One out of many / Andrei Rodin; Isomorphism, identity, category theory

0:00 And now it's a great pleasure for me to introduce Bob Hooker. He used to be one of the pillars of Foundations of Physics and the Center of the Apostle in Brussels. Now he's, I think, one of the pillars of Komlach, where he's doing Foundations of Physics and Wolfson College in Amsterdam. I think this was supposed to be a little busy session this morning, but two out of three I have standing there, so I need to keep up the stage on my own. So my title of my talk is one out of many. The context basically is like, there has been a lot of pathologic research about 78 years since Berger von Neumann basically threw out his own brainchild together with a dead father when he denounced quantum mechanical formalism in the home space. And since then there's been a major effort, like, is there some better high-level structure or more perceptual structure which enables us to reason better about quantum mechanics? Now, if you look at mathematical practice reading quantum mechanics today, nothing is very visible in any textbook. You don't see the stuff. So it's nowhere. and after 70-80 years it doesn't seem to have had any impact on what people have been doing. So what went wrong? So basically the way people approached this problem was they thought of quantum measurement as a sort of prime ingredient of what distinguishes quantum mechanics from classical physics. You cannot just observe something, measurement is like a dynamical process in which you disturb the system. And there is little ways of articulating this. Some people try to do this in terms of non-distributivity of the language propositions, for example. And there are other paradigms which have been around. Basically the idea is from the concept of measurement, so from the concept of measuring the physical system, you basically try to accept some structural essence about that system. And so the idea is you have your concept of measurement, structure and after that you declare what is a single system which comes together with this predefined monolithic structure and then you try to say what many systems are. And so there is, you expect some compositional structure to emerge from your single structure on your system which you extract out of the concept of measurement. That's what people have been doing. Actually where it really has been

2:30 going wrong all the time was when extracting the description for many out of the description of one. It was something we should just review in all instructional approaches to how to describe a single-clothed system in a more conceptual or logical way. There's always In the local space terms, what you need to justify is this. So this is what you want to justify. Let's go to the task of the two local spaces. Take one local space, take another local space. And this is an essentially different beast than, say, putting two local spaces together with the Cartesian product or without each song. And this is basically the thing which is, as identified by Schrodinger already very early, the thing which is very typical for quantum mechanical behavior. Old phenomena you know about, non-locality, that things can be sort of in some weird way at two places at the same time. This all follows from the way how you combine the description of one system with the description of another system. And people just, the only way people succeed in one of these more logical approaches to do that is really by saying we are working in Hilbert spaces. Basically, there was no level of abstraction there, there was no level of conceptualization anymore at the moment you want to talk about many systems. So, well, if that goes wrong, then you would assume there is something wrong probably with the whole start of the program. That you start with the concept of measurement, and now with the concept of measurement, you try to derive the structural description on the system. Okay, so what I'm going to talk about is the opposite. So, first you try to think, what does it mean to be many? So what does it mean to be compartments or togetherness or the premise of many things without giving too much attention on what are single things? So we actually are trying to modelise the concept of interaction. So from compositional structure, from compositional structure, we then expect that the monolithic structure of a single system should rise. So it's a completely different way around. And so, something else together with compositionality, which comes sort of hand-in-hand, is the notion of process. Because the moment you start thinking about two systems, well, there's one easy situation. You've got one system, you've got another system, which don't do anything with

5:00 each other. And that's not very interesting. Now, the moment you're talking about quantum mechanics, and you know that the effect of an action on one system can affect the state of the other system. This is a process. I do something here and then something there happens. So you're immediately in like a dynamic language where processes are going on rather than saying some static description of what the state of the system is and trying to sort of summarize all the essence of the system in something like state, where the static notion of this system, this is what it is, and then hoping that you can derive how things will evolve, how they will interact with other systems. So no, we start with the concept concept of process will be the concept of interaction. And basically, measurement will not be an A-parallel, but a definable concept within the structure. Now, what is the sort of mathematical universe in which I'm going to go with category theory? Why categories? Well, there's a lot of people, a lot of people will tell you a lot of different stories why they would like to use category theory. Typically it's because it's some better way of doing mathematics, or if you come the algebraic geometry is really a way to solve real problems. So, my answer is very simple. It's just because it's the algebra of interacting systems. It's like on-the-nose algebra of interacting systems. And I'll describe this now to you. So, we're going to think about types of physical systems, and we give them names, like A, B, C, and this can be an electron, an atom, in a more fancy language, qubits, or classical data, meaning you're doing an experiment, you read your meter, you get a number, that's a piece of data, so I want to be able to describe this in my language. Of course, then I've got operations or experiments on my systems. Say I can do something in one system, like I'm acting a force field act on it, or I might do a measurement on a system. Now, this is already a very funny distinction from the traditional Hilvers space description of quantum mechanics, where a a joint operator on a Hilbert space. It's like an endomorphism. It goes from one space to itself. Now, if you do a measurement, you know very well that what's at the end, the outcome of the measurement, is not the same type of system as at the input, because you're actually measuring something. So you get a piece of data, you've got a meter reading of something, so that's a piece of data which you want to do something with. Sometimes you want to use this to control your next operation on your system. So a measurement is typical of this type, from A to B, where A would be

7:30 the type of image system be the type of the output system. It might be the same as this, but to get you some bit of classical data. Or if you destroy the system, this would just be classical data. Of course, so if types match, you can compose these things. You know, for example, just the classical computational computing, which you're like processing your data, which came out of CERF, is an operation of this kind. So unlike in quantum mechanical formalism, where the moment you talk about classical outputs, you go outside of the formalism, we expect these things to be part of the whole story, everything. All the data systems should be in one story. So of course you can compose this operation, like if I've got one process and the type of the output matches the input type of another process, I can compose them, I can think about some process which means doing nothing. And finally, whenever I've got two systems which I want to make interact, I want a symbol, this is just a symbol now, to say I'm considering two systems instead of one. There are operations which apply to two systems, given an operation on one and an operation on the other. But there are also some more interesting operations where this thing is incomposable. Where this thing would be, so you take, I've got carrots, I've chopped them up, I've chopped carrots. So that would be, I've got salads, I've chopped them up, I've chopped them up. And I start mixing them with mayonnaise. That would be much harder to sort of get them around. So this would be an operation which is indecomposable. You get something which is not more of the type of like, say, monolithic piece of vegetable, but of the type of salad. So that's... That's because of the mayonnaise, you see? Yes, yes, glue, glue. So here I will get my vegetable, and I will get my vegetable, and here I get something of the type of salad. So these are the more interesting ones. So there is a concept which is called symmetric manual category. If you look at this in a textbook, it's quite a fancy definition. So it's very involved. Sorry. Can you distinguish them? This from the other one. What is... So this is like a separated case. So what I write down here is a really more interesting one, but I'm really mixing up things. Now, one example, so this can be anything C. C can actually, for example, be B tensor A, which means I just swap two systems.

10:00 In quantum informatic terms, this could be something like C knot gate, which really creates an entangled state out of two. So it can be a lot of stuff. It can be a lot of stuff. This is very terrible. Basically, when I talk to you, it's not just about physics. You can apply the hidden. The whole story is, say, computer programming, where each type is a data type, each operation is a program or a process, and then you can have programs which take two pieces of output and then just multiply them, say, and get something in some other space or whatever. So people use this in computer science language, this is like semantics for programming languages, especially for interacting systems like distributed computing. People use this in proofs theory. People use this in proofs theory. This is a proposition, this is a proposition, this is a proof. And as you've just pointed out in cooking. And as you've just pointed out in cooking. I'm cooking, and there's much more spice examples, if you want to. But it's just a general theory of interacting processes. It's very basic. This is the basic mathematical structure, and this... Is this related to what you call convolution? In a way, there is something which, at a later stage in this example, corresponds to convolution, but not here yet. Because convolution has something non-linear in it. There are a bunch of axioms. There are a bunch of axioms which tell you how this thing interacts with this thing. If you go home and you don't want to take a blind, you see there is a stock of rules I'm going to show. But there is sort of the obvious rule which just emerged from the operational idea of having two operations which you compose in parallel and an operation which you compose after the other. So there are some interaction rules. And this exactly corresponds to what is called a symmetric monolithic category. It's like a concept which if you look in a category theory book, you find very much at the end of the book. So it's not sort of a basic concept, although it arises in such an obvious way in nature. So there are historical reasons for that, I'm not going to go into that, but ask me maybe at the end. But this is sort of, but it's so canonical. Now this is for example one of these rules. But this is like the swap operation, and so you won't assume if I've got system A and C, and then I swap C and A,

12:30 then if I wouldn't do an operation F on A and G on C, then this should be the same as, let's say, A first do this operation, and then swap, or first swap, and then do the operation in swapped order. It's obvious, it's obvious, but you have to make this explicit. You have to make this explicit. It's a rule. Here is another rule which is called by thing to reality. And which says, first doing f on a and then g on c should be the same as first doing g on c and then f on a. It's called by thing to reality. It's something you can prove for the Wilbur space data product. But nobody just realizes this usually. Now we have to make it explicit. So there is this set of obvious rules which gives you a symmetric amount that is not applied to basically everything. Now, so this is such a general theory that we have to say something specific if we want to go and say something about common games. So, I'm a physicist, really, I'm not joking you. So I'm going to impose some additional structure, and the physicist is a phenomenological, so I'm going to do phenomenology. I'm going to look outside in nature, see what I learned about the physical world, and I'm going to impose this on this basic structure as an additional structure. And then we're going to start reasoning and see how far we get with how little. That's the game. So, Phenomenon 1, we know that a panglement state exists in nature because people have been doing stuff at one side of the lake of Geneva and the other side of the lake of Geneva, and clearly there is clear experimental evidence that this phenomenon of a panglement exists. So, we were putting a panglement in a very sort of conceptual abstract way by hand in formalism at some point. So, that's the first thing which I would do, and I would call this a dull state just for historical reasons. Phenomenal 2. So this is something we know very well. If you've got a scientific result, say you work in the lab, you do some measurements, we write a paper and we hope that it gets copied a lot of time and a lot of people read the same result. Something we expect from our scientific results to happen from our measurement readings. This is something you can't do in the quantum system, for example. You can't copy the scale of the quantum system. So, a phenomenon which we know is that classical data can be copied and deleted. Okay, so this is something I will need to put in, and there logic comes in. The people who know about linear logic see, if you ever know about linear logic, you see this sort of game I'm starting to play with. So, I'm going to bring in classical structure, think classical logic by the ability to copy and delete premises.

15:00 It's classical logic, you never worry about copying and deleting premises. And that's why here's some piece of logic about classical logic coming in. Pictures. So, in fact, although I've been talking about physicists in hyperspace, they don't use hyperspace really in practice. They use this thing which is called direct notation. Who never saw his life, this sort of notation, direct notation. Anybody who never really saw this? Yes? Some people. So you think of this as a state, it's called a cat, then if you put it like that, then the same thing, then you call it a bra, and if you compose it, then you call it a bracket, and you think of it as an inner product, as a number. If you compose it, you don't know the way you call it a projector. And it turns out to be a very convenient way of computing, and it's a very different thing than computing with vectors. It works very nicely and physicists like a lot, but it doesn't really have a formal basis really. It's sort of ill-deprived, which is funny. Now, on the other hand, there is a physicist's design from pictures like Ben-Rosester and Firestone. It turns out that these categories I was talking about, so first I could use this operational innovation to save your dog categories. It turns out this specific case of symmetrical categories, they come with a graphical language. And this, the horizontal direction will correspond to the tensor while vertical direction will correspond to the composition. You'll see later on the words. And there is a highly non-tribial result, a highly non-tribial result that says that whatever is covered from categorical axioms, which I didn't all spell out, is perfectly in some graphical language. The graphical language I will spell out. So that's why I didn't bother to give you a definition, because I will give you the graphical language definition, which is much easier. And from now on you only want to see any formulas of symbols anymore. Just only pictures. And it turns out these pictures are actually the representation of two languages. So okay, so we think of this as a process from A to B. We think of this as doing nothing. This is the composite of two processes. G after A, this is F of B, G of C, and this is more composite thing. Well, if you read it upside down, it would be my salad example. So, and these are then sort of, you remember that I had this commuting diagram saying first do f and then g should be the same as first do g and then f. I also had this example of doing f and g and then swapping is the same as swapping and g.

17:30 And you see that you've got some sort of very intuitive diagrammatic reading of these goals. It's like flowchart diagrams. So you think of information flowing through these wires and being processed on by these boxes. So that's going to be the logic of the system. Now, we've got some special operations which... So there is a special type which we call nothing. This is important. Sometimes somebody tells you here is the state of the system and you don't care where it comes from. Nobody specifies how you actually produce it, they just give it to you. Like the Sun out there, we didn't produce. We just observed it. It came out of nowhere, for instance. And we're going to represent it by these triangles. Then you have to take the sort of dual to this. And if you compose the two of these, you've got something without an input and without an output. And we will think of this as a number. Why do you need to dual? It's just an intrinsic part, because I've got to type nothing. So I have to also consider the process. This could be a destruction or something vanishes or destroy it. That's the way you think about it. Now the analogy, the analogy is very simple. So for those who know this cat, you just close down this and you get a triangle. For those who know it's brown, you just close down this, you get a triangle. And this is your bracket, you're in a product, you close up, and you get a complete triangle. For those who know, bracket rotation will immediately see why this thing is really meaning instead of physical language. This is the bracket rotation, but I've got an additional dimension to it. So there is this ability, like I refer to, that you want to, that you have something, say, of this type, and there are also duals, and in direct notation you do this all the time, you turn and get into a trap, so we assume we have an operation which reverses our boxes upside down. Okay, so this is the general setting. Now, let's try to say something more specific about quantum mechanics. Let's try to say something more specific about quantum mechanics. So, I said before that I want to implement the idea that there is entanglement. So, I'm just showing you this to just convince you that there is a formal way of defining these things. I'm not going to go through this diagram, but this is like the definition of what I'm now going to write down is a fiction like that. So, this is in a peggled stage. From nothing to two systems. So think of it as a bell script. This is one side of the lake of Geneva, this is the other side of the lake of Geneva, and one way or another you create and stay there. It doesn't really matter how you do it or which resources you use, but you know it's there now. That's this thing.

20:00 And then I'm asking this very weird behavioural strain about it. So this is something you can act, so in Hülper's space, in Hülper's space, in Hülper's space, for those of the Hülper's space formalism, that thing would be basically always called the bell state and looks like this. Well, it's not really a state, it's an operation, so what I do, I send the number one as an element of the complex numbers to its bell state, which is of course the same thing by the end. This is basically the triangle. And then I have this rule, and this rule you can actually prove in all the states that it is true for this one. You can prove this, it's a single complication. The thing is, this is a very simple action, and we'll see how much comes out of this. Now, I want to understand a bit more of the logic of this rule before I'm going to start using it. And therefore, I'm going to do the phonic trick. I'm going to sort of change the representation a bit, and I'm going to think of this as a little cup and a little cap. And if I can't forget the triangles, then you see that my rule is nothing but stretching a wire. It's sort of a topological rule. I've got a wire, like here, this one up there. I've got a wire and a structure. So this is going to be my reasoning mechanism, nothing else. Okay, can we do anything with that? Well, let's see. So I've got these cups, and I've got these caps, so I can define something new. So I've given some box F, I can define this new entity. I can, for example, also define this new entity. This adjoint refers to putting the thing upside down, remember? the autistic part of the language. And that turns out that there are a whole bunch of things relating this to, let me humanize it. So let's see how we're going to do this. Yeah, so I start with F adjoin, yeah, start with F adjoin, and then I do this. So basically what this thing is, is what I call F-lower style of the fish.

22:30 Then I'm going to apply the left-hand side, which is putting another one of these here, putting another one of these there. I know I can stretch these things because of my axiom, so basically this thing is nothing but this. And this thing is nothing but this. I applied first the right-hand side operation and then the left-hand side operation gives me this edge one itself. So what I mean is like, if you first do this and then that, then you get this one. And now here is a nice geometric representation. I already said that this edge one is like just putting something upside down. If I now geometrically represent this guy as just stopping something left-right, then you see, first I apply this one, which means I go here. Then I apply that one, and then I'm going to correspond to a 90 degree rotation, to a 100 degree rotation. First I come here, do a 108 degree rotation, I come there. So the geometry of these three or four pictures which I depicted here basically turns out to implement all the rules you can prove from this little Yankin thing. So basically everything which is there to say algebraically of this little system is encolored in the geometry of these four things. What? The mirror syllabus, yeah? Can you go back one slide, please? Yeah, no, one more. one more yes this operation there described on the bottom of the page to the left that's a state that's a state this is this having this low state available ah and the fact that you have the two because you have this this this was part of my language there was like two things upside down typically this will correspond in practice to a measurement a projector okay okay so we'll see You can see physical processes out of that. You'll see later, hopefully, or you can see physical processes out of that. Now, here is the game which you saw. So this is sort of what I just proved. And it turns out now that if you take these boxes to be linear maps, that this operation, this one exactly corresponds with transposition, and this operation exactly corresponds with complex quantification in the other spaces.

25:00 So at an abstract level, we now capture the transposition and complex conjugation, just in this topological manner. So there's a topology behind the idea of conjugating complex numbers and transposing matrices. Now, here, I'm going to prove you the lemma. So this was this thing which I called the transpose. This was this thing which I called the transpose, which is this guy. This is this guy. So this is this thing in the middle. Now, we can do two things. We can stretch this wire and we can stretch this wire. wire. And basically, this gives you the following results. So, I get this if I stretch this wire. I get this if I stretch this wire. And now you see it's as if I take this thing in the middle and just sort of slide it along these wires. You see? It rotates like one of the data degrees there. And it rotates like one of the data degrees there. So, I got this sliding mechanism. I can slide these boxes along these cuts and these caps. And Basically, this is the theorem. So I can sort of like, it's like you see, you've got these pearls on this chain, you can swap them all to the end. Now, here is an exercise. So this is the theorem about linear algebra. Don't you get this. This is the theorem about linear algebra. I wrote down what this little bit of wire means. It means this linear map, where it goes. And this, this is that joint, and these are just hard linear maps, just sitting anywhere else with wires. and I just proved that this is equal to that. Now, you can try to do this in a week explicitly, but probably you're going to want to finish it by the end of the week if you want to do this in linear algebra terms with matrices. So it's like a high-level way of reasoning about calculating with tensors. Now, let me go a bit quicker. Yeah, okay. So let me go a bit quicker here. So, basically, the geometry, which, yeah, I'm not going to say it because I want to go further. So, what is important this year is, is that in traditional quantum logic, this was the paradigm people used. I've been working on quantum logic as compared to classical logic was basically about having no deduction as compared to deduction. So, the paradigm there was having no distributive lattices as compared to distributive lattices. And distributivity is basically equivalent with having a deduction mechanism, a distributive mechanism. And so what we are actually trying to do here is to see what stands the point of theory as natural deduction stands to tables. As a matter of fact, and I can't go into details, but this stretching of values is actually a fault in two deduction rules.

27:30 So, these all four words have a corresponding syntax, of course. So, there is this categorical language, and there is a very hard-guard morphism, which relates category theory to sequence calculus. And the stretching of this wire actually corresponds with two deductions. So, I can't go into detail there. So, every time I stretch a wire, there's a really thing that there's a deduction going on. Like a logical deduction. Two, actually. In some funny way. Okay, now let's do some physics. So this just follows from my scheme. Now if I pick F such that it's inverses, it's adjoint, which means it's unitary, then it would fall away. I introduce two triangles there, which I'm allowed to do because that's what I want to treat them. I give them two names, Alice and Bob, and this is quantum teleportation. I don't know whether you hear about this, but this is exactly quantum teleportation. How does this work? This is one side of the Lake of Geneva, this is the other side of the Lake of Geneva. We've got Alice and Bob, which share this special entangled state, which I postulated to existing names because it has been observed. There is Alice which does a certain measurement there. Bob does another sort of operation there. And the net effect of everything is that there is a flow of information from Alice to Bob. So whatever state you have here as an input will end up with Bob. That's from the third phase. What is the nature of Bob's operation? Well, in practice it will be a unity operation. So it's like just a dynamical change of the state. Now that's a very important point that it needs to do something here, depending on what happens here. And as you all might know, like quantum mechanics measurements have uncertainty in it. So you try to do something, but you're not sure what you're going to get. It's not an observation that you really see what the state of the system is. Like you get some outcome, and the outcome is actually what the state of the system has become during the measurement process, not what it was before. So this thing is actually non-deterministic in this sense, like whatever f is going to happen here, we don't know in advance. So basically what has to happen in practice is that Alice does a measurement here, some of these f's happen, and then she communicates through Bob which one happens.

30:00 And depending on which one happens, he corrects. Now the nature of the phenomenon is that this thing which is underneath here, this quantum state, is typically continuous data, in some banks of the Hilger space, which is continuous. The number of outcomes they have to send from here to there is typically discrete. It's finite number. So basically you're communicating continuous data by only communicating finite data. That is sort of the magic of the protocol. But something else is you can't... If you don't have this communication from well as the bot, then nothing is sent. Nothing is sent. And you can check that what actually the information he gets here is just completely mixed. So it's a very delicate mechanism, but people don't really understand it very well in the structural way. This quantum teleportation, the way it comes out of it, just this is how you find it in the textbook, typically. It took six eminent scores, sixty years, since the birth of the quantum mechanic of four months. So, yes? And this is improved? What? And this is improved, so this is the description improved? Ah, you improved this in the outer space. This is just, you can write down, you can write down this statement just in matrix terms. You can write down this, that's how people do this, by just composing matrices and stuff. And then, so the description looks something like this. And then we need to bring vector spaces in groups. The thing is, of course, in this language which I'm presenting, this is sort of an obvious trivial thing from the Ansatz, that you can do these things. In vector space language, it took 60 years to find it. And, well, six people, 60 years, sort of the last coincidence. There's just one six missing a few. Okay. Are you one of these six, or...? I'm the third six. So anyway, what is completely missing in this language, which is actually missing in welcome form list 2, but which would be nice to make explicit, and I hope I'm still trying to be able to discuss something about that, is where is this classical data flow I was talking about, this line? It's not part of the picture you saw. As I say, there is an implicit correlation between this and this, because they carry the same name. But it would be nice to see

32:30 Because you see, this is a theory about flow of information very much. You think of data flowing through these lines in some way or another. It's very important. So you want us to be careful because this looks as if the information is flowing backward in time. So I don't want to go too much in the discussion about this, but this is the way it looks. But I would like to see also this information flowing from here to there explicitly. This is the next step of the development. Whatever I've been talking about now, I call quantum structure. of compound systems in bottom mechanics. What I want to get out now is what I call classical structure. It's a description of classical context relative to the quantum world. And in the third stage, which I probably will get, then you can actually find measurements within your language which unifies quantum and classical. So I already alluded to this, that quantum data cannot be cloned nor deleted. Classical data can be cloned and deleted. And we're going to treat the first as a non-feature and the second as a feature, which is completely the opposite way people usually do it. They think of this as a freak thing and this as obvious. So we're going to turn it a little bit upside down. We'll treat this as a feature. And we'll specify what classicality means by this ability to clone and delete. So a classical data comes together with a cloning from one to two, and a deleting operation from two to nothing. Pictures from one to two, from one to nothing. And, okay, so me and my friend who worked this out, then we need to figure out what is the sort of actions we have to impose to really get classical behavior and reason in the same way as we can reason on classical behavior. And then we get like, so categorical admissions, they know immediately what all this means. But, okay, it's like a bunch of actions. So this. Now, given the time, I have no detail to talk about this. But this is a little bit like, in some way, this is a little bit like associativity. Because you see, I've got A, B, C. And if I first look at my B and C, and then A and the result, it should be the same as first note applying A and B, and then the result of C. The only difference is that I read it upside down. Now, it's co-associativity. It's something you need Kabynick theory for to even express. Because it's like the operation of composing in the other direction. So you start with the output, then you go to the input.

35:00 There's no concrete way of course to testify. For a copying operation, it's sort of obvious. If I've got A and I make two copies, and then I make two copies again of one, it should be the same as making a copy of the first. First, you see, if I want three copies, it doesn't matter in which way I produce it. If this is a general copy of the collaboration. If there are two copies, it doesn't matter whether I... This is something like commutativity. It's co-commutativity, commutativity upside down. This is like having a unit. So if I've got a group and I have a unit, and I multiply the unit with any element, then I get an identity, I get the initial element. This is again a unit upside down. It's called a commonoid structure. And then there are these two very funky actions which we also need. So, you see, this is one of these very weird, funky actions. It's called Frobenius one. It's due to, it's a very recent sort of mathematical thing which people discovered when they were trying to exunitise what is a relation. Because category theorists spend an amazing amount of time thinking, what is a function? And they develop topics, theories, and all this business. These people were thinking, what is the relation? The relation is completely different in this kind of function structurally. Nothing to do. The relation is much closer than two vector spaces and linear map than a district function. And this is one of the rules that they actually introduced to describe relations. So, okay, so given this Frobenius rule, now I do the following. I plug in my little triangles here and there. Then you can go back. You see this unit long here, which means I get this and now we get back to the first picture of my thought. Now you find these big triangles here again. You see this is the roof we start with all in the beginning. So this axiomatic structure which I introduce in the classical world basically defines what I introduced to describe my linear quantum structure. to think about this semantically, formally in vector space terms. These things, which look nothing like anything to do with the arousal graph, these things, they actually define a base. This is the definition of a base. If you express them, and this is quite a possible truth, if you express them in Hilfer spaces, this is exactly a base, an orthonormal base. This is what it is. And so what I've expressed here is that by expressing

37:30 in some way, you actually capture a base because you can't copy and delete in quantum mechanics in Hilbert space. The only things you can copy and delete are vectors of a base, of a chosen base, which is your classical world. What is the classical world in quantum mechanics? It's a base, a chosen base within your Hilbert space. And this can be position base, this can be momentum base, there are many different bases, and each of these bases corresponds with two with a trapezoid and a little triangle of this kind. And this is the structure which completely captures this. And you see, a base refines the structure of Hilvers base. That's what's shown here. Now, one of the really neat things about the first part of the story was like a simple mechanism of the Anken Rhoads. It was a very nice basing mechanism. And now, I've got five equations here. I've got what is called Frobenius Comonoid. So, is there They're a way to make a simple graphical formalism for this horrible fiber equation. It turns out that there is a very simple graphical formalism which refines the yanking of a rope. So, is there something finer than yanking a rope? Just, I'll give you three seconds to think about it. Yes, there is. Okay, so, I open this box in the same way as I found these cups and these caps, and I find the dots in there. Just a dot here with three wires and a dot there in one wire. And then we'll do like the most beautiful notational convention you've ever seen in your life. We'll never have got a configuration of this kind, you see? So here, this thing, this little bit is something which came out of the interior of this box, huh? So this is a notational convention. Whenever you see this, you'll rewrite it as the spider. This is a spider. And then you can prove the following theory, which is combinatorically horrible to prove, but you can prove it, that each big network involving these copying operations, these deleting operations, upside down, the upside down ones, I'm not going to explain what they mean, but they mean something very clear, you have to do this fantastic behaviour actually. They always can be rewritten in this normal form, as a spider, provided your network is connected. So if you combine these operations, you've got some very crazy configuration of all these things, and they are connected, you can always write it as this. So these are sort of, for people who have looked at the literature of things like topological quantum field series, you see similar things, but then they are proving this stuff with like two-dimensional manifolds or three-dimensional manifolds.

40:00 You've got similar normalization results. So I said that the yanking of the rope is actually, that this refines the yanking of the rope. So here we start with the yanking of the rope. I can decompose this like this. And now we apply the spider law which says that there is a normal form which only depends on the number of inputs and outputs. I see one input and one output. So I normalize it like that. Or you could think that there is a dot inside here, a dot you just throw away, a single dot on the line. refines this spider reasoning, refines this yanking-of-road reasoning. And last, now, so what is the logical context? This is like slightly brutally put because it's a lot more subtle than it is. But in fact, what I've been doing here, like, if you think in terms of logical terms, since second-calculus search, then I've been comparing classical world to quantum world, our classical logic stands for linear logic. Classical logic can actually be refined as two components. Linear logic, which is a logic in which you are not allowed to copy or delete your premises, and the ability to copy and delete. It turns out that this is a very beautiful logic with a nice geometric proof structure and all that. Well, this is a horrible logic. Whether you believe in it or not, it doesn't matter. This is just much more pretty, structurally. And it's much nicer to look at classical logic in this factorised way. It's just more pretty. And it's more useful, and it does actually a ton of applications in computer science, by the way. And this is what I just did. I represent the classical world as a quantum world plus the ability to copy and delete. Like the grey boxes which I introduced on top of my big triangles in the beginning, there was actually the ability to copy and delete. I guess, am I running out of time, or, yeah, so let me just end quickly, to sort of close the circle, so I start in the beginning by saying that in all the other approaches to quantum logic, people start from measurement, and then, derived from it, how would we structurally describe a single system, then tried and failed to describe multiple systems, so I start with multiple systems and their interaction, So like, for example, that rule that the yanking involves three systems, eh? The yanking rule is three systems involved, eh? It's a multiple system rule.

42:30 And I want to get now to the notion of measurement, which is basically a flow of information from my quantum rule to my classical rule. How the... how the...