POAMs / breaktime conversations (contd.)

0:00 What is going to do with... There's a nice way to think about it, topologically, and so I thought the best thing would be to just review everything, more or less review everything. Now, maybe I should remind you while you're staring at this little... The idea is that you would like to transmit a quantum state from point A to point B, and the people who are thinking about quantum information theory, and I haven't got my history straight, so I'll just say the people, because I'm not quite sure who invented the teleportation procedure that I'm about to talk about. So anyway, if you know the names, say them, I'm sorry. Anyway, teleportation intends to transmit a quantum state from here to there, but you can't copy quantum states, so you have to do something a little more subtle. And this invention is quite nice. The idea is that you take an Einstein-Rosen-Podolsky pair, an entangled pair, and the protagonists in this scenario are called Alice and Bob traditionally. So Alice wants to send a state to Bob. And Alice has one end of the pair and Bob has the other end of the pair. And that pair gets used in conjunction with a measurement by Alice to change the state that Bob has. You could ask what measurement could Alice perform if she succeeded in performing it, to make sure that Bob's state ended up being exactly what we want. Now you can't always have that because you can't always predict the result of a measurement. But the simplest quest that you can do, and we'll do that eventually, is to think about what measurement should succeed for Alice in order that Bob would have the right state. And then when Alice doesn't succeed, she could find out what she did measure and send a little information to Bob, and then the whole thing works. Okay, so that's the way this works. And if that made no sense to you, then never mind, because I'm going to do it slowly. I was going to ask a question. I suppose the teleportation was paranormal transfer.

2:30 Oh, no, no, I'm not talking about anything paranormal. This is the paranormal transfer. This is the intent to transfer quantum state from one place to another. They count the words in science fiction. But they have actually done it. But you have to transfer passive information as well. So I want to remind you of basic... Quantum principles. So here's a quantum state in two dimensions for two possible outcomes, zero and one, and traditionally called the cubit because it's like a bit except it's probabilistic. And as a quantum state, it has coefficients that are complex numbers, the sum of whose squares is equal to one. And upon measurement... On measurement, one state or the other appears as indicated for a person who measures and appears with probability of the absolute square of the coefficient in the state. And this is part of the more general situation of a quantum state where you may have some large number of mutually perpendicular outcomes which constitute all of the possible outcomes, and each one has a complex coefficient for someone whose squares is equal to one, and the probability of landing in one of those states when you measure is the absolute square of its coefficient, and when you measure in this So that's the notion of measurement. You don't know which one you're going to get. You just have to probably go. So that's what I was saying a moment ago, that I could think of just the right measurement in some cases, but if I'm being careful, I have to think, let's measure and then see what the outcomes could be. Okay? So that's the notion of a qubit and different information. I'm intending to kind of go slowly here at the beginning. But the whole thing is not more complicated than that. Here's quantum computing or quantum information in a nutshell.

5:00 I've already reminded you that a state is like this, or Cupid is like that. If you apply a unitary matrix, a matrix whose conjugate transpose is equal to its inverse to such a thing, then the length is preserved. And consequently, the total probability is preserved, the sum of the squares, and so if you were trying to invent something that would preserve this sort of probability, you'd think of unitary matrices, but in fact, modeling quantum mechanics is exactly that, that a process, a quantum mechanical process, is a composition of some unitary transformations. And a measurement consists in projecting into some direction, and the probability of getting that is the absolute square of that inner product. So in the case that we were looking at, where I want to project into the directions of zero or one, and when I take the inner product of zero with b, I just get alpha, and if I take the inner product of one with b, I just get the beta. And the absolute square of that is the probability. Now I'm assuming that these zeros and ones are perpendicular to one another like that. So that's the set. And to connect with AMPA considerations for a moment. Not that, yeah, yeah. It's fun to look at this formula from Tom Eder's eyes and I thought we could just do that for a moment. This is a Tom Petter diagram of the probability calculations. If you just look at it, I'll just break it for a moment. Here's the this multiplied by its conjugate. And when you take its conjugate and think about what computes that, you can reverse vector and evaluator and get a little negative star in there like that. So that's the product of the complex number and its conjugate. But if you think of them being put together that way and make a diagram of it, you see the sign.

7:30 All of this information is connected into a U, and it also gets connected into a U star, and psi prime also gets connected into the U star and the U, so that the preparation and the measurement are connected into this circle containing the U and the U star. And if you were to remove either preparation or measurement from this picture, then the U and the U star would multiply each other and nothing would happen. So that's a kind of an insight into the structure of the amplitude, the structure of the probability, and the insight as I tried to write it here is that without both the preparation and the measurement there's no event, the quantum event interlocks the preparation and the measurement of the past and the future in that sense, and without that divide of past and future there's only the unitary matrix and its conjugate which multiply each other. And Thomas said things like that are similar to things in this paper here. Diagramming matrix multiplications is also practical and we'll be doing some of that. But I think it also helps in a way to think about that in relation to the meaning of quantum physics. In any case, there is this idea of computing, and if you're thinking of computing, then you like to have naught, which changes 0 to 1 and 1 to 0. And that isn't any kind of unitary transformation in and of itself that you would want necessarily, but, well, actually, let's see. I mean, you can think of that. You can think of that all by itself. But a particular kind of data that people like is the control model, and that's also going to, if you decide that I'm zero with another input, the gadget will not do anything, but I'm one, it will act like not. And you write down the matrix for that. Now I'm dealing with two qubits. Then you see that you have a little unitary matrix there. Quite a useful idea. So that's control math. I'm just giving you an example of this notion of building computers by using unitary dates, which constitutes this attempt at the field of quantum computing, which is perhaps going back to the field of computers.

10:00 So, and I've also introduced here the fact that we might want to think about two qubits, and I will use the usual notation of putting the two in a little bracket here, but you can also think of this as lines like this, and the lines correspond to the elements of a tensor product, right? This is one vector tensored with another vector. So lines parallel to one another correspond to tensors. I'll use that again later. And similar than here, these have tensors also. So if I write zero, one, you may wish to think of it as zero, tensor, one, which is just, tensor, of course, is just a way of combining things into other things. I don't like products, but... But when you're doing the linear algebra, saying tensor is a way of explaining how, if I want to apply a certain pair of matrices, say A and B, A tensor B, A will act on this one and B will act on this one, and then their products will distribute over one another. So it comes up. It's a formal product. So you can take formal products of matrices in the same way, letting one act on one another. One more part of this sort of little background lesson in thinking about this sort of thing. Here's what's called a Hadamard transformation. This little unit area on the left, the square is equal to the identity. And this has to do with the relationship between matrices and thought experiments, really. So in the thought experiments below, I have a half-silvered mirror, which H is talking about. There are also a number of different types of equations that can be used to calculate the number of times the number of times the number of times the number of times the number of times the number of times the number of times the number

12:30 So the matrix language lets you pack up possibilities in some gadget. And I have another gadget here, a half-silken mirror, which doesn't transmit but reflects, and when it reflects, just like this one, it changes the state as well. So that's a 0110 matrix. You put in a 1, you put in a 0, and you get out a 1, where the 0s and the 1s that I'm speaking are a little confusing because they're not the same as the 0s and the 1s that make this point, but this 1 stands for the second vector, which is 1, and this 0 stands for the 0 counts first vector, which is 0. Everybody clear on my 0-1 speaking? Yeah? So these are the, so the little kets are the prepared states. And a bra, like bra zero, is the evaluating state, and so it's functional. And really that little matrix in the square, that is the representation. Right, so this little matrix is a representation of the action of the half-sloping mirror, and the Hadamard is a representation of the action of the, I'm sorry, it's a representation, the Hadamard is a representation of the half-sloping mirror, and this guy here is a representation of the mirror. And if you then put these together into this little gadget here, then you can use this instead of the double slit experiment if you want to discuss elements of quantum mechanics. Here's the photon zero coming in with this spin, and then what happens, right? Well, it goes into this superposition of being transmitted or reflected. And then I put the mirrors in so I could do another atom on it over here on the right. So then we have these possible pathways, reflects, reflects, reflects, or transmits, reflects, or transmits, and transmits. And you have to take the superposition of everything that happens from beginning to end, assuming you make no measurements in between. So I've drawn it all in a chart, right, all possible pathways are drawn in a chart. These, as you see at the end, these pathways add up to zero.

15:00 So there's a complete interference at the top. And this is a little reinforcement down here. So that says that if you send a proton in here, it will come out the bottom and it won't come out the top, unless some measurement happens in between. And what does that look like in matrix calculations? It looks the same when they're done with matrices. The matrix calculation, the concatenation of the three matrices is the same as adding up all the different times in all the different ways. So that when we multiply these three matrices together we found 1, 0, 0 minus 1. This first column corresponds to what happens if ket 0 goes into the system and says that ket 0 will come out. So that's a nice example which can be discussed in more length, but all of that gives a picture of pretty much all of the principles involved, right, the idea of summing over all of the different possible pathways, the idea that things can be represented by matrices and so on. Comments? Then there's entanglement, and this is the most complicated slide, and then I'll get into things that are a little less condensed, but this one is admittedly extremely condensed slide, but let's look at it for a moment, because it's got a lot in it. A little more down. So I want to put the, well, there's a moral at the bottom of it, but we can go this far. The eigenvalues plus and minus one. And if they were classical, then you would expect to just see plus and minus one each time, depending on what you made the measurements. But as quantum, then, a measurement will have eigenvalues plus and minus one as well. But you have to compute your expectations and probabilities using the rules of quantum theory if you want to do it as quantum. And then this very particular expression. So this is a nice exercise of a few pages of little matrix calculations, but if you do that, you get some nice results, and I'm just going to tell you what the results of the exercise are.

17:30 So I'm starting with this state, two-qubit state, with coefficients a and b. And I compute the expectation here. If a and b and c and d are real, then I don't have to worry too much about squaring things. And that's the calculation I have to get. You can carry it over complex numbers. So this is the square root of the probability of respect. And this is the expectation. And what you get for it, if you compute this. Now, what does this mean? I just want to tell you how to do the exercise. So this means Q acting on the first tensor factor and S acting on the second. So that means that if you went up here and applied qs to b, it would mean that if you took the first one, q applied to zero, well that's zero. And then s applied to the second factor, which is zero, and that would be minus zero plus one. Sorry, minus x0 minus x1 over x2. And then you multiply them together. And you add all this stuff up and compute this. And that's the expectation, and the expectation you get in the end is this quadratic kind of expression. And then on the other hand, what is the expected value, expected eigenvalue if everybody was classical? Well then, either you're going to get eigenvalue 1 or you're going to get eigenvalue minus 1, everything is very simple in classical physics. So you want to add up and you want to get 1 and 1 and 1 minus 1, you get 2, or you get minus 1 and minus 1 and minus 1 and plus 1 and you get minus 2. So the expectation classically is directly plus or minus 2, the expectation is less than or equal to 2. And so what you have is that if you compare the classical expectations of what people get from certain states, you find that it's not the same. So for example, if the state you started with was 0, 1, minus 1, minus 1 over 2, that means that b is 1 and c is minus 1. And A and B are zero. A and B are zero. And B and C are one and minus one count. So that's, this is, sorry, what is one and what is minus one? So this is one. And that's minus, what was the mistake? No, no, this is, yeah, two of them count. What?

20:00 You've got two to the left. There's this two. And that four. I wanted two and four as six. It looks like I made a mistake in my arithmetic. Thank you for your attention. So it comes out. And if it didn't, you could change the sign somewhere. And in many cases, you see, the point is that it ended up greater than two, and on the other hand, it was classier than two, so it shows you that you can get different, quite different kinds of expectations out of the quantum situation than you would out of the classical one. And this is the, this inequality situation is what people call Bell's inequality. And then you can ask, well, what if I had a general state and I wondered whether I could or could not violate the inequality? Well, where are we here? This over the square root of 2, greater than 2, is the same as this greater than 2 root 2, which is what I wrote here. So you want this expression to be greater than 2 root 2. And then you play with this a little bit. But it's the same as saying that root 2 minus 1 over 2 should be less than this. This is squared, negative, so that's going to be now. And here is the determinant of some little matrix, right? So what you found was that if you went back up to the state, there's a matrix associated with the state that you should consider, and it is the matrix whose entries are A and D and B and C. And that matrix is telling you something about when you will get these violations. The matrix is A, B, C, and D. So this is one reason to think of this matrix associated with the state not having anything to do with anything other than the subject at hand.

22:30 But if you're thinking of a two-qubit state, you really should think about this building two by two matrix. And if its determinant is non-zero, that could unbalance this equation and create this violation of the inequality, which is in fact exactly what happened here. In the case we were considering the determinant was minus one, or one, and it ended up unbalancing the inequality. So, if this determinant is non-zero, then it's possible to violate the Bellian equality and have a quantum state which behaves different from a classical one, okay? And, as I said, this is a very concentrated slide, but I recommend it to you if you want to try to do the rest of the exercise because it contains the lore about the Bellian equality. If you just do the rest of the exercise, you have all that for yourself. And the key point is, what does this mean, this determinant non-vanishing, okay? Well, it turns out to mean something algebraic that's quite clear. You'll see on the next slide. Suppose that I take a state that's a tensor product of two single cubit states. So now I'm using that tensor product. I take the tensor product of two single cubit states. I multiply out one way and I see I get AC to go 0, AB to go 0, 1, AC to go 0, BB to go 1, right? There it is. And then you could ask, well does that have determinant on 0? No, it doesn't, see? Because when you multiply the endpoints, the determinant was the product of these two minus these two, right? And when you multiply those out you get ACBB minus ABBC. So it's zero. It's turning into zero. So this state is said to be unentangled. And a state is said to be entangled if that determinant is non-zero, and equivalently that means that it cannot be written as a tensor product of two individual states. So people often just say states are entangled if they're not tensor products, but I thought it would be a good idea to actually tell the story, tell the whole story. The reason why people say that states are entangled if they're not tensor products is because if they're entangled they could violate their inequalities. For some choice of operators, not necessarily any choice of operators, but it's possible. And so that's where the spanking comes in.

25:00 And the other moral is that one should think of this two by two determinant, sorry, this two by two matrix in relation to Puget's statement. Okay, so now let's turn back to matrix algebra and topology. First a little matrix algebra. In diagrams, well everybody knows the formula for multiplying two matrices is this, right? The sum of the second index and the first, and the first index and the second, you multiply them together, and you get the product of two matrices. And if you draw simply two boxes, one for the, where the, in this case I'm going vertically, so this is the first index and that's the second index. And they're tied together, first index, second index of A to first index of B, then the concatenation of these two boxes can represent the product of two matrices. And the only convention for you if you want to play this game is that you have to sum over the index that corresponds to where the two boxes are tied together. And if you have a free end, then you don't have to sum over it. Aspects of matrix algebra from this point of view. For example, the trace of the matrix is obtained by tying the bottom to the top, or the end to the end. And you see here that the trace of AB must be the same as the trace of BA, because you can just push one around to the top or around the loop. Or if you want to play games with quantum mechanics or formalism, you can. Like here's a famous formula for the expectation here. I'm going to show you a picture of the trace of Newt Rowe, when he was doing unitary transformation, and then I'm evaluating back in alpha. And Rowe is the so-called density matrix, and that's this guy here, the Kep-Raw product. So that's a matrix because, you see, when you just form this...

27:30 We haven't summed on anything, and you've got two free indices. So that's a matrix. And this is a model of a coin building. So that's another example of this. And then here's a general idea, which is not strictly speaking part of the talk, but I wanted to put it in. Because I think it connects across the way I'm thinking and the way of Tom. If you have some network that really corresponds to the matrices you see, then going into that network and putting in a density matrix like this corresponds to computing an expectation. And so you would think that you should go in somewhere and cut multiple links in the graph and put in events and images like that that will correspond to doing a measurement. And so you can think that way about spin networks and other objects like this, where you have a network that represents the quantum process and then measurements correspond to doing things in the network, whereas evolutions correspond to just allowing the network calculations to happen or possibly putting it in unitary somewhere. Anyway, you can think that way. You can make the diagrams into a kind of space and the physics is happening inside the diagrams if you like. Now let's shift all the way back to simple topological considerations without any quantum concepts for a couple of minutes. So I'm thinking about very elementary topology. I'm thinking about things like One-dimensional membranes, as Cliff was talking about this morning, two dimensions, and the fact that I may want to take a curve like this and form a curve like that. And I want to think about how to turn this into algebra. Now, the reasons why I would be motivated as a topologist to turn such a thing into algebra has to do with the fact that, on my side, that it isn't just the curves, but I can also represent knots and other entities by this, but this is the base topology that lies underneath the knots, and I'm not going to talk about knots particularly. I'm just going to talk about the curves. And the curves have an enormous structure in and of themselves, as you can see, as soon as you...

30:00 And all of that structure of the curves in the plane comes really from the fact that You can have maxima and then you can have minima. And then one very complicated Jordan curve like that guy is some kind of composition of forms of maxima and minima going around and eventually connecting back up with themselves. And in fact, let's just digress for a second. There's the famous Jordan curve there, which says that if the pen has ink in it, Well, if you draw a curve that is void in appearance, then there's no need to do anything about it, right? So there's some illustration for you. The Jordan Curve theorem says that any curve that I draw on a plane like this divides it into an inside and an outside. And in fact that the inside and the outside are each topologically simple. Let's say, just to learn about the inside of a disc, the outside is a plane with a disc removed. Simple as that. And of course, in this case, you can just look at it by eye and see, yes, indeed, I see the inside disks and the outside disks, the point disks involved. Or you can even do some counting to figure out where you are. For example, it's quite clear that if you're outside and you run a line through the curve, then you will run it an even number of times. You see, like, you go from outside to outside, you cross them twice. So outside is an even intersection number and inside is an odd intersection number. If I'm here and I want to know where I am, inside or outside, I send a line out as far as it will go and I see that I cross it once and even an odd number of times.

32:30 So odds and evens tell you whether you're inside or out. And the theorem says that this can be deformed into a simple circle. And one way of understanding the theorem, or even proving it to yourself, is to understand the following, that there is this elementary deformation like this, of a maximum and a minimum canceling one another, a nearby maximum and minimum go undo one another, just in a little hole in the black screen, and it goes away. And so you could look into the curve and search for canceling maximum and minimum. And you see various examples of them. For example, there's a chance of a pair, and so I can just locally replace that with a straighter line without eliminating one maximum and one minimum. And I go along like this, clipping off pairs and eventually reducing this to something which has exactly one maximum and one minimum, and then it will be a circle. So that's the proof of the Jordan curve theorem from this point of view. But it could go wrong, or it could give you some problem, right? Let's say you went to this one, which is a minimum, and then you hit a maximum. That's alright, that cancels. But suppose you went from this maximum to this maximum, and they don't cancel, right? So you could go from maximum to following maximum and find that this happened, and then they wouldn't be canceled, right? So if you're searching for it, you might start searching here, and I'm looking for it, and it doesn't pair with that, and then this one didn't pair with that one, oh, but this one did pair, and you see, the only way that this process could go wrong, if you start searching for a maximum and a minimum there, is if you spiral. You see, it might happen that you started here, and then this didn't pair, and then this didn't pair, and this didn't pair, and this didn't pair, and this didn't... And if you were able to spiral to infinity in there, then you wouldn't be able to get out, you wouldn't be able to cancel, but if there were only a finite number of these maxima and minima, then you won't spiral to infinity and the whole thing will unwind. So I just thought I'd digress and talk about the Jordan curve here, but if we want to make algebra out of this, then we need to make an algebraic image of the maxima and the minima.

35:00 And that's what I started doing on this slide here. So, see, if I make a matrix for the maximum and I make another matrix for the minimum, so I'm just telling you what a topologist might do in this kind of situation, and then we'll get back to the long explanation. So, I make a matrix for a maximum and another matrix for a minimum, okay? That's two indices. And then, if you concatenated a maximum and a minimum like this, this is supposed to be the same as this, which should be the identity, right? There shouldn't be any change as you go from here to here. So that should force the lower index to equal the upper index. That's going to be an identity matrix. So, maybe I should write that. That this here, AB, is going to be the identity matrix. That's AB. And then we have a matrix equation corresponding to a topological deformation. m lower a i, m upper i b is equal to delta a b. And of course you have to sum on i in order to make this. It's the same as our rules before about matrices in terms of diagrams. So then this would all work very nicely, and I have a way to represent by matrices what happens with curves, and if you have a more complicated curve arrangement, you could have a more complicated matrix problem, like this one, you see? These are from M over KB and I guess you missed 1M. And I might mechanically write them down and send them to you and I would say, well, what happened here? Could you simplify that? And instead of looking at the picture and simplifying this apology, you can look at my algebra and say, well, let's see, this is something on I and these are inverse matrices, so this is just a delta. Here's another sum on K and these are inverse matrices. That's just another delta.

37:30 In order for deltas to multiply to a final delta, which is just delta a b, excuse me, you could tell me without looking at the picture that this was dense, right? So that's how the matrix algebra and topology interact with one another like that. Of course, if the matrices are not inverses of one another, then we would just get them crossed. And what am I telling you? Well... Let's make a little transition. First of all, there are other ways of making the products. I guess I just might as well do this as long as it's on the slide. Here's a circle, just the bare circle. And in that case, you see the two matrices that are tied together are the lower one and the upper one, and they're summing on both of these. So this is the sum of the products of the corresponding entries. And that's the value of the loop itself. And any loop I draw, like this more complicated one, will evaluate exactly the same as this one, because of the fact that the matrices cancel each other out. So any Jordan curve that I drew would evaluate like that. So that's another thing you could do if you're playing algebra versus geometry. You draw something... And turn it into algebra and hand it to the computer, and the computer gives you a value that's different from this, and you know that whatever it was, it wasn't just a Jordan curve transformed itself. And there's an analogy here with albaras and kets, which I've drawn. You see, you can think of one of these as a ket, and the abra, the other one is a ket. And in fact, I quite intend to do that. I want to think of this as a preparation, because I'm thinking of things going up there. And I want to think of this one as a measurement. And I want to think of two together as a combination of preparation and measurement. And we do that, and we combine that with our earlier notion of associating a matrix with a two-qubit state. Then we're going to be in the position of having a nice diagram thinking about self-protection. So remember the previous idea, it's coming up in the same way. I want to associate a matrix with a two-cubic state.

40:00 I'm thinking of the matrix as diagrammed like this, just like we did for the topology. And the matrix is exactly the one that I would associate with a two-cubic state in the usual way. The zero-zero energy force finder, the zero-zero state, and so on. So that means that this preparation, you can think of it as a little Feynman diagram which says that In the course of time, we created this situation, and the situation is a superposition of these two-qubit possibilities with these coefficients. That's the creation. And a measurement is a dual vector, like this one with some upper indices on it, and it's also associated to a matrix in the same way. And then we want to figure out the following quantum information. What happens if I start with a state here, one qubit state. I prepare this two qubit state. I measure it here in the first two tensors backwards by some two qubit measurement. And I want to know what the result is. There may be some transmission of classical information from here to here, and this is intended to be exactly the kind of thing that happens in the teleportation scenario, only it's got a lot of topological pictures so you can by now actually see into it what's going to happen without doing the calculation, but the calculation comes on the next slide. So in that teleportation scenario, here's Alice over here. Alice can make this measurement. She has control in the first two terms of math just to make the measurement. This is the EPR state. You'll see a diagram of this in a moment. Alice can contact this end of the EPR state. The other end is over the bottom, who might be very far away, so maybe this diagram really should look like this, just to indicate that the EPR state has gotten its two pieces separated from one another into space. Alice has access to making a measurement over here. So that's what we want to think about. Now think about it before I do any more. We know, we know that if we were to just multiply these matrices according to the topology scenario, this was one matrix and this was the other, this would be their problem. It might be the identity. If it were the identity, then you would expect that what happened here would end up being equal to what happened here. That would be direct transmission of a state.

42:30 If it isn't the identity, then you'll get some matrix like this. And what I'm asserting, and you can almost see it, I think, is that the matrix multiplication that I'm doing in the topology is actually exactly the same as the scenario, prepare this state by using its matrix, and measure these two factors using that matrix. It's exactly the same as what I was doing in the topology where I wanted to find out the resultant of the problem. It's exactly the same. If you're used to these things, if you've seen it already by now, and if you're not, that's okay. It's fun to take a look anyway. So we're going to do it according to the quantum components. So here we are in quantum capitalism. This is the preparation, and we know that for indices J and S, you will get the coefficient of that state that is represented by M. And for indices I and J, where I and J is set equal to I, if you separate J and I from this M right here, you're going to have the co-indices for M. I keep putting that arrow because somebody has to tell Bob what happened. And so, here it is. Here's the tensor product of Psi with preparation N. And here's N evaluating in the first two tensors. So we'll just write it out as usual. Here's Psi. Psi is Psi k k. Those are called special signs. It's a minus sign summation convention, so you don't have to look at summation signs. And N is MRS, RS. And N is NIJ, RJ. IJ evaluates into KR, right, the first two factors. S is left alone. Now, if you go back to the diagram, and that makes sense, you see, this is evaluating into the first two factors, and the last factor is left alone. And then we have I is equal to K and J is equal to R because that's the only case that anything can happen here. And that forces K and R to be I and J.

45:00 Then you see that you have NIJMJS, that's the product of the two matrices, and it's been applied to psi on the left, so there's a transpose there. And so you see that you get the product of the two matrices applied to psi. And that's not unexpected after all the geometry we were doing as a diagram, because what underlies this is just a little wiggle in the way those two matrices in the wiggle compose. I could have done the whole thing with a diagram. I thought I'd better go back to algebra and make things clearer than measuring intensive factors. So here we are. Alice is over here, and Alice has the ability to measure into that part of the EPR pair. This is Bob's part of the EPR pair. And then the simplest successful teleportation would be psi time equals sine. Now let's think about that for a moment. That would mean that n times n would be the identity, right? Because the resultant here is n times n applied to sine. That means that M, M, M has to be invertible. And now you see the utter consistency of this whole thing. M has to be invertible, which means it has to have a non-zero determinant. There was no way to take an unentangled state and do this procedure. It can't possibly happen unless the state is entangled. Then you could do it, and that's just a matrix thing. Let's see what else is on the slide. So Alice and Bob are sharing the APR here, and this would be an example of a successful measurement when these two are together. What? What? We're standing. Oh, I'm standing. So that's just summarizing what I said. Okay? So the simplest teleportation would be when the end state is equal to the beginning state after the successful measurement.

47:30 And that could happen if the measuring state that succeeded had its matrix the inverse of the EPR pairs matrix. Now, the simplest example of that would be if the EPR matrix was the identity. That's a good one. That's 1, 1, and 0, 0. That's intangible. So we could do that. And then that would mean that the best success would be measuring in the identity matrix. Oh, I see I'm repeating myself a bit here. We're running out of time, too. Oh, that's okay. I'm almost done. So, here we could take m, and we could take m to be 0, 0, 1. That's an ICP out there. And it's the identity matrix, and then the best possible measurement would be the identity matrix. And then everything is sent through. And the remark about the determinant is the same, so let's quickly go on to the next slide and figure out what we need to do in general. So, in general, we're going to get n multiplying sine, where n is the measurement. So now what we need is a good basis of measurements. And now we come to the point where if you're reading a book on the teleportation procedure, they will tell you they use the Bell basis or something, and then they check and everything works. But I'm setting it up so we can invent it. Because we know that these other guys should be rotating the original state. And so we need some nice matrices in order to rotate. And then take the corresponding states. One of them is the identity, and the others are known to you. And each one has associated with it a rotation, a unitary map, which, whose inverse will undo this set up. And then you can have the agreement between Alice and Bob about a little code that tells Bob, I got this one, or I got this one, or I got that one, and then Bob knows what unitary to apply over at the other end and rotate it back and get the original state. And that's the teleportation procedure. But you see from this point of view you could almost have invented it yourself. I was looking for a way to tell myself that I could have invented this thing because it was such a clever thing.

50:00 And it turned out that the topology was a way for me to understand it. Now that of course is not saying that topology is fundamental to the teleportation procedure. It's just showing how somebody who's used to topological language might think this through. And yet, on the other hand, there's something kind of interesting about the fact that lying in the back of the formalism is just a wiggle, which is showing how things are connected up. And since I don't have time, I won't speculate further, but I will... I want to point out that we can generalize this. There's a long, long paper by some other fellows, I'm sorry I didn't check my references, that points out that you can teleport entire transformations, and indeed you can teleport entire transformations because this whole procedure was about transporting a transformation, right, a matrix transformation. So if we wanted to, if our aim was to have a higher dimensional vector, And to apply a unitary transformation to it that we have selected. Then we could also play the same game, look for the successful situation that will apply to unitary and then extend to a basis and get the other rotations that will do the compensations and thereby create essentially a teleportation quantum computer or process maker which will, via teleportation, cause certain unitary transformations to be applied to vectors. So you can make quantum computers entirely out of measurements and certain possible bit transformations like that if you want to. And the little topological wiggle informs that. Now there's more that one could try to do and we intend to. You see there are other procedures which have patterns that are topological like this is the procedure for what people call, this is the picture behind what people call entanglement swapping. There are two preparations and two measurements and you can make up all sorts of complicated interlocked preparations and measurements and then analyze them quite quickly by picking this way. So let's go. A couple questions. Isn't the main thing a crystal piece of security because, in a sense, what you're really interested in keeping secret is the thing you teleported.

52:30 That's right, it's very important that we're not measuring that thing with teleporting, right? That's why you have to go through all these fascinations, because as soon as we measure it, it goes away, and I want to keep it as a quantum state, and maybe do another transformation on it later, or take it somewhere else, and move it around. So that's why, that's the reason for this. And people who are thinking about abstract quantum information design regard teleportation as a fundamental element, because it's going to be very hard to know how to get quantum state as quantum state from here to there in any other way than by teleporting it, and you're going to have to move them around inside this possibly distributed machine. Well, okay. Thanks, Lou. I should mention that, of course, I'm really interested in seeing some really fundamental relationship between topology and entanglement. And these things are certainly interesting. They're very much on the surface. No, that doesn't happen. It just appears to be true. Yeah, so how many... The problem is, what's the problem? What's quantum computation? Quantum computation is certainly a nice metaphor for talking about fundamentals of quantum theory. And it may turn into something real. But I'm not in a position to do that. The last time I heard of it. The last time I heard of it was when Lewis was at least six months old. He built a time computer with four bits and a factor to ensure that 15 factors could be defined. Oh, I see. The source algorithm is pretty fast, and if we manage to try for 15, you can do that too. Well, the thing is, if you give it a big number, then you start to notice differences, because the quantum computation is an exponential speedup, and that cancels out the difficulties of factorization.

55:00 The whole factorization thing comes down to, you produce a key, an encryption key, which is a product of very large primes. You can publish the product, the total weight, yourself. You want to send a message to Al, you want to send a message to Bob to publish the product of your crimes. Anybody can send a secure message to Bob. But in order to crack the message, that's a factor. That's a very big crime. That's a computationally difficult algorithm. When's the end of the crack? If you get out of it. You get out of it. You get out of it. It's a very difficult time, though. And then once you... So you get exponential speed up. But it's quantum. You have all states. And so that allows you, again, a chance of cracking very large. Now the next step is quantum priming. Well, the teleporter can be used to send messages, as a matter of fact. Oh, yeah, exactly. And you can also tell somebody you saw. It's kind of a cool thing to do, because if there's anybody listening to the message, the key around it is destroyed. Yeah, and you can tell somebody, if somebody's keeping it, you can tell them. It gets very cute, if very expensive. It's kind of a good thing to do. As far as the speed issue, there's an old-fashioned idea on analog computers. It solves differential equations. You can put Schrodinger's equation on an analog computer and solve it. Is that fast or slow? It's the same. It sounds a lot cheaper. It's very much hotter. I used to do things like that in the week, but me and the computer have a very small room at the top of the building. No, the problem is the stability. The stability of the analog components means that in practice you can't do anything very complicated. And you care for practical work. Yeah, I mean, it's a question of precision. Precision and stability. As soon as a thing runs, you'd be a slightly different answer to this article than whether the one that's open or not.

57:30 And that's very hard. You have the infinite precision of the real life at the same time. You want to know exactly which point on the line you got, and you're out of luck with it. Okay, well let's stop there. The next speaker is Lynn Clare, who is going to tell us about further developments in her analysis of the technology. I guess so. If you want to take a three-minute break, we'll be right back. That's a good idea. Thanks. Without that world blowing it up, two fundamental things, aspects, particles, whatever you want to call it, blow out of it. One is blowing one way and one is blowing the other way. I would like to have you consider that one is water, what I consider to be salt water. The other is what I consider to be a diamond, the gym that's where it's structured, so you'll see the name in much more detail in a few moments. And then what you have is absolute nothingness. And that's why I call this something, S-U-M, something from not-thing, which is applied on the word checkbook. So, all of a sudden, out of the nothingness, you can see a point in the line. There's always more way. As they come closer to you, all of a sudden your perspective changes and you see a line over a point. If you change your perspective again, if you are the observer in observing, all of a sudden you see a line under the point. No, you as the observer are changing your perspective. It's like if I point my finger to you, you're seeing a point over a line. Okay, the computer looks at my finger, he sees a point under a line, but you see a point over a line.

1:00:00 Right, that's what I meant. The finger looks over a line. That's right. Right. Exactly. But again, if we change our perspective, if someone else is looking from the bottom of the floor, from the ceiling, they see two lines over a line. But in fact, what I want you to consider is that we're so far away that we can't see what they are, and they're actually spiraling geometries. And I'm not going to go into the detail of those spiraling geometries except to say that one is significantly larger than the other. And the water, which I would posit as the gold, is much larger than the red, which I believe is the diamond. Now, now we're going to watch how these two come together and interact. And what they're going to do is form embedded sphera. What you just saw was the water closing at the south. The diamond you can see is inside and it's still forming and is about to move into the center. It turns into polyhedra. It's snakes. It's polyhedral snakes. If you fold up the geometry, it becomes spheroidal geometry. It becomes spheroidal. We can demonstrate this. We can prove this. And I'm just showing you the fast way through this. Now, earlier today, we can demonstrate. We can unfold it into a flat, into a planar geometry. Now, this morning, Dirk was presenting his membrane theory, and I'd like to ask you to consider this to be a membrane, because it's like a bubble. And where Derrick was having the masculine penetrated, he was using an almost concrete example of a penetration. What I'm showing is on the inside, and here what we have is the feminine, and this is a feminine principle, not gender, but a principle, and I'm going to show you why it's feminine. It's because it's breathing, it's taking energy in, it's utilizing the energy, and it's putting it out.

1:02:30 This is a mathematical principle, as you're going to see, and you're going to see why. So, now that we have, and I want to show you with this, this is 120 faces, this is 144 faces. Both are unknown. Both are unknown column patrons. It took Bob Ray three months to write the computer software just to do the 144 in the center. Can I ask you to hold your questions so we can get through this because it's really, I'm moving fast and I know and I'll be here the rest of the week so we can talk. But what happens, here's the 144. Again, it's unknown. When the 144 comes together in the middle of that structure, the hands are a perfect model for it, there's three edges that have to come together in that geometry. Two, three. Where does the energy go? Because it's like a diamond. It's a spark, a spark, and a spark. It rolls in. It implodes. It's an imploding energy, and it goes into the center, but it can't contain it. And when Dirk said this morning, like a little volcano, it is going to erupt to that energy out into the system. And here we see how it sends out 48, because that's how many vertices there are in that 144-faceted polyhedron. It sends out 48 rays of light. They're clustered very unusually. Bob Gray, when I showed him, I said, you've got a little bit of a storm too. He said, well, that's not logical. And I said, well, that's the way it is. I want you to know it's the only way it works in this geometry. Once that light escapes, the 144 is done. It's like the masculine principle, it's not complex. You have the orgasm, the man just sits back and has a cigarette, just a spin, and waits for the feminine to do her thing. Okay? Okay, so the light goes. So we have another polyhedra here. In March, we just mapped what that polyhedra is. I've always called it the focusing sphere.