Abstract Quantum Physics — A Category-Theoretic View of Quantum Computation

0:00 So, a little mic, now in advance of working technology. Let's have Bob Cotto talk to us about how it's going to be self-contained, I believe. So, from what perspective you can think of this as making quantum physics more suitable to do for information like quantum computing or all these things. But basically you can also see this as just only doing quantum mechanics. I want to be clear why. So, why do we want to do quantum mechanics? So, this is something you find in a physicist's textbook. You don't have to understand the scale, but this is something you might find on another page. And this is actually a description of form of teleportation. So, what you see is a lot of words. Now you see things like, then she sends the two big arcs in XZ2 both using the classical channel. So, it's a bit messy. It's not something simple. So, what we are basically trying to do is to replace this whole stuff by something like this. So, something very synthetically which actually has clear significance. These are just morphisms in a category, all the specific types which you need to have the sort of kind of operation you are doing. This, for example, means producing EPR pairs, it means spatial relocation, perform a bell-based measurement, do classical communication, do immunitary correction, and the composite of this is then the teleportation protocol. Now, okay, you can save slides if you try to find a nice description for these things. But this is another method. You have to prove these things. And this is like a sort of proof you find in the book of physicists for the same thing. Actually, it's in the big log, but that's the way it looks. And this is our proof. So, or is it the sticks on? Um, I turn it. Here's a pen, right down there. Yeah, yeah. I'll put it down here.

2:30 You read this as some sort of like a flowchart diagram. You see this line, you think of that as a flow of information, you first pass this thing labeled by beta i, then you pass the thing labeled by 1, then you pass the thing labeled by beta i minus 1. So basically, 1 is just identity. I use 1 to denote identity, so I can get rid of that. I can just straighten the wire, And in life, these things, they are inversed, so they imitate each other, so what you get is like an attempt. Just need to measure it like that. So that's the way you breed this thing. Okay, and now, this was like a sort of problem, okay? Just to justify that. The next part, well, the main part of the thought is like working for basic quantum mechanics, two formulas of quantum mechanics in which you can do these sort of things. So let me just quickly review the view that is important. So I'm doing this in the usual big qubit size, just to make things easy. A bit admits two values, you can think. Then I say it's really readable. What I mean by that will become clear when I actually make a difference between a qubit. And it admits our bit of transformation. Basically, a qubit, instead of just having two values, can take values on a whole sphere. So on the other hand, in some way, it's spanned by two values. Because whenever you take two values by just making the combinations, you can find the other ones. And a measurement of it has two outcomes. So again, we find this very binary nature. But what is important is that the measurement changes the value of the state. Like, if initially we're in this state and we perform a measurement which is always characterized by two mutually autonomous states, then this one will change either to this, or it will change either to that. The things which it emits are essentially things which prefer to sort of the angle of the sphere. So you can just rotate the sphere like you can run mechanistic rigid bodies in physical space. The really important part is the part of the measurement. So let's look at this again. So we have the sphere. We think of this as the initial state of the system.

5:00 We think of these blue ones as representing the measurement. the other one we call minus. And what is actually crucial for this measurement is this change, that when we do a measurement either this one will change to this or it will change to that. So this transformation we can actually represent as a map, just a constant map essentially, which sends whatever initial state you have to this one. And here we can also, this transformation we can also represent by a map, constant map, which sends whatever initial state we have here to that one. projections. These are projections. They are partially defined because here, here, these are completely impossible. And here, here, these are completely impossible. So basically, these p's, these things which I denote by p, are partial constant maps. Very simple things. Very simple things. And just keep that in mind. Whenever you see p, you have to think of partial constant maps. And so, what's also the case, of course, is that the probability in which, in the measurement, you would go from here to there, or from there to there. There will depend on the angle. The closer you are, the more probability you have to make this transition. You can further away and progress. Okay, just to give a plain summary of what comes in again, this is in a nutshell. So, systems are described by vectors of an inner product space that we see. To describe We use a tensor product, operations are unitary, but crucial is the part that measurements are described as self-adjoint operators. Which is, technically speaking, a sum of these P maps, of these projectors. But this matrix does not represent a transition itself, it's just a way of encoding things. These values here are really not important. What is important is that you've got a family of these of possible transitions. That's really the important part. Each measurement is characterized by a family of this piece. Okay, we're going to look at an example now. Is this clear? Yep, this is clear. This is all you need to know about the cosmic answer here. So here, one teleportation. We've seen this yesterday. Basically what you have is, it starts with an initial state. Then you can actually, the type of thing you start with is like three yoga spaces. H, tensor, H, tensor, H. The three yoga spaces.

7:30 So this one is initially in the state 5. And this one is paired to a cubic cell, paired to a cubic cell. The initial state is denoted by this. I'm not going to say what it is here. So then, we do one with these p's, which is the measurement, the family of p's is the measurement, so from these two. So that's the measurement, this is family of p, x, lambda, y, x. And then it turns out that you can make the appropriate unitary transformation here, depending on which p happens, because you do get informed about which one happens, to actually obtain the initial function. So you set continuous data through two classical bits. Because actually the only thing you have to communicate are two classical bits which characterize the outcome of this piece. Now, what causes the magic? Well, what causes the magic can easily be as formulated as follows. That this thing is not just a pair of states, but it's a function. It's actually a function from here to here. You think of that state as a function. You can find in quantum mechanics textbooks that it's a function. They say it's an element of a tensor product. It's very easy to see that there's nothing else on a linear function. The tensor product of Hilbert's space, how do you generate it? You take two Hilbert's space and then you make combinations of pairs of base vectors. You take base for one and base for the other. And all combinations give you the base of the whole thing. Now you can write it as matrix. Each linear component of this gives you a matrix element. linear map. So I can indeed think of these states of these compound systems as functions, linear maps. Yeah? And there's a very important example of that. If I just take the simplest map I can imagine which is the identity, then I get what is called the Einstein-Roth-Gerodon thing. It's just an identity. So I can think of a measurement since I want to take the I think of these compound states as functions, I think of measurement as being characterized by measured functions. Because I had four functions, because I used the names of both base vectors to measure several functions. So I can label my trees of my measurement by functions.

10:00 Which is a little function to another one. Constant mettings on functions, sparsal constant mettings on functions basis. That's how I want to think of these trees now. For example, the identity, the p labeled identity of a woman as sometimes of a woman, would be the constant function which sends everything to the identity, which sends every function from q to q to the identity. So, what I argue here is that you can think of these entangled states as functions, but what are functions good for? They are good to compose. Do these functions compose? They do. And so, what I'm now going to do is, we're going to identify some compositional behavior within ordinary quantum mechanics. What I'm going to do is just ordinary quantum mechanics in Hilbert's place. Then we're going to abstract over that. So we're going to identify some theorem or some properties, some compositional properties. We're going to abstract over that. And it turns out that after this abstraction, you can still do everything essentially you want to do in quantum mechanics. You get notions of scales and values and all that out of this purely compositional behavior. Now, just to identify this compositional behavior, what is this? This is sort of the same setting as this, after three hidden spaces. One, two, three. What do I mean by this box? By this box I mean an operation, an operation, an from H tends to H, P, F1, the thing I've learned by P. It means it's a projector of the entangled state which you can mathematically encode by the function F1. So F1 is from H, F1, which is the number of two strings, let's see, H2, H1, H2, H2, H2. So F1 will be typically a function from h2 to h2. So this thing has a type of h2, that's h2. So you think that there are, sorry, do the speed and partial constant function. Then I do a little bit here. But then I do one layer by h2, then I do one here, then I do one here. So it's a physical setting. Now, physically you cannot really do this piece. With certainty, you can only do that probabilistically. So, because they are part of a spectrum of measurements of life, of the set. But just forget about the non-determincy, just think about the action of the piece.

12:30 That's what we want to be getting. And when it turns out, so what do I ask now? If I give you that this is the infinite state here, I ask you what comes out there. That's my question. Can you give a simple expression which expresses this one and turn out this one? And the result is this turn out. So it turns out that the output there does not depend at all for what comes in here. It actually does. Up to just a scalar log. But qualitatively it doesn't. So it doesn't depend on what comes in here. It seems that first you actually function F2. Now you have to be careful here. F2 doesn't mean that you use P, F2, which means the function which labels that state appears here in this external discussion. Not the project, the function itself, which is not physical, which is just a label for the state, it appears here. Then F1, this is a bit weird, because physically we first did P F1 and then P F2, but here we first F2 and then F1. Then find F3, H1. F2 again. Weird. We only do PF2 once, but it appears twice in this excursion. F4, F2. This you can verify. This you can verify. This is a simple calculation. Now what's the logic behind this? This is the logic behind this. So, see there is a line, a bit like the one we had in the beginning on this line. First test is F2, F2, then F1, then F3, but in the opposite direction, F3, F3, then F2, then F4, then F3, that's exactly these things. So this is the theorem. And what does the theorem say? The theorem says that if you've got a configuration of these T's labeled by functions, and these are objectives of all the integral states, constant functions of elements of transform states. If you've got an arrangement of these things and you can write down a line which follows these rules, it's actually very simple if you get it. at all. So this direction is time. That's time. Time goes up. This is physical time. So the geometry of this line should be as follow that if you go forward in time in at

15:00 one of the legs, then you have to go out at the other one backwards in time. But I don't mean that you have to go backwards in time. The line has to point in the grass backwards in time. And all the other passages are up. But if you've got that, then it turns out that if you start here with a stake that whatever comes out is just the application of the labels which label your projections. No, these are not physical operations. These are not, they don't have to be unitary or anything like that. Yeah? So your input is just the leftmost wire and I would say just the rightmost wire. What? The input, the input is this one. I mean, you have three wires. Yeah, there are three of them. There's three of them. But I just didn't draw them all here. Because we don't care about what comes in here. I just dropped it. Because it doesn't matter to the outcome. Except for... Because this is constant function. This is constant projection. So it just produces F1 at the other side. Whatever comes in shouldn't be orthogonal to F1. That's right. What are you saying about the outcome? It's just arbitrary. maps from intensive product to intensive product? Well, they are from home to home. So like this, for example, F1 would be from H2 to H2. F2 would be from H1 to H2. So, they are in the... The basic thing they are, they are elements of H, of the space of linear maps, Let's take the whole set. There are elements of, if you take five dimensional vector spaces. So this is H1 by H2. And of course this is isomorphic to H1. The star, the dual, the tensor, H2. So I'm just writing down these elements through this isomorphic. Instead, we are used to call back low steps of getting in, to make good elements of thousands of S functions. So each one is actually an arrow from the space of dual steps, or an element of the tensor type. And when you wrote F2 up here, you didn't make two F2s. So there was two steps, one was going from a map to the element of a potential product, the next is going to a projector. Yeah. It will project after a long-term subspace. So I've got a projector, initially you have a projector of some fire which is an element

17:30 of a potential product. Just forget about you, forget about you. It's going to play an important role. So initially you have something like that. And then I replace this notation by just the f, which goes out of this as a motion. So just relabeling, relabeling of the projectors by not giving the integral state in which you project, but it was only for the type of a box. For example, the box for f1 is going to be h2 tensor h3 or h2 tensor h3. That's what it would look like as a projector. So the f would be the h, h, 2, h, 2, h. That's f1 would be 2, 3, 3. So basically two steps. The first step is you enable. So you rename each integral state by a function, which has multiple. And then the box means I take a projection, the projection on that function. So then the F2 box is actually B tensor B to the scalars, and then another one, maybe I'm hitting dual spaces, and then another one, B tensor B, B scalars up to B tensor. Very good. Yeah. Yeah, yeah, yeah. So then you know how to trace it. Yeah, yeah, yeah. Good example. What? Okay, so that's basically, that result. Well, this is you can already reproduce things which didn't seem to be that easy to find out. For example, teleportation protocol took like, I don't know, 60 years after quantum mechanics to be discovered. Well, it's actually nothing more than this. Where are the two bits? Well, that's the next slide. No, just, if you want to get something from here to there without changing it, you put an identity box and an identity. This is not an identity, this is a projector on the EPR state. So it's definitely not something, it's a complicated thing. But in this language, it's called identity. These identities, of course, identity and identity compels to identity. So that's the geometry of teleportation. Now, of course, where is the classical information? The thing is that I can't repair an EPR state, but I cannot deterministically enforce this outcome of the measurement.

20:00 Because there are four possible outcomes. So this is just one of the possibilities. So basically what I do is, I build the base of four functions, and I do the appropriate direction and the answer. This is really unitary now. This is really unitary. This is not all of these p-boxes. This is just the unitary matrix you apply at this side. And the composite of these things gives you the identity. That's your classical information tool. There's some other examples. There's things which are called logic gate teleportation, where you actually, instead of just teleporting, you want to apply a certain function. This thing is universal for computing. Basically, you need this computation relation to really have this function. And this brings you then issues like the clipper group or swap. Another one is, it's funny, it's entanglement swapping. So what you do basically there is... So you've got these two are entangled by an identity and an identity. I call them one, two, three, four. So that's the same. So one is entitled, two, you need to have your three, and four. And then the swapping says that if I measure here, then this entitlement is going to vanish, and you get this one. So it swaps. And you see this from the geometry. Initially you get these two ones, and at the end this one behaves like an identity, and that one behaves like an identity. Okay, so, now, the crucial bit, this is the crucial part of the result actually, and you can write it down more explicitly. So this is basically something you can show, and this sort of germ of the whole behavior, that if I've got this box, so here I write it down explicitly. This expression is really these two boxes now. See, I've got first this pf2, which I apply, tensors be 1, then 1 mu, then this pf1. I compose them, I apply that to an input, don't care what's there, what comes out. I don't write down what's there, but what comes out is this. And you see there is this weird change of order, that while physically I first apply this one labeled by 2,

22:30 and then the one labeled by 1. At the output, at first you want to apply and then two for some of these things. Now, the question, of course, you asked is how typically is this for a Hilg space? How typically is this for a vector space or whatever? Well, it turns out that if you replace a Hilg space by sets, linear functions by relations, and that's the product by an ordinary Cartesian product, then you can show something like this. You just have to see the analogy with the expression. What I did here is I defined the notion of the projective of a relation. And this is not an accident. The reason that this works is that the category of finite dimensional building space and the category of relations with Cartesian product distances are very much the same kind of categories. And if you turn out that that sort of structure will be crucial of doing everything we did before, but a lot more, a lot more than I expected. So basically what we're going to do now is we're going to try to abstract this geometry of this line, see what we get, and see how much quantum mechanics we can still do by just relying on this geometry, and turns out we can do a lot. So just as an introduction, So, the original for the form is a form of mechanics, at least since 1932. But, somebody said in 1935, I would like to make a confession so I need you to ignore it. What? This is exactly, this is exactly my question. I think the word probably was anymore. I know, it's like that, it's literally from a better, I know. It's a consolation. It's a fabulous consolation. This is from a paper of somebody who's studied the other times, whatever. It's paper by a red egg, maybe you should find this. I'm trying to make it a little bit spicy, but I'm supposed to have more in it. Yeah, so okay, so this is the annoying one. But nothing happened to me. He tried all the things, but nothing really happened to people. I'm still using the same content mechanism as that. We're just saying that he wouldn't be up to now. You wouldn't? I don't know. Yeah. So, this is sort of a problem you want to actually solve. Like, quantum mechanics in the open space is actually much too low a level, and you want to make it higher level. For the purpose of application, you're also just for the purpose of quantum mechanics

25:00 itself to finally become a decent theory, which doesn't look like I showed it in the previous slide. And there are some clear problems in quantum mechanics. I'm not going to go, but like, types do not at all reflect kinds. If you've got something of a type age to age, this can be either a unitary transformation, which is a data transformation, this can be a mixture of clustering of a state and ignorance about the actual state, which is called a mixed state at the same time, and this can be a measurement, which means a transition, a non-deterministic change of the system together very reproducing classical information, like now. And they don't understand. If you compose two of these measurements, it doesn't mean anything. While a measurement is really a complex or physical operation, if I've got one measurement at hand, I've got one measurement represented by hand, then the composition has no difference in every one. It doesn't mean anything. While I can't really compose them, so the theory is going to be compositional. So classical information flow is not addressed. And then there is this important thing, like this geometry, which I just exposed, is of course not present in formalism. Why would it be? Otherwise, otherwise they would have known teleportation before they were in pomba mechanics. Just to think about building up the formula. Okay, now we're going to go to high level pomba mechanics. So, here we're scanning this category. Category 3. Or, of the sequence, I don't know. So, we're going to do the same thing as they do in prog, why people use categorical semantics in programming. So, origin logic. So you think of objects as propositions or types, and you think of morphisms as proofs of programs. Although in our case it's going to be that, like the cubits or the variables will be the object. There will be other types of objects too, objects which combine classical information and quantum information. there will be a zoo of operations. The process, like measurement, like sending classical information on these things, they will be mortises. They all will be mortises. And the type will in that case reflect what kind of operation we are actually doing. Compoundments, ah, we know this thing here that this is very good. We have been using in this sort of diagrammatic the fact that you can represent these states of an entangled system by function spaces.

27:30 Which is good in categories theory, because we have function spaces. So of course, it's useful to have a connective tensor, if you then want to correspond, of course, with the function spaces. So, basically we're going to assume that we have a symmetric monoidal category, So we definitely don't want this thing to behave like a product or a co-product or anything like that. So we have the obvious transformations, but the crucial part is that we don't want the whole to be the softer components, as you can see. Because if you have products and co-products, then you've got morphisms of this kind. You go from A, you have a diagonal, so you go from A to A times A. And this would mean that you would be able to copy it. You can project away something. That would mean that you would be able to delete. But there is this thing which is called no closing and no deleting. You cannot copy it and you cannot delete it. The states of quantum system. So you definitely don't want anything from project like or code like. Now we're going to see what sort of structure we want to distance ourselves in order to be able to do quantum mechanics. And so we go back to our projector. So basically, what I write down here is this. So I have these boxes, yeah? These boxes which I lay in F. There's an input here, an input here, an output. And they sort of behave as if there is a flow going through, and then it went back, and you had an action of that. So I'm sort of decomposing this behavior. I think of this, the journal of returning the flow as this triangle. And then there was this action of dysfunctional. This thing exists at both sides, this flow. I could sometimes have flows in that way and also sometimes flows in that way. So I need to add two sides to a projector. So I'm decomposing a projector in an input side and an output side, so to say. And it turns out that you can actually do this formally in outer space. And that's the thing I'm going to do. Forget an expression, I'll tell you what this seems actually are in the universe first. This is obvious. This is obvious.

30:00 This thing, the type of it is from complex numbers. And actually this is the following. This is the map in which you map 1, 2, the sum of the base, base 1 and base 2, and so on. This is actually the EPR state. This is the independent state which corresponds to the identity. That's why I denoted my identity with a little bit. And it turns out that the other side is actually exactly the edge one to that. And what is the edge one to that? So, this is equal to the H1 of this, and this is tied from H1 to H2, to see, this is in the product, this turns out to be in the product. You can check this, and then you can, and if you compose this, so this is in the context, it's in psi, so, maps to, that's actually what it is. So you can, you can, you can make the calculation, you can verify this in a vector space and we'll check if you find a projector. This will be, and this will be the ingredients for our conservation. So this is the Alcatraz entanglement. This will be. So for people who know star-atonomous categories, if you take star-atonomous categories, if you take star-atonomous categories, And you ask for the tensor to be self-viewed, you get a compact load schedule. But this thing is, this turns out that compact load schedule can be some extra structure which deals with the involution of complex numbers. That's what this thing turns out to be. So it's, we call it a strong compact load. It's more than compact load.

32:30 So, so for each object A, you want an object A star. How do you think of this? Like, if you have a line going up in a flow diagram, and this would be A, then the line going down with that type A star. just a witness of the direction. For each pair of a and a star, we have like this germ of time reversal. And then we also have this one, because that's just the edge line. The third part of the This is actually a correct morphism, so f from a to b because fh1 from b to a. So these are the pieces you create half a theory, nothing more than this. And we have one equation. This equation, what is this equation? This equation is this. You have this, and then you pull it. So here you've got the stride, and then you've got the stride on the signal. This is just a symmetry map. So this is the environment in which you are going to do quantum again. So it seems very weak. It seems very weak. It seems a very weak setting. You get a lot out of it. So in logical terms, this is sort of the generated linear logic. Sorry, this is a question, but the idea was that This, yeah, so this little point means the type is i, the unit, complex numbers in the case of the... This is, yeah, this is i, I should say what is i. So the i is the unit for the tensor. So in case of vector space, these are complex numbers. I mean, it's just that the notation was a little bit confusing because those little path brackets you're using sort of look like they're... Yeah, this is the thing they are sitting near me. So this is the thing. So I can write the equation down if you want to. It looks ugly, but here I can write it down.

35:00 So, I start with A, then I plug in one of these things, so that's an identity test. This, you know, a little happening. So my type is now A also. Then I do a symmetry. So yeah, I should introduce an identity. So here there's a nice motion to be introducing this, this, this, this. So here there is symmetry on these two, so that's sigma A, comma A star, that is a log of A, this is a log of A, this gives me A. Well, I think, I changed my comprehension, my start should be here. It's okay, then it's this, and then we do the actuality. Here. That's what we get made, and we should be here. So, this is like a latitude thing. So, the point is that because of the adjoin, we can press the two triangular identities as well. I think co-productions are not strongly compared close, I think, because the other side could start. If they're oriented, then... If they're oriented, then... Yeah, they just come back to the insertion. But a lot of these examples of this, for example, sets, relations, and Cartesian product. The actual one is just a relation. Converse, a star, doesn't do anything. And this here, this is like, yeah, I used it. So here, this is not this, I mean there's a sort of change of theory, but I should have denoted this differently. This is like this triangle, that's how it looks. Other examples

37:30 are of matrices in any involutive pair and summary. Now, what is funny about this setting is that it looks completely qualitative, but it's not. Because you've got this unit, I. So I is the unit for the parents. So we have I, so A times the model for A for each object and at each I. So, now we can look at the atomomorphisms, right? And you can prove that the multiplication here, the composition, will always be commutative. Just from the fact that the category is monoid. So, this is commutative, commutative monoid for a start. Now, assume that you take an element in here, S, and then you've got a morphine, F. So what can you always do? You can always introduce an identity and an identity if I ask more. Let me put this little scaler there. So this is going to introduce an identity. And this whole thing, we think of as scalar multiplies with scalar multiplication. So these things will be scalar. Now, what kind of structure do these figures have? The point is that the S is a natural transformation, so it can be taken as a composition. Of course, we have the notion of uniterity. Uniterity just means We've got the notion of in a plastic product value in this thing, which will not be nothing

40:00 that I've been taking things of this side, taking upon the edge line. So we think of this as like what they call the brass, and that's the cat. So a brass, the edge of the cat. And then we have these following equations. This is like the defining property of unitary transformation in Hilbert's equations. We get it from the abstract here. This is the defined property of an edge line in real experiences. It just follows from the abstract. I mean, you can identify very special scalars, like, of course, the identity will be, you think of this as the number one. You think of possible grills as things multiplied with the edge line. Scalars multiplied with the edge line. You can think of some other ones. You know, these end up being representative for the dimension of the space. So we can have sequential dimensions. Now, so just a little rotation, the only extra thing we need to have sort of a grasp on is like plugging these different pictures together which are labeled by eye. So, in order to do that, we take also an additive connected, which is just bi-products, which is really thinking things as pairs. And you get bi-products, so bi-products is something which is at the same time co-products and co-products. It's very well-behaving. And then you can plug the piece together using padding records, and you can, this is a cubit measurement. So, the idea of defining a projector as something, remember that in the case of the triodes, so how do you build a projector? How do you build a projector? You take something of this, Take this, and what comes in is this. This would be projected P identity. Now, if you want another function, then here at the end you plug in F tensor 1, and the same thing you do here.

42:30 Now we have to do one tensor. That's how you build a projector. So this is actually solving any set choice. in this way. This is something and its adjoint. And this idea we're going to push further to defining projectors by something and its adjoint. And this is the type you get. This would be an ordinary measurement where you have two possible outcomes. If you want to have a destructive measurement where you just get a number but no systems when you write it like this. And the interesting thing is that if you get byproducts, they distribute over the tensor. So you've got the notion of communication. These are two possible outcomes, and I distribute them out of two systems. So what is the result of all this? Well, you get a big dictionary. You can define a lot of things. But the main result is that then you prove the folding theorem, which is actually just commutation of a diagram. This means, like, how teleportation works in the abstract and does what it's supposed to be. It gives, like, this is the composite of all these operations. You've got unitary corrections, which means that in each branch you do a different operation. Distributivity, you distribute your classical communication. Then, these two things actually, you see this triangle and that one, these are these triangles. Actually, you can use the axiomatic of the strong compact load of there, the pulling of the row, to sort of cancel out this and that, such that this one will be annihilated by that. So this is just, this is very simple. You prove it very easily that this is good. So this is like the end of this branch. So, what's happening in the background is this. This is what happens in the background. You can see the blue things here. One, two, three. That's where I see it. here. So we use the, now finally, you direct the polymer, which is the root for the mobilities

45:00 of modern mechanics. I'll just do this. I've got an input vector here. I've got one of I decompose it, which will be something and its adjoint. And this will provide maybe the scalar. And then you prove the following. The sum of these little things look like being the adjoint, giving 1. So possible branches add up to 1. It is very surprising that from this seemingly very qualitative step we actually get quantitative results. So, I don't know anything yet which you can't do here, which you can't do in one week. I don't know which... Basically, the thing is, of course, relations are an example of these categories, but you cannot do this for relations, for the simple reason that relations are not rich enough to define this method. So if you can define it, if you can write down this diagram and you can't read it, then it will be true. Then this will be true. If you can't write it down, then it will be true. If you can't define this thing, then it will be true. If you can't read it, then it will be true. So it's just a question if there is something which has enough scale. For example, this thing. This thing, in the case of complex numbers, is exactly isomorphic to the complex numbers. Because these are like linear maps from the complex numbers to itself. So they are completely characterized by the image of one. So they want one correspondence with C itself. So basically you're actually doing something over something which is much more general than the complex numbers. This would be sort of a field, but it's not a field at all, it's generally, it's a semi, it's really semi. So, yeah, that's it. I'd like to point out that I have a paper called Teleportation Topology, which is not

47:30 anywhere, you can find it in my name, Coughlin, which is approximately the same, very similar to the first half of your talk. It wasn't thinking about the category theory aspect of teleportation as a quote. Actually, it hasn't told me about it. And I wanted to comment that it's interesting in this point of view that the thing you needed the EPR pair to do was be an invertible matrix as an amortism. As an amortism, it has to be invertible because then it goes out. So, the fact that entanglement and infrutability are equivalent at the level of two people in the States is why, you know, I've had entanglement, entangled by the teleportation. Cool. Any other questions? This story has been built out of a couple of papers. The first time actually, I think in the CTCS paper, there was already something. The main reference to which everything is He's the rich side. So the study paper started from 2001 to 2001, basically. Go to your work and finish it. Yeah. So, any other questions? So, go back first one. That's a general fact about by-products. You have the chronic adult laws between injections and projections, and if you sum, on the other hand, the homogenous things where you first you project and then inject so you're getting all the subspaces if you then sum all of those that you get those two things together exactly and then another beautiful thing is that from

50:00 having byproducts you get an additive structure intrinsically positive you can actually define the addition so just from the type theory to get out of it you only get there in a sort of linear algebra just from this type thing. I can also view this as a kind of logic, I mean, not an other aspect that I was emphasizing in this talk, but I mean this kind of straightening out of strings that we've seen as a implication of logical proofs as well, in this because of this degeneracy the self-duality of the tensor in this elimination, you will get loops. But the loops will exactly evaluate to scalars, so it's kind of very much in the same spirit as what we were doing in the talk. It's in the same spirit as the Penrose kind of tensor notation. I don't know if you've looked at the linear logic literature at all, but you can make it, once you get to a sort of a tensorial kind indeed you can't sort of copy and delete and so on then a diagrammatic notation for proofs really works very nicely but then indeed to do logical simplification logical path elimination by yes and this connects up with the idea of treating proofs as kind of deformations of paths in the homotopy category I mean there are people I understand who are working in homotopy who have been trying to connect this with some ideas has improved theory to do this. Yes, that's getting... Yes, I mean, in fact, people haven't looked at Brady, but... ...which is... But, I mean, here, of course, we're just at the sort of... It's all symmetric kind of Brady. So, much of the behavior is here for some things like Kelly and McLaren, No, I mean, well they have the fact that the monologue at scale is miscommunity, but they have a...

52:30 No, no, there's not much there. It's only the fact that the monologue at the unit is miscommunity. And the way they find a monologue at scale is what we do with the monologue at scale? Well, the composition in the pre-compact closed category, yes, I didn't see what I was going to do, it's scalable. Oh, no, no, no, no, I asked. The one who was here, he put out the first exit. Yes, yes, the fact that you're getting into the parts is there, I suppose. So, up until now, people were in general logic and pre-intensive time for it? That's relatively recent, but I think there were always some intuitions. So I think one of the sources of the linear and linear logic was linear algebra and Hilbert spaces and so on. That was there quite early on. I think there was a hope for those here connections, but maybe there hasn't been active work until the last few years. I'm curious how this works in cases of the political position, like, uh, for interest on that, and so it's, uh, Yes. We mainly looked at, in this work, the protocols of the teleportation. So it's because they brought out the soldiers. Yes. Well, there are lots of things to do, of course. primitives now rather than being building up a circuit of regulatory gates that we're really doing the computation are doing small maybe long bits and two bits measurements so this is this is so that that's one direction to be looking at it's and of course one of the points there is I'm sure you can map the classical circuit problems into that, and that would be one thing that I think might come up with as well.

55:00 Of course, I'm sure some people are going to be interested when it comes up with a new killer algorithm. I don't see my relative to identity, which do, can be proven in the abstract of the language. Oh sure, yes. Yeah, it appears, I mean, it looks like there's a lens that we're going to salvage. Right. because we have a lot of time because we have a lot of time it's really because we have proven about the relationship between quantum reflexive and classical reflexive and classical reflexive and that's