually is an interesting way. If you have a situation where you're going to see all these different knots, then maybe you should think of a superposition of those knots rather than just one knot. So that leads to a not fully formulated idea of a quantum knot, where a quantum knot would be a superposition of classical knots. Wow. I have difficulty doing classical knots. I'm not going to have enough trouble with classical knots. Okay. Let's just wander a little bit. here's a well-known way to make bigger borromean rings. My colleague, my collaborator and colleague, Roger Fenn, who teaches in Sussex, pointed out to me that there's a wonderful realization of this particular link in the Sutton Coupe exhibit in the British Museum in London, where they found these ships that were buried and there are chains. There's a beautiful chain in that exhibit. You'll find it if you go there because it's a small exhibit. There's a beautiful chain which is actually made exactly this way. Chain links brought like that and the next one brought through and bent and so on. And finally blocked at the ends by rings in some way. But of course you see how this works and how if you dropped any one of them the whole thing will undo. so this is a generalized form and ring and here's a generalized GHC state so you can do that you can ask yourself how to make corresponding kinds of objects so one way to do that if you wanted to make this is another problem along these lines if you're trying to make the correspondence as a kind of correspondence of language I mean at this point all I can say is that I have a language of entanglement I can think of talking about the pattern and entanglement of a given length, and I can talk about the pattern and entanglement of a state.

20:00 And it may be useful to draw an entanglement picture in the topological mode as a picture of the way the state is entangled. So that's it. It's at this stage. So our language is fun to play. And so you can ask, well, how do I make these? If you check the Berolian rings, you can see that it itself is a braid going around an axis particular to the plane, coming straight out of the plane. It's always weaving in the same direction going around that way. And I can redraw the picture so that it looks like this, so that you can express it as a braid, that is as a closure of an element in the braid group, where the braid group is generated by little switches in successive positions across a series of lines like that. So this is sigma-1, one of the generators in the braid group, signal one inverse, and this is signal two. That's just orientation matters, which way you call it signal one or signal one inverse. So the Boromian one is signal one inverse, signal two raised to the third micawatt, as you see. And so that's the Boromian braid, and it helps to have the braids around because you might want to design, as I said, more complex ones with various properties, and you can do that in the following inductive way. Suppose it's Boromian rings you want. Now of course we already saw how to do it with linked rings as in the Sutton Hoo exhibit but here's an inductive braid construction which you can then modify to do some other things. You take your enfold Boromian ring and then you take another one and put it in the inverse form so that if there was nothing no glue going through there they would just unwind each other and then you drop another blue component through in that fashion. And then you think about it, and you realize, well, if I drop blue, the whole thing will align. And then you think about it a little further and see that if you drop anything else, it will align. Now, the reason I bring this up is because you might want to make other kinds of entanglement patterns, but you can start with these and then put some extra little linkages in between braid strands and get the other things to link the way you want it to. So you can make, you can design your own entanglement patterns this think about what you've got you can also then face the problem of whether or not the closure of the thing really has that entanglement pattern or whether it unwinds which is just possible but as a braid you can make sure that it has the

22:30 right property so those are some things on topological construction side of things now let's let's step back again and ask could there be anything deeper in this relationship between topology and state entanglement? And is there anything that we know that is something like that? And we do know something that's something like that, that everybody I think would agree is deeper and has the right character of an interrelationship between physics and topology, so I want to mention that. And that's the following thing called the Dirac string trick. Now, on the level of just cut and paste or a topology of bands in space. It's a pretty simple thing to understand. You have a ball here which is attached by a ribbon to the ceiling. And the ribbon is attached rigidly to the top of the ball. And the ball can be rotated, but you also imagine that the ball is somehow suspended in mid-air so that once you've turned it say by 720 degrees as I did here, then it just hangs there without rotating okay if I were to perform this and I can't because I forgot my belt and I don't feel an extra belt and I don't want to take mine off but you can perform it by exchanging hands using one of your hands as the ball and then you're gonna have to exchange hands in order to get the ball get the arm out of the way right but you can imagine just imagine it suspended in midair like that. There's 720 degrees of twist going down the band, and I'm not going to move the ball, but I relax it a little bit, so there's one full 360 degree twist, and then slide it down until it looks like this, and then slide it back under until it looks like that. And then you see that what you've done is you've taken the 360 degree twist and turned it into the reverse opposite sense, a 360 degree twist, so that when you straighten it a little bit, you have negative 360, positive 360, and it all unwinds and respond. So 720 degrees of twist on the band goes away completely. Now, what does that have to do with thinking about the physics of something? Well, you could point out that there seems, as people did, you could point out that there seems to be some analogy here between what happens to observe a fermion and going around

25:00 the fermion by 360 degrees or 720 degrees and observing again, because the wave function of the fermion should change by a sign when you go by 360 degrees, for a reason not mentioned yet, but mentioned in a moment. And so when you go around twice, you're back to where you started. Now, in this case, it looks very similar to what we were talking about before. happens in the topology and what happens in the physics. Except that we know that what happens in the physics is mediated by the fact that rotating around in space is to be carried by a representation to SU2 because of the properties of quantum mechanics. And that SU2 covers rotations in space twice, a fact about the structure of the groups. And consequently, when you go around twice, when you go around once, you're in one sheet of the double cover, and when you go around twice, you're in the other. And then, furthermore, if one thinks carefully about the properties of what this thing is, it's actually a depiction of the structure of the way that double covering works. So that you can go more deeply into this picture and see that it's actually talking about what happened in the physics. and what it amounts to is saying that as you move through some rotation like if you move through starting and not moving at all and move through 720 degrees of rotation continuously then you could map that into what is happening to the band as it gets rotated and then watch what happens as you start moving rotations through rotations and it corresponds to moving the band around so the whole thing actually has some depth and talks about what's really happening in the physics The hope here would be that there would be something like that that would help to understand quantum non-locality or quantum entanglement in terms of the topology. But I can't say that I've found that yet. I'm only exploring, and I don't see yet anything that has that kind of nice click and actual correspondence that one could start to work with. I think there might be something there, but I'm going to tell you a bit about the exploration from my computer. So one thing, this is actually another side of the one, but I can't resist making it.

27:30 You see, if you're thinking about bond states, qubits, then you're thinking about something like this. The state is a complex number times zero plus another complex number times one, and the sum of their absolute value squared is equal to y, because you normalize it to have length 1. So that's a point in a three-dimensional sphere and a four-dimensional space, often made to be complex numbers. And then you don't care about quantum states except up to a constant multiple, so you can divide by the action of the circle phase, the global phase. And this is often called the Bloch sphere and corresponds to the information that might be in a single qubit. Now the mapping from S3, which is SU2, by the way, to S2, not to S3, but to S2, is also 2 to 1, and given any point, there's a circle line over it, the phase circle that I was just talking about, and given two points, you get two phase circles and up in the three sphere are linked like this. And that's called the Huff Fibration. And so the Huff Fibration, which is a real piece of topology, is sitting right there any time you're looking at a couple of cubits. But I don't know what to do with it. So this is the other side of the thing. I mean, it's quite clear that this topological gadget, which is quite useful in many contexts, is sitting there. But I don't know what its significance would be for thinking about the information structure of cubits. So I tossed that one out as well. There, it's not a question of the analogy. It's just part of the mathematics. But is it of some use to the physics? Maybe it'll occur to some of us. Is it Bloch? C-H-O-B-L-O-C-H. C-H, yeah. I think. Yes, it's a map. Yeah. Now, the order of things could be different here. But since I'm wandering among analogies, I couldn't resist another analogy about non-locality. And this is the well-known Penrose Triangle. And the reason I think this is worth looking at is that, first of all, it gives you yet another category.

30:00 Maybe three categories are better than two for making analogies. and you can make an observation on the Penrose Triangle just like you can make an observation on the link but it's a little more striking in some ways here's the Penrose Triangle this imaginary entity constructed by your brain you're looking at it and then there are three real or realizable entities that you see over here something which is made like this with 90 degree angles and so on And it may not be entirely obvious, unless you've thought about it recently, that the imaginary object is carrying those three real objects. But as soon as you put something over one of the corners, it becomes quite clear that the imaginary object is carrying the real objects. This kind of interesting perceptual thing. But that corresponds to observation here. just like dropping a link, you can put your hand over, you can obscure part of the imaginary object and it falls into the real. It falls into the real in three different possible ways. And Penrose, I guess it's Penrose's father who invented this, if somebody can correct me, is that right? Invented the Penrose Triangle, Roger Penrose's father. But Roger Penrose wrote a paper about it. But you can think of this observational situation with regard to the Pentos Triangle as cataloguing an element in the chronology. It can measure the entanglement, the non-locality of impossibility. Is the impossibility of that triangle limited to a three-dimensional situation? The impossibility is because if you try to construct it partly, then as you continue with it, it will miss. Would it work in four dimensions? Oh, I don't think it works at all, does it? I mean, because it's a geometric object, and these guys are going off at 90 degrees from one another, and they aren't going to meet. As for the invention, I mean, there are these pictures by Escher, which might be fairly early.

32:30 Escher parlays the same illusion of, I mean, it's, you know also that you can make this entire thing up to a certain point so that you can look at it from the right point of view when you see that. So, it's in many museums, the three-dimensional object, which, when viewed from just the right perspective, becomes imaginary. I think that it suggests that this may be pre-pandemies. No, actually, we should have never followed you each other. Yeah, yeah. But one of the reasons I thought it might be interesting to follow that analogy about the non-locality of impossibility, and notice that is, it's another non-locality, right? The impossibility is not here, or here, or here, because, after all, if it were here, then if I obscured it here, it would be real. So Penrose says, well, realize part of it, realize part of it, and realize part of it, and then try to put that together in three spaces, and there will be some distances that you can't resolve, and you'll break down those distances in the right way, which I won't bother you with, and you'll find you've got a cocycler in a chromology group. of the co-cycle is measuring the impossibility of the figure. So it's interesting. I think it's just nice to juxtapose it. Now, let's see how we're doing in terms of time. How much time do I have? 20 minutes? 20 minutes to go. Pardon? You have 20 minutes now. Yeah, that's fine. And without conscripting it, that's . Well, I have more topological things I wanted to say. I want to say this next one very quickly, because it's fun. And that has to do with intrinsic nodding of graphs. Now, this won't look like a, I'll try to draw the moral of this in a moment. But a graph can be intrinsically nodded. It means that no matter how you put it in space, it has an on it. A beautiful example is this complete graph on seven vertices. So this is the complete graph on seven vertices. I've just thrown them in order here. Complete graph on five vertices. Every plane is connected to all the other four. And then six. And then seven. And in seven I have read

35:00 point of connection to everybody at six. Now, what I tell you is that in 6, there must be a link. And this you could probably see if you stared at it carefully, that there's a link in there like that. What's a little harder to see is that there must be a knot in K7, no matter how you put it in, and I put it in in some way. And I didn't bother to work out where that knot is, so I don't know where the knot is either. But there is a knot in there. That's the theorem. K7 graph into three-dimensional space that would be anonic. That's the theorem of Conway and Gordon. Now, on the other hand, K7 occurs, this is really a little digression, but I have a reason for it. K7 occurs in another context. It occurs in seven-color map on the torus. That is, there's a map on the surface of a torus, which I've just drawn here, torus, and the plates, hexagons, each one touching six seven-colour map. It has to record seven colors in order to color it on the surface of the torus. Now, that, if you put a point for each face and connected them up, would be a K7 sitting in the torus. We know from Conway-Borden theorem that the K7 must have a non in it if it's in three-space. You normally think of that torus in three-space by attaching the edges to one another and taking it in a torus like that. There must be a non on that torus, which is a non in three-space, walking through the K7. So the knot in the K7 is actually sitting in the paving of the map, the seven-color map on the torus. The seven-color map on the torus has to twist around in such a way that it has a torus knot sitting in it. And that one's easy to see, and I've actually drawn it here. I've drawn it halfway to the knot. So there's a K7, which is easy to locate the knot. Always has to be there. And the fact that something had to happen in three space tells you that there isn't any way for you to draw them out on the surface of the torus which isn't necessarily in 3-space except to have a twist. So that a topological phenomenon in 3-space could implicate a property of something which wasn't intrinsically in 3-space. The same is true of K7 itself. K7 is intrinsically nodded. It has a property that has to do with 3-space speaking a denizen of three space nor is the seven color map on the Taurus so

37:30 topology can have an indirect effect and so so sorting through so so sorting through various ideas in topology those are some of the ideas that I think are worth thinking about here but the topology can have an indirect effect it does in the Beltran and it might well in the case of thinking about entanglement But entanglement is curiously hard to think about because it's so simple, and just uses the basic principle of quantum mechanics and very little else. Now, we decided on another way to think about it, and that way leads to some connections, but they're fun, and I don't think they're so deep. They're just interesting ways of thinking about the connection. The connection is to look at the way braiding works and turn the braids into matrices. Now, this is something that one does when one's working with this apology anywhere. You see, you can say a braid is like a matrix. It starts here and it ends here. It maps from here to there. And I'll associate a matrix to the braid. And I want to associate it in such a way that equations about braids turn out to be equations about the matrices. So if to a one little crossing you assign a matrix R, which took a tensor product of two vector spaces to a tensor product of two vector spaces, you could think of it as operating on those vector spaces. You could think of it as operating on little quantum bits and taking them to another tensor product or some tensor products of quantum bits. And then if this operator is unitary, that would be interesting to think about in a quantum context. And if it satisfied the Brady equation, which is expressing that this braid is the same as this one, which is just a little shift, why then it would be interesting in the logical context. So I can make an algebraic gadget, which can be interpreted in one or the other of the two contexts. It could be looked at as something which is measuring things about braids or representing them, or it could be looked at as something which is working on quantum states. And then you could ask, do these operators that you try to construct entangle states and is the way they entangle states related to the way they represent braids and play

40:00 back and forth at that level? And so we've been doing that for a little while and there are some examples that are interesting and I'll show you a couple. I just wanted to go to a concrete example, here's a matrix, it's a little 4x4 matrix, And if A and C are unit complex numbers, then this matrix will in fact be unitary. And on the other hand, you can use it to measure some things about braiding and linking by various schemes of which I've illustrated one of them. In this scheme, what you say is, well, R applied to 0, 0 is A times 0, 0, and R applied to 1, 1 is A times 1, 1. And R applied to 0, 1 is C times 0, 0, similarly here. So if I walked up through a little braided link like this one and had it start and put some labels on it, like zeros, well, you see if it had zeros here, it has to have zeros here in the zeros program. So each component gets labeled by a zero. It's a very simple labeling scheme. every given component is either labeled by 0 or 1. There are four possibilities. But then once it's labeled, the weights from the matrix come in, and this one would get A squared, and this one would get C squared, and the other one gets A squared and C squared. You write out all these different contributions, A squared, C squared, and so on, and you find that if you normalize it in the right way, it tells you what the number of times one of the curves links around the other one is. And it does that as long as C squared over A squared isn't 0. But then if you analyze this same little operator and ask what does it do to states and when does it entangle them, you find out that it's the same condition, that C squared should not be equal to A squared. So this guy entangles states exactly when it can detect linking. So that's a hint or something, maybe. I hope so. It's an example. These examples are, it's like solving crossword puzzles. It's not so easy to make the examples work.

42:30 That's kind of fun. I'll give you one more example, a more recent one. We kind of forgot about this one. It's an R matrix which satisfies the braiding condition. Quite unitary, real. It takes the, it's all known in quantum computing. what's called the bell-lists, all of these little entangled states that's used for teleportation and other things. But it satisfies the young answer equation. It's a braiding operator. And you could wonder, well, what kind of an invariant does it give? You make an invariant out of it in a way that's very similar to what I just indicated, and you find out that it can detect some things. It can tell you that the trephal might isn't trivial. It can tell you that the Borromian rings isn't trivial. But curiously, once you've normalized it, so far, on my computer anyway, only returns values, the invariant returns numerical values of 0 or 1 or minus 1. That's all I get so far. I don't know why exactly. Maybe it's obvious in the end that it will only have three possible values. So it turns out to be an interesting thing to think about. A puzzle related to this operator is the following. Yeah. Now, a very off-the-point question, the Romian ring, as you were repeating in the beginning, I want to have spelled it, and in this slide you have two eggs. Oh, yeah, of course. So I couldn't tell you which is the correct spelling, but I think two eggs. Two eggs. Yeah, I think so. Look it up on the internet and see. And Google will say, I'm sure you didn't want it. So anyway, this is an interesting example because it does some topological work and it's also a maximally entangling operator. He said, do you... Oh, sorry. Okay. He said, look it up on the internet. It says, do you mean this or do you mean this? I never know whether you're allowed to say yes or no. I guess you're allowed to say yes or no. You just may be taking off into the wrong domain, you know.

45:00 You might have them in bondage. I don't know it. The one that sends you the right way is the right spelling. Hopefully. I mean, I have this problem with Kauffman. How many Xs are? Right. How many Xs are? I thought. Let me get back to our questions here. Here's a long-own operator called CNOT in quantum computing, controlled NOT. It's known that all the unitary operators that you need in quantum computing are generated by CNOT and one qubit unitary operators. So you can do that to get a dense set of operators. It's good enough. So you can say quantum computing is generated by CNOT I don't know whether quantum computing is generated by this guy and one-cubin operators. Maybe it's obvious that it isn't. If it were, then you could say, well, grading is good enough to generate quantum computing, which would sound good, and still might be a bit content-free, but in any case, there are puzzles. If you start doing this algebraic level of correspondences, then you have the two categories which meaningfully are mapping to the algebra in their own ways, and then you're doing the usual kind of mathematical correspondence in that you have a mathematical domain which applies to more than one application and then it helps to make the comparisons. So we think that's a good way to go, and we're still working on it. So that's a summary of what we've been doing. Now, I had a couple more things I wanted to say, and I guess I have a couple of minutes, so I'll say them a little bit in each order. One of them has to do with putting braiding into context. Some people, Friedman and Katayev, want to do anionic quantum computing, In an anionic quantum computing, you have to imagine there are some particles which can be created and annihilated, like in a quantum Hall situation. And there's a bare bones for this kind of quantum theory, a particle theory, which is a favorite among topologists anyway, because it turns out to have topological applications. And in that bare bones, there are some creation and annihilation operators,

47:30 and there are also some specific particle creation operators and fusions. And then the brain interacts with them in some understandable way, mathematically. And then it acts naturally in the sense that you get nice topological pictures about how the particle interactions and the bradings work with respect to one another. and I almost didn't mention, you see, if you, for example, created two particles and then you created two particles, you might have done it in a different order and if the same particles came out the top, you would want there to be some kind of transformation between these two, this is a recoupling phenomenon, like an angular momentum theory So you want operators that do that. And as soon as you're in that kind of a domain, then of braiding, interacting with those sorts of things, then certain patterns have to come up mathematically. Like the fact that if you start re-associating things, there's a pentagonal circle that takes you back to where you started. You start, something's associated, then you re-associate it, and you re-associate it, and you re-associate it, and you re-associate it, in the right way. I'm thinking that this is associating because this is A times B times B times C. It looks like a different pattern of brackets. That's right. Patterns of bracketing are patterns of associating. And if you re-wrap it, and that's a re-association, if you have a product of four, then you go around in a circle of five. And it goes two to three and corresponds to some formulas in angular momentum theory, for example. It's a fundamental pattern. It comes up. It has to come up. It's in the mathematics of the thing. So that's one kind of thing where it has to come up. And then another kind of pattern that has to come up and either be there or not be there is the way the braiding interacts with the associating. And there's a hexagonal pattern of that. kind of ground of things that can go on, is a hexagonal pattern. And those two patterns, pentagon and hexagon, if you assume that they're exact, that you come back to where you started under both cases, that's what people tend to call topological

50:00 quantum field theory in that dimension. And so the braiding actually can be thought of as sitting in some context like that, and And there are people who hope that some quantum computing would be built on that structure. Whether it will be or not is another question. Yeah. This is, to me, practically fierce. Fierce. I fear to ask this question. But we know how the homitive matrices that can be divided, can be analyzed, in the weighted sum of projection of ratios, and the weights of the eigenvalues, and the projection of ratios themselves can be represented as the outer product of the eigenstates, however, by whatever vectors you represent those, depending on the bases you're using. And so there is correspondence, according to what you're saying, between the eigenvectors, which are in one-to-one with their projection operators, you can just manufacture the projection operators by doing an outer product. And these are, according to what you're saying, in association with certain knots. So each term in this thing is associated with a knot, presumably a different one. Is there any meaning to the idea of weighting those knots? For example, the simplest weighting you can give them is all ones, because if you add all the projection operators up, you get the unit matrix, so you get back to identity. Now what would that mean in knots? I can't sort of visualise it, but it should have some kind of operational equivalence, shouldn't it? What would we mean for the knots to add up to? You're thinking about one state and a lot of different projection operators can be applied to it and each one is giving some entanglement to knots. Yeah, but the simplest thing you can do is take the orthogonal projection operators, which correspond to the eigenvectors of a particular operator,

52:30 and let's say that's just a 2 by 2, so there are only two of them, and they sum up to 1. They sum up to the unit matrix. Now, firstly, there is likely to be, if there is an association between one state and a knot there is the same association between the projection matrix and that knot but the other projection matrix will have presumably an association of some kind of different knot and then you have to ask yourself the question if I add them together I get a unit, a matrix domain What does that mean in the, in the knot domain? This is just sort of very tentative. No, I agree. It wouldn't be obvious. You might, on the other hand, in the knot theory, people do things like this. They might say, well, I have a vertex which, for various reasons, can be thought of as resolving into either this or this, and it's quite common to use the sign in fact. You could use arbitrary coefficients. And then if you had a graph with a lot of vertices, then this would be the sum of the lots of knots with plus and minus signs, and the source is the graph itself. Whereas the sum is where the sum, the plus sign, is merely a convention. It's just saying you're combining these things in a certain way. You're calculating in a certain way. I think so this is just like a derivative. But it does struck me that this sort of very basic process, which you probably don't need, you can see it all ago, would be a way of analysing what the hell this correspondence means between knots and states. If I went back into my original context, it would be not necessarily the line of the one, the different possible projections that could give me different knots and making an algebraic bag into which they all fit. For example, these projection operators are orthogonal and that is if you apply one after the other you get nothing and if you square them you get one. Sorry, no, you get the same thing, A squared equals A.

55:00 Just, I'm wondering what the hell that means in knots, you know, in anything. But if you can marry all that up, then you've got a total correspondence. Now, whether you can use the knots more precisely to calculate quantum mechanical consequences, I don't know. I don't know. I wish you'd talk so long about that. As you know perfectly well, I'm not a visual, but I have very great problems with knots. I can't visualize them. I think knots are actually very hard to visualize. They're not so hard to notate in one way or another. That's right. They're definitely not visualized. Perhaps that's not so bad. I'm going to mention one more thing and keep it to two minutes or something, In the course of looking for others, looking for some papers in my files of things, I came across the following paper by someone, some of you may know. I've never met him, but I think he lives in London. His name is Henri Bortoff. How many people have met him? Some of you know him? He's Donna. He worked with Basel a long time ago, I guess. He wrote a paper back in 1971 called The Ambiguity of 1 and 2 in the description of Young's experiment, which I saved. I have a copy of it if anybody wants to see the paper. He makes a beautiful distinction in that paper, which I think is relevant for thinking about non-locality. The distinction is between compresence, which means objects which are together but not necessarily in a whole. They can be removed individually. And coalescence, which is like looking at the pin through the lens, and there is no way to separate anything there. You are observing the pin through the lens. and it comes out as a poem. And then by thinking about that in this long and beautiful essay, he comes to a phenomenological explanation of the double-slit experiment, which is either compelling or not, depending on whether you believe or follow this phenomenology. So I won't go into any more detail on that. I did make some slides thinking I might have time to show you what his argument is, but we're out of time.

57:30 And the basic concept is coalescence versus kind of presence. And coalescence is non-local, after all. The pin and the lens are out here on the table. The pin is over here, the lens is here, and the observer is over there. But when it's shifted around and you're in the position of that observation, then it's all in one place and there's no separating it. So that's phonological non-locality. And perhaps the non-locality is the common ground here between something as apparently prosaic as non-theory and something as apparently arcane as the quantum theory. Well, I think we've let that conclude the meeting. But hands on the other. Well, I could just get to the people who've got important questions to ask them all, but people will be free to disperse, say we. Anyway, thanks for listening. It's really been enjoyable. Ahhhh