Symmetric Bimonoidal Categories & Quantum Systems — Part 1

0:00 It's a research program which sort of started in 2004. And the idea is to really have new quantum mechanical formalism. Not just to sort of categorify it, because there's people who have very much categorification. It's really just plainly, can we do quantum mechanics better, and what is the right mathematical structure to do it? And that turns out to be monoidal categories, for a reason I will be showing. So there is no motivation like category theory is a better way of doing maths. That's not where I start. I'm just trying to do quantum mechanics in a better way from an operational physicist's perspective. So, what does this mean? First of all, a better formalism should show the important things up front. Like, if you look at the mathematical structure, you should see what is really important about quantum mechanics. So at the moment the situation is, if you look at the quantum mechanical formalism, you're staring at matrices. Do matrices tell you something about the fact that in the quantum world things happen non-locally, like there are very special correlations happening between things far apart? But matrices don't really tell you that in a very precise way, because we use matrices for a lot of things in mathematics. So it isn't very specific of quantum mechanics. So that's what I mean. The reason to redo quantum mechanics is to see the essential features of the theory very much up front at a high level. So there is an analogy with programming languages. You could program with zeros and ones. And then it's very hard to understand an operating system. On the other hand, you could use a language which is very much crafted to specifically understand features of, say, operating systems. And that if you look at the language, you immediately see we're talking about an operating system and the important features that you've got different modules and things like that are very much up front in your language. That's what we want the quantum mechanics to do. We also want to make it sort of closer to the person who's actually practicing physics in the lab. Basically, there is quantum mechanics system. I'm a physicist. I'm interacting with the system. And this is actually how I see quantum physics, by interacting with nature. And I want to see this very much up front in my formalism. I want to see this operational structure where you have things interacting and where there is a connective which tells you immediately now this and this is interacting.

2:30 I'm working in computer science department, so this needs to have something to do with computer science. The idea is to have a theory which you can automate, which comes with a programming logic, such that you can just tell a machine how to do physics. At the moment this is completely impossible with the current quantum mechanics of formalism. I'm not going to go too much into it, but the current formalism of quantum mechanics is actually not comprehensive, In the sense that if you are studying a physical system and you start to calculate, then at some point you will do an observation or a measurement. At that point you have to do things like diagonalizing a matrix and then writing down another piece of paper which your values are in your measurement. And this is not within Hilbert space anymore. You go out of Hilbert space, you go out of your mathematical formula and start writing things down on pieces of paper. comprehensive formulas which gives you the whole process of measuring and experimenting and all that. All this stuff is not part of quantum mechanical formulas. It's a series of rules you apply but at some point you just start writing numbers on a piece of paper. This is not really part of your mathematical structure anymore. So you want fully comprehensive theory which we can automate. And actually finally if you want to sell this thing to physicists who have been using the same theory now for like 75 years, so to say, then it needs to be very attractive. And one way to say, very attractive means user-friendly. So it must look simpler than what they've been doing for 75 years. So that's a very hard task to convince people about that. So these are the four key features I want in this new formula. And the way I'm going to present it is very much along this line to convince you that there is something going on here which looks very different than what you will be seeing in any textbook on quantum mechanics which you will ever read. And it will look very different than what you will find in any textbook on category theory too, by the way. There's a big problem with category theory at the moment. Like a lot of people who like this sort of stuff and they say, where can we start reading about this? I don't know. Honestly, the sort of category theory which I'm going to be using, I don't know

5:00 can sort of point to and people say, start studying there and you'll get there. And there's really a need for, I would say, a textbook which is specifically crafted towards physicists, which is very simple, which is very intuitive, and which explains to them how they could use category theory in a sort of very end zone operational way. I think there is a need for something like that at the moment. So first, why categories? Like I say, I don't come from some school where they taught me that I need to use categories to do mathematics in a better way, so I need another motivation to do that, which forces me here. And this is the following motivation, which is very much the reason why people use categories in computer science. So I'm just thinking about different systems which I want to experiment on, physical systems, which could be an electron, an atom, a number of qubits, like if you're talking quantum informatics language, classical data. so I make an observation and I write a number down. So these will be my types of systems I care about and I'm just giving them names, symbols. Secondly, I want to consider operations on this system. So I've got something here on the table and I want to study it, so I'm just going to interact with it by maybe applying some force field to it or measuring it. Like I take a big device and I measure it. There's one thing in quantum mechanics, measurement is a very involved process. It's not just looking and you know what it is. It's like a very big interaction with the system. And so we represent this by processes. And of course, for example, if I measure a physical system, what I start with is a certain physical system, say an atom, and after the process, what do I have? I have some value, say the position of where this atom is, or some other physical quantity, or the energy it carries, or some physical quantity, which is a number. Classical data. So the type after my process is different than before because this would be the system, this would be maybe the system together with some data. So you can have this change of type here. Then, of course you can compose operations when type match. And finally I want to consider the following thing. I want to consider situations where I've got for example two systems. And so I need a symbol, I need a symbol to write down that I'm looking at two systems. For example, like here there are two systems, and I'm looking at them at the same time, and maybe I'm going to apply operations which sort of mix them up.

7:30 The simple example which you can think of is like you want to make a salad, and you've got some lettuce, you've got some carrots, so there are operations where you shuffle the carrots, there are operations where you shuffle the lettuce, and then there is also the operation of mixing, which you can't read and so then everything is sort of intertwined so you need the notion of tensor and in that case you would do that like that one veggie one veggie two so I'm looking at two vegetables I mix and I go to the south but then the question is there really a natural way to distinguish between this case when you you don't change type right from A to A and when you does change the type or it's kind of conventional or how? So the thing is you have to give names to things otherwise you can't talk about them and if you there are there are clearly physical phenomena where you start with a system of one kind so if you've got two in an electron and a positron meeting each other and typically you get two protons after the interaction. So you've got a change of type. Now, the place where the type becomes slightly more vague is in quantum field theory. But still, you want to consider, so there you might think about regions of space as your system. Regions of space and whatever you observe there. And you think then of the occurrence of a particle like, because even in quantum field theory, if you go to the pictures of CERN, then how see these sort of things. You know these were in the bubble chamber, you see this picture of the sun, so these are positions of the system, so you know something happened in this region. So I would think of the type of whatever happened in this region, so there are ways to deal with that. But for like plain experiments which people do in the lab, like where you check bell inequalities or things like that, this is like the sort of thing you're working with. This is like your language, and of course this is the monoidal category, because of all the other sort of rules of a symmetric monoidal category, like symmetry, associativity, they are sort of just bureaucratic rules where you take account of which system is where. For example, if I've got a system here, I've got a system here and a system there and I decide to swap them places and then the operations should follow the swapping that's basically what this naturality of symmetry tells you it's like purely a bureaucratic rule it doesn't

10:00 have any physical content it's about keeping keeping track of where your systems are and maybe like associativity means if you've got three systems so basically now I'm actually focusing on these two and maybe at a later stage I'm focusing on this one that's what associativity Now, of course, you can always go to the category where you just throw away your brackets and where you say dematural isomorphisms to be strict, except for symmetry, then, of course. So how do you think, so what is the unit, the tensor unit of your somatic monocategory? The tensor unit, well, you're joined to any system, and you get the system itself, so it's nothing. I think about it, no system. But it actually has a very important role in this theory, you will see. But a priori, I think no system, or no system specified, if you want. I don't specify what I'm doing from somewhere, I don't specify where. For example, this type would mean from somewhere, I'm not specifying where, I get something. How do you think about something like that? You think about that like a state of a system. So somebody provided me with the state of a system. I don't care where it's coming from, I don't care from what it has been produced, like here is a state. That's something which appears in any physical theory, a very important concept. The concept of let's psi be a state of a system. That's how you read this. From somewhere where you don't say it, from nowhere or from nothing, you get something. So that's what it means, a state. This line is ignorance. You can think of it as, I don't care, I don't know. I don't know. Unspecified. Yes. That's one way to think about it. So, but anyway, the main thing I want to say is that, like, the definition of symmetric manual category is sort of a priory, almost, in the way I'm talking here. And people in computer science who think categorically about programming languages have more or less the same philosophy. The tensor is about composing programs, the types are like the data types that you take as input or as output, so it's a little bit the same philosophy. People in proof theory have also the same approach to these things, where you think of your types as propositions and your morphisms as proofs. So, it's all a little bit the same story. By thing to reality, it's like a very weak assumption on the nature of compoundness.

12:30 It means, basically, that if I've got two systems, so here at the same time, here I've got space, I've got two systems far apart, I've got a system here, I've got a system here, I've got a system there. What biocontariology basically tells me is, like, if I'm operating on this system, then it's completely in the final of operating on this system. that they interact in some way. So, I can basically... Now, there is something very important there, because if you know about relativity, this sort of is weak enough to accommodate relativistic spatial-tentral structure on your events. Because in Priory there is no interaction of things far apart. So you think of them as being outside each other's light cones. If they are within is each other's light color, and in direct, of course, you can specify this, but a priori is this, this is a sufficiently weak notion of tensor to accommodate for logistic phenomena. So it's, it's like the most primitive basic setup you can ever think about. Now, so what I'm going to do next is, so this, this, this language is just any operational theory I can use it for anything, where I'm thinking as an experimenter and interacting with the environment, be it as a cook in my kitchen, or as a physicist working in the scientists programming some some network it's all the same this is the language you speak and there is no philosophy there is just like so such basic setting now I want of course I want to specifically now recast quantum mechanics so I need to introduce additional structure in my theory and the additional structure I'm going to be very pragmatic about it will be will be purely phenomenology things I have seen things which have been observed in labs and we know about, and I'm just going to add as part of my structure. And actually in this talk I will only consider two things. So one thing will be the fact that entangled states exist in nature. So basically it means people have been observing nature and see that sometimes you've got these non-local correlated phenomena. So the best experiments they have at the moment is they're standing at one side of the lake of Geneva, they're standing at the other side of the lake of Geneva, correlations between outcomes which can't be which are faster than light and can't be explained by

15:00 local statistical theories so there are these special states of photons where while they are far apart in some way they communicate to each other it's in a very delicate way because you can't send information like that but still there are correlations so we know these things so i will put this in my theory as something which i call a bell state because people typically the name people give to it. Another phenomenon which we know is we can copy classical data and we can delete it. We know this all. You can basically, if you have a scientific result, you can write this in a piece of paper and this can be distributed and everybody can read this. So you can copy this as many times as you want this observation which you did in your lab. And you can of course delete this. If somebody has done something and you want to be the first one and you kill him and you just erase all the evidence that you have found this result and i don't know and then you get famous so you can do all these things this the reason this sounds a very silly observation but the thing is for quantum mechanical states you can't do these things you cannot if i've got a quantum mechanical state here which has been given to me in the way i talked before then there is no way of reproducing a copy of it without like destroying it or disturbing it or so I can't just make a copy of it and I can also not delete it in a certain way. So this is very specific about the quantum mechanical structure and so a second piece of structure I will be introducing is the presence of the ability to copy and delete data and this will be specific for my classical theory. So one important thing about this setting which I'm going to define is that you've got both classical and quantum within one theory. Both types are there, which is not the case in the open space, for example. And I will show you certain protocols where you've got an interaction between classical and quantum. So they all take part in the same formula. So, now which categories are we going to consider? Of course, in the literature the most important case of categories people have been thinking about are Cartesian categories. And you've got a natural diagonal there. So, now you can think of a natural diagonal as cloning. It's like copying something. I've got a system A here, I've got a diagonal map, and here I've got two systems A. So, if I would have a state F. So, think of this as now. Okay, let's take A to be I, let's take B to be the state space of a system.

17:30 I've got a state here of my system, I copy, and then here I have two copies of the same state. That's what basically this naturality condition tells me. So as far as you can, by the plurality, it just kind of applies the Cartesian approach? No, no, no, no, nothing. I'm just asking the question now, what sort of categories can we look at? And the first candidate, I open my textbook on category theory, the first monoidal structure, and see, of course, the Cartesian category. So I'm asking the question now, is this useful for me? My answer is going to be no, of course. So that's what I'm showing. So for a specific situation of I'm going to pick f, I've got my i here, I'm going to pick f to be a state, so then this will be just i times the i, which is i itself. So, this naturality tells me that here at times 5 times 5, so, and here is this copy, this copy of the case. So, so basically, this diagram tells me that I copy, that I can copy states. Now, and I already told you, you can't do this in quantum mechanics, and, and here is the proof in Hilber space. So, I take here the trivial Hilber space, complex numbers, this is a tensor product of two of them, which is the same thing. I take the state of the physical system. So, by this I mean, this is the, just, I've got two base vectors, 0 and 1, and I assign the number 1 to the sum of 0 and 1, the sum of the two base vectors. Down there you have b squared. What? Down there on here. Oh, sorry, sorry, yes, that's fine. So, I take the state, I copy it, so 0 will go to 0 times 0, 1 will go to 0 times 1. So, now, if I go along this side, I see I get something else. These two things are not at all the same. Because here you've got a pure tensor. So I can factor this in a state of one and a state of the other. Here they are what they call entangled. This thing is called the bell state. You can check this. You can never write this as a pure tensor of two things. This is one of these entangled states, which are very specific to the tensor product. If you make a tensor product, you get a lot more than just system one state and system two state.

20:00 You've got sums, and this is such a sum. So some of a tensor and an other tensor, and you can't reproduce this as a tensor of two other things anymore, because the tensor product is of course much bigger than the direct. And the presence of this entangled state violates this commutation, so there's no way you can dream, you shouldn't dream about trying to do this in any Cartesian setting, it's just not going to work. Now, look carefully on the specific maps which I write down here and the spaces I picked. Because this phenomenon is not at all typical to Hilbert spaces or vector spaces. For example, it also happens in the category of relations. So I am looking at the category of relations, sets as objects, Cartesian product as tensor, morphisms as relations, not functions. Suddenly, what is the diagonal, what plays the role of the diagonal in the category of sets and functions, ceases to be natural in the category of relations. We've got a completely different... So it's sort of funny that... I'll come back to this later, and I'll refer to work by Carboni and Walters, where they studied the axiomatics of the category of relations, and how can we axiomatize the category of relations, and you're doing a completely, completely, completely different chapter of category theory, as, say, everything people have been doing by looking at the categories of sets and functions. It's like a completely different world. On the other hand, it's a world which is very close to my quantum mechanical world, category of relations. A lot of things which you encounter in quantum mechanics will be present in the category of relations, although the objects are the same as the category of sets and use the same temperature. So we'll come back to this issue later a couple of times. So, okay, now, so basically we're going to talk monoidal category. Now, there is this, so I think in the early 70s, so when Penrose was studying general relativity and he was doing multilinear algebra, you start to use all these pictures, like in a sort of intuitive way, to represent certain calculations in multilinear algebra. So that this thing has been formalized in the early 90s by Joellen Street and also Fried and Getter have been working in this direction. And basically what you can show is that there is a graphical language for monoidal categories,

22:30 where you think, so, well this is a convention of course, where I will think of the horizontal direction of the redirection of the tensor and the vertical direction of the direction of composition. So you've got this two-dimensional categorical structure where you've got two ways of composing. I can compose systems, and I can compose operations. One system besides another system, one operation after another system. These are two dimensions, and you can represent this graphically. And so what Joellen Street showed was that basically that whatever is provable from the categorical actions of a symmetric monoidal category And there are other categories which have to, like, trace moral categories, where you've got the same sort of nice story. And we want to stick in this realm of graphical languages, because that's something we can sell to physicists. When you say prove, will you mean, like, internal language, or...? Just from the... well, there is no obvious internal language, but, I mean, just from the actions of the theory. It's like, you write down your actions of symmetric mineral category, if there is an equational statement I can prove from this, then I can derive it by some topological graphical manipulations. I'm not going to be very precise about the rules, you can find them in the literature, but they're very intuitive. It's sort of, the rules are what you expect them to be. If nobody would tell them, that's the ones you would pick. So, I don't know whether anybody here ever heard about this direct notation. Because, of course, quantum mechanics is a Hilbert space theory, but if you read like a sort of plain, hands-on physics tech book, you will see a lot of these sort of guys. Cat, bra, bracket, projector. So, it's this weird... So, basically, you think of a vector as something like this. You think of the dual. So, this thing is living in a dual space, and if you compose them, if you compose these two, then you get the inner product. On the other hand, if you compose them in this way, this would be the projector on the state of psi. So this is very intuitive language to reason with, and that's the thing physicists use. So basically, my claim is that this two-dimensional notation of symmetric mineral category is actually this Dirac notation, that's how this thing is called, Dirac notation in two dimensions. So, mathematical physicists usually hate Dirac notation because it's not sort of, it never was really formalized, it was an intuitive notation, and even Dirac himself admitted that it was how we felt that the mathematics was, but it was more like, in the same way as Penrose used these pictures.

25:00 So it turns out that symmetric monochromes are actually the semantics of direct notation. A little bit more, by the way. It's in two dimensions. And the reason why mathematical physicists don't like direct notation is because there are some seeming contradictions, or things which are undefined, and it just arrives from the fact that you've got a two-dimensional situation, which is the tensor and composition, which you try to represent in one dimension. You feel that at some point this will lead to contradictions when it's not clear anymore whether a composition is actually a tensor or a composition is actually after the other. So you can find examples there where it's not really consistent anymore. So anyway, so this is the graphical language. I think of the physical process as a... I'm going to represent it by a box. So this would be like a morphism from A to B. I represent like a box. So you think of the box as the machine which is doing the operation. be a line. This is the composition of two operations, doing G after F. This is doing F besides G. So on system B I will do F. On system C I will do G. And here is some sort of combinations of these things. And like, for example, this is bifinctoriality in this language. It just means you can slide these things beside each other over the lines. And this would be the symmetry. But you can have crossings and things just slide along the wires. It's very intuitive. So what about the tensor unit? we represent by nothing, because we don't say where it's coming from, anyway. So from nothing to something, so this will be a state. This will be the dual of a state, from something to nothing, like destroying a system, or just throwing it away. And if I combine these two, I get something without an input and without an output. So that will be, I will think about this as a number, a probability. So this will be something like a probabilistic weight in my theory. Now, this is the rack catch. If I close this triangle and turn it around, I get my triangle. Take this, close it, I get the other one, and if I take the two of them, I get my numbers. So this is just what I've been saying. This is something physicists I've been using for 75 years. It's a bit expanded now. So now the additional dimension is a tensor dimension,

27:30 but it's an expansion of Dirac notation. So there is nothing new for them here. I can just tell this to them and this is something they know already. Okay, so one of the essential things is because I've got two vectors, If I've got two vectors, phi and psi of the same type, then you want to compute in a product. It's very important, you want the mechanics to compute probabilities, so I need to have a way to turn, to turn something of this type into something of this type. Of this type, yes, and then I can do this. and this is something from the endomorphism unit, from the tensile unit to the tensile unit, so this will be my numbers. In the case of, say, complex numbers, if you look at all the linear maps from the complex numbers to the complex numbers, they are, of course, isomorphic to the complex numbers themselves, because if I take the image of one here, and this will be, say, say, a number, complex number C, and the whole linear map is determined by the image of 1. Just looking at all the linear maps from complex numbers to complex numbers is the same thing as looking at the complex numbers. So this is a general, so in the case of Huber's space, this will give me a general model. So I'm assuming, in addition to this metric monomial category, I'm assuming I've got this operation which sort of flips the type. suggestively by the adjoint, because in linear algebra this will be the adjoint. So okay, so this is my language, so I've got, this is my background structure which is called, which I call a symmetric monoidal dagger category, or a dagger, actually better, a dagger symmetric monoidal category. And dagger is just an involution on your category. You see graphically, I take a box, so the reason I do this is just to introduce asymmetry in my box so that you can see when I flip it. That's the only reason I put this thing here. So you see, if I flip I get this, I flip it again, I get this one. So it's an involution by its graphical representation. It's an involution by its representation. Okay, so here is the next level. So we want to say something about quantum mechanics which is non-trivial. We want to say something non-trivial about quantum mechanics, about things people have been recently discovering in the literature, but in a much simpler way.

30:00 And for that reason, so this is basically where we started this program. It's in a paper in the conference, Logic and Computer Science, which was called the Categorical Semantics of Quantum Protocols. And the sort of mathematical structures of which it's going to show up are compact loss categories, which trace back essentially to Kelly's work in the late 60s even, I think. Yeah. So, okay. So, this is what I call a quantum structure. So, I'm going to show this. Let's do it like this first. So, this is what I call a quantum structure. Quantum structure is a special state, so a straight state or a Kethos triangle, so a quantum structure I call a special state, which involves two systems, which you think of as one at one side of the lake of Geneva, the other at the other side of the lake of Geneva, possibly, and it satisfies special axiom. so this is this is this one upside down so this is the edge one yes and I say if I make this configuration then it should be the identity so formally this will look like this so what I call a quantum structure is like a system with a certain name together with a special process from nothing so because it's a state to a pair of systems I will refer to this as a bell state so this is what this is this special state which in the example of naturality I got which was as a pure tensor like this so the typical bell state which you find in the literature would be zero zero plus one one and then they typically do one over the square and two to normalize it so that's what i'm trying to axiomatize but really more general fashion and this is this diagram here people who know compact close categories will immediately recognize this this is this one so So you can see, I'm starting, let's just go through this, so here, I start with system A, so I start with system A, I join a tensile unit, then I introduce this belt state, eta, here, you can see. this is happening here so I'm going to use this yeah so then associativity I use associativity here associativity happens here I associate from here to there and then you see the next thing which I do is I apply the edge joint of this etha map tensor P1 this is happening here

32:30 So this is the same thing in a formal way. Now, what is really nice about this, you will see you can do an enormous amount of stuff with this action. This is amazing. So some people might be a little bit confused because they would expect here to be a star. I'll tell later why there is no star there. So I'll talk about this at a later stage. So for those, so there is a reason there is no star there for the people who know compact loss categorism. I'll have to claim that later why. So, I do the following now. I want to find a more intuitive reading of this action, which looks a bit weird, a priori. And the more intuitive reading I'm going to get is by putting a little bit of wire in here. So you see there's a bit of wire in here and a bit of wire. And now I'm just going to forget about these triangles. And what you see now is just that I've got a wire which is bent on which I stretch. So it is like a very intuitive graphical reading of this action, which immediately brings you to, like, when you've been reading about mock theory and these sort of things, you see that it's becoming in the vein of these sort of mathematical theories. OK, what can we do with this thing? So this is our language. Note that as an axiom, this is a local axiom. This is not an action which involves any other object in the category. It's not that I say, for all other objects, be blah, blah, blah, blah, blah. So this only involves the object to which I assign the structure. So it's really assigning more structures to an object. It's really not about properties here. It's just really a local structure which I assign to an object. And it's really important to stick to the sort of simplicity of the whole formalism. So, it's a bit amazing, because these compact-close categories, how do you get them? So, take a symmetric monoidal category, then you assume that it's closed. So, you've got an exponential, so you introduce some notion of exponential, which, defining this thing, of course, involves all objects of your category, because it's an adjunction. Then, at the next stage, you maybe want to have an even stronger category, Symmetric, Monoidal, Close, Category. The next stage I'm going to ask to have a star from this category. I'm going to start on this category. And basically this means that I can be right so that this guy, this implication, arrives in the following form.

35:00 So I can write it... I'm going to start here, pencil in the middle, and I'm going to start here. So, if I've got an involutive functor, a star, so that I can write this thing like that, then I've got a star autonomous category. In a star autonomous category, due to the star, you've got some sort of more than due to your tenses, which is the path. What star stands for then? Star is an involutive functor. Oh, okay. It's not that important what the exact definitions are I'm putting down here. So, first I join this closeness, which is a global operation, which involves my whole category. like some sort of negation logically it's like this is linear logic this is linear logical negation because because so i can do more than due to my tensor from my star and then what is a compact close category a compact close category is a logically insane thing where your conjunction equal equals your disjunction now the magic which happens is the magic which happens there is that this very involved definition is very important definition which involved all my objects of my and all that, now reduces to this very local definition, very simple local definition. So it's suddenly sort of... Now, there is a definition of Saratomas category which is a little bit like that, which is in terms of what is called weakly distributive categories, or linearly distributive, I think they're told now. But that's not that important. So it's just such a beautiful definition for something logically, from the logical perspective, well, from categorical logic perspective requires a lot of work to get there. But in itself, it's a very simple and beautiful structure. So, and that's why I also made this remark about the star. Typically, the star would still be around, but I will assume that A is equal to A star. So a little bit more degeneracy. Now, what can we do with this structure? I must say, the structure... So, I started by assuming that there was this adjoint, which gives me actually a little bit more than a compact loss category. So what I'm going to do now is more than what you can do just in a compact loss category. So I'll probably allude to that. So as part of the structure, we've got this little bit of thing. So I can define the following thing. For each morphism F, I define a morphism F star as follows. I can do this. and I can do the following two given each f so now this is upper star, this is lower star

37:30 I can define the following one so instead I take the adjoint and then I construct this thing so these are two things I can just construct given the language of my theory and then it turns out that I can start proving stuff so given one we already knew we had an adjoint which we represented like that and I'll define two new things which I'm going to represent like this going to represent that one by taking my original morphism and flipping it like that. And then you can actually prove that the other one will be obtained by applying the adjoint and then applying this operation. It's easy to see that because how does it go? So suppose I start with F, then I construct this one, which is this thing, yeah? And if now I flip this whole thing upside down, which is applying the adjoint, then I exactly get this one, yeah? You have to remember it's symmetric, so I can just take this y from one side and do it. So there's a whole bunch of rules between all these things, and basically they're all captured by this graphical representation. So I think this graphical representation is due to Peter Selinger in some paper. So it's very compact and it's very easy to calculate it. Now just to say, this guy you don't have in a compact loss category. This guy really arises from having this adjoint. So in a compact loss category you only have this level. So now you've got something extra and it's essential for what we're going to do. So one of the rules which you can, which is obvious from this operation and this operation, and you apply, they will make up the adjoint together, yeah? Because from this one, the upper star is like going along the diagonal, and this one is going like this. So if I first go along the diagonal, and then I do this flip, I end up in the adjoint. Same two. Now what would this mean? If you look at the category of Hilber spaces, well this guy will be the transposed and this guy will be the complex conjugate. So the two ingredients of your matrix calculus being transposition and complex conjugation, they are actually captured by this language. So the very important part there is the conjugate. The conjugate means that there is some

40:00 involution on your underlying field over which you build your matrix calculus, like complex numbers of a non-trivial involution. So it sort of shows up at this level in this language, which is really crucial. OK, so I'm going to prove a little lemma now. So this thing has been very smartly chosen, I say, this sort of special geometry. So you see, going from this to this is like a 180 degrees rotation, yes? OK, let's start with this. So the thing in the middle, the yellow be this f star so which i represent by 180 degree rotations of the original one so now i'm adjoining this to this and adjoining this this there so i'm doing this here and there i can use my action i can stretch this wire or i can stretch this wire yes so i represent this now so if i stretch this wire i get this if i stretch this wire i get this and what you see now is this is so this is my is that i can take this box and it's rotate as if i just follow the I am here, I am there. So this is just all in the pure... So basically what I show here is, instead of just sliding like this, you can sort of slide along the bands. This just follows from the structure. Okay, so this is a theorem. If you would do... You can prove this in linear algebra if you want to. You can prove this in linear algebra, but it's going to take you a week. So, it looks a bit stupid, like, is it of any... So, okay, the idea is just to take everything out and put it to the end. Now, I'm just wondering, so, why is this helpful in any way for quantum mechanics? Well, one of the very essential things in quantum theory is that you've got projectors. And projectors typically are on the form, like this, like it's a Kexibra. and you see inside here you got something and then something upside down so these are actually I sort of chose them very carefully they are projectors so this is this is what how you can think of a projector this is projected on two systems because this is like in my triangle location it's like this and this would be then just the state on which you project yeah so so how does this work I've got some input here these together make up a number and then this is what

42:30 So this is how you act with a projector. So what is nice about this representation of bipartite state, and it's here where you will see, so I said we start with the method in all closed categories. Here you will see that the closeness sort of shows up in a very nice way. If I've got any operation, then I can turn this into one of these, just by joining this. so you see this is just internalization so it's here and the closeness is sitting and I cannot of course go back how do I go back I do this one and then I stretch so I've got a bijective correspondence between physically operations and states on two systems and then there is this one too of course we have my head joint so there is some closeness in here I'm going to skip this Well, maybe I shouldn't say it, yeah. So just let me say a little bit of history here. So in 1932, van Neumann formalized quantum mechanics in this book. So this was the first time that the formalism was actually publicized as a coherent mathematical theory. So then three years later, he wrote in a letter to Birkhoff, like, I would like to make a confession, which may seem a moral I do not believe absolutely in Hilberspeits no more, which is amazing. and now we're still stuck with this this thing. So this is also folklore that then Birkhoff von Neumann went on to do this thing which is called quantum logic. So that's basically my, that's where I, this is my background, this is like the area I did my PhD in and stuff like that, this quantum logic stuff. So several of these programs merged and this is like, this is like the major, the paradigm of quantum logic. Like quantum logic stands to classical logic, as no deduction stands to deduction. So it's very funny, you characterize some new theory like quantum mechanics by saying what it does not do, or what it fails to do. So it's a very weird starting point to study something, saying to somebody who is not. So just to go back to the motivation which I said at the beginning, The sort of mechanisms which we have here, the sort of mechanisms which we have here, is obviously this is a deductive system, and this wire stretching is logical deduction, because it's like unfolding closely.

45:00 So you can implement this thing on a machine, and we have some people who are implementing these sort of reasonings on machines. So this is like genuine logic, but it's a very different kind of logic than you would imagine, say, the internal logic of a turpos, which would be your classical logic. So here you're dealing with a very different, weird sort of genuine reasoning. So we're really trying to look what would be the analog for quantum theory, as, say, natural deduction does for true tables. So this is all encoded in this sort of geometry of the statue. We're going to see a lot more sophisticated mechanisms later, once we have an interaction between quantum and classical, but for now, it's just the main. So, okay, so here is the deal. This looks all very silly, but here, this suddenly becomes a non-trivial result. You have to look at the picture. So, assume you do an IQ test with some bright, smart prodigies, and ask them, what's inside given that this is true. This is a hard problem, I think. It's not obvious because you have to just check where all these things ended up, like green, green. It's very hard to see any logic there now. Just started this picture. It's very hard to see any logic there. So, it's not that obvious altogether. So, okay, let's prove something now which is physically relevant. So, this is obviously, so I just take this one and I slide it there. And I'm going to choose this guy such that it's inverses its adjoint. In human spaces, it would mean you've got a unitary map, which preserves the angle. So, you can assume that the inverse of this is the adjoint, so it vanishes. Now I introduce my triangles again. I introduce some names. And what you see now is called quantum teleportation. It's like a protocol which was discovered somewhere in the early 90s, the birth of the quantum mechanical form. This simple thing, which is conceptually like a very big deal. You see, you've got Alice and Bob, they share one of these special states. This is left side, this is one side of the lake of Geneva, this is the other side. So Alice does something here, some operation, and then Bob does some operation there. And as a result, whatever, whatever every unknown state here would have been sitting here, will end up in the ends of Bob. So you get this flow of information from Alice to Bob along the Lake of Geneva.

47:30 So, this contemplator took 60 years to be discovered, which is, of course, conceptually a big deal. It's not like you have to think about this physically, it is a big, big, big deal. So just to go about the details, now this thing is part of a measurement. Actually, you can think of this as a projector sitting here, and projectors occur in quantum mechanics when you do a measurement. So you've got some cell of a joint operator, which represents an observable. You diagonalize it, you get a spectrum of projectors. During a measurement, one of these projectors happens with a certain probability. So this is what would happen. So you count, this is not a deterministic process. So that's why there is a variable here, f. will happen. So Alice has then to communicate to Bob which one happened, what was the outcomes of the measurement, so that it can do some correction there, which is correlated with what happened here. Now, the quantitative aspect of this is like, this would be some state, say it's described in two-dimensional hyperspace, so which means it's continuous data of two dimensions. Typically, this would be like a four-outcome measurement, four outcomes, So, two bits. So, you send continuous data by communicating two bits. So, that's the sort of magic of the thing. You have to communicate something. This is like the very weird thing about quantum mechanics. There's no way you can really send information faster than light, because if you don't send these two bits, no information is sent. But once you send these two bits, you actually send the continuum of information. Where are these two bits here? So the two bits are in correlation. So there are a number of f-books. So this is a non-deterministic process. This is a measurement. So this would typically be a four-dimensional. So say this is a two-dimensional Hilbert space. The two together is a four-dimensional Hilbert space. So an operator would have typically four eigenvalues, four distinct eigenvalues. Those are the four outcomes. And these are your two classical bits. And so you have to say what your outcome is from here to there, and then here we'll do this operation. So this here, this is the closeness, which is in operation here, because here I've, well, it's the co-closeness, it's co-closeness. Here I've got some operation, and here this is internalized as a co-state, so to say, as a co-element. and this realizes this flow of information so while I'm not going to go

50:00 so this is more involved one I don't know whether yeah let's let's sort just just this is not just to say that these things like I said before these things have been done like in by several groups of people so these things are realized Like I said, one side leg of Geneva, the other side leg of Geneva. So this is another one which has been realized. So again, you see a simple picture here. This just goes there, and then you stretch some wires. And I introduce Alice and Bob. And this is, for example, cold entangled and sloping. So it's very much the same spirit of things. So I've got this thing and this thing and this thing and this thing. This thing is initially entangled. So basically then you perform a measurement on the middle part and as a result the two endpoints suddenly this system and this system become entangled. So basically, I'm going to start with this, then I apply a measurement here, as a result, this connection breaks, this connection breaks, but miraculously, now there's a connection here and a connection there. So this is another one of these things, and you see it's just trivial from this way of reasoning. Now, so what is completely missing in this picture is that I have to explicitly explain the correlation between those two things, which is what's happening classically, it's like this classical two bits you're sending, and it would be nice if this would become part of the formalism. So that's what I'm going to do now. The next step, which I'm going to do, is like, can we make this implicit correlation explicit really as part of the formalism? So I need to introduce the right categorical semantic, and also I have to figure out a way to sort of do this graphically, so stick within my graphical realm, because otherwise I can't sell it anymore to the physicist. Okay, so the first thing which I called quantum structure, so obviously adding this line will be called classical structure. So while the previous sort of story very much traced back to paper to work by Kelly and La Plaza,

52:30 basically the only thing we formally added was this adjoint, which was necessary, but that's the only thing we had to add. so now the world was very much straight back to like Taper by Carboy and Walters in 1986 which I think deserves a lot more attention than it has been having so far so what they basically did is let's now do the category theory of the category of relations instead of the category theory of the category of sets and that's a completely new chapter people have not been giving a lot of attention to I would say which extremely beautiful fits in in this story so basically quantum data cannot be cloned nor deleted, I told this already and I showed this in open spaces and classical data can of course be cloned and deleted that's the Cartesian structure so I'm going to treat this as a feature and the other one as a non-feature so our previous theory, actually for the previous theory which I showed, you can prove that there is no cloning theorem so in general abstract in the general abstract setting if you assume that there is some morphs category, somewhere which behaves as a cloning operation, so the assumption you ask is that it's co-associative and co-symmetric, then your category completely collapses. The category completely collapses. This is the result due to Abramsky. So you cannot have, so you've got no cloning theorem in the sort of categories I was showing. So now I want to say, now I want to be able to specify that there is something classical, some classical type somewhere, and And the way I will characterize this is by equipping it with the ability to clone and delete. So, this will be, now, an object together with a cloning operation from X to X and to X and a deleting operation. So, these are the pictures. So, we've been figuring out what sort of actions... So, we were working with these protocols, figuring out what sort of actions do we have to assume on these things, such that we can actually abstractly prove or diagrammatically prove protocols which involve classical quantum interaction. And it turned out that this is what we needed. So we needed a commutative commonoid together with these two actions. This is now you typically refer to as Frobenius structure, and I'll explain later what this is. So, okay, let's see what I mean. Let's see what I mean. So this is the commutative commonoid structure.

55:00 Everybody here knows what an internal commonoid is? Yes, no, I can't repeat, yeah? So an internal monoid in a category, like, think of, so you think of a monoid. So what you've got is a multiplication and a unit. So read now the picture from top to bottom. So this would be a multiplication. I've got input A, I've got input B, and you multiply them. And you get some number A times B. And this little triangle I think of as the unit. So what do you have? I multiply it with the unit, I get the number itself. This is the associativity. And this is symmetry. So if I put now the picture upside down, I've got a co-monoid. And it's sort of... The thing I'm trying to axiomatise here is a copying relation. So, of course, if I take an input and I make two copies, I can swap them at the end because they're supposed to be the same things because they're both copies of the first one. And if I want to produce three copies, it doesn't matter whether I follow this procedure or this procedure, get three same things and if I delete one of my copies then I just get my original one. So they are very obvious actions in what I'm trying to set up. Now these are the two other ones. So this thing is called Frobenius law and this, if you read this paper of Carboni and Walter's Cartesian bicategories, this is like what they identified correctly in fact as the essence of relational calculus. What you do with, like, the category of, like, you can say categories of sets and functions, axiomatization is a topos. Like, this is what they identified as the main property of the category of relations, which is a completely different beast. It has nothing to do with, say, Cartesian structure. Now, check out this work. So, you think of this as a copying relation, from 0 to 0, 0, from 1 to 1, 1. So, this is diagonal in the world of relations. In the world of relations, you've got converses, which is like the adjoined in my form. So this would relate 0 0 to 0, 1 1 to 1, 0 1 to nothing. Relates to nothing. See, it's very essential to relate to nothing, that you're working not in a world of functions here because it's so okay so zero zero zero zero zero zero zero zero zero zero zero zero zero zero okay zero one zero one one nothing zero one nothing so it's true it holds for relations a very weird identity holds for relation so so we will get an interpretation of this later at the moment it's just a piece of magic which which happens to hold and this this is very

57:30 this is some sort of normalization. So, in the physical term, my physical interpretation to this would be a unit term. Basically, you ask for the thing to be an isometry. I want my copying operation to be an isometry, which means preserves the length, preserves probabilities. So, this is not that essential. Okay, now, this is the great trick. You see this one? So I just multiply this with an input, and I put one there. Now, so this, by the unit law, is the identity. This is the identity, yes? So this is the identity, and bang. We find our original law, bang, which we start with at the beginning, when we were sort of axiomatizing what I call quantum structures. Basically, what's happening here is, it seems that what I call classical structure is refinement of my quantum structure. So, because if I've got this copying operation and the adjoint of the deleting operation and I've plugged them together like this, I've got something which exactly behaves like my bell state. What I call the bell state. So, how do you have to think of this? How do you have to think of it? This is very different to how people usually think about quantization and things like that. This is you take a quantum world and you introduce a classical context, which could be an observable, like physical space. You've got your quantum systems which are interacting, and suddenly we're trying to look at them from our physical space. And this is how you do it, is you take your big triangle and you chop it up in two pieces. Formally, I'll show this explicitly, this means introducing a base in the category of space. So the object which comes instead of with the big triangle, with these two little pieces, is a hidden space together with a base. From this base I can always construct this thing. I'll do this explicitly later. So what we do is, we take a quantum world and we classicise it by adding a classical context. So it's not that we are just looking at classical data. No, we are looking at classical data adjoined to the quantum world. That's the formal interpretation. You refine your quantum mechanical structure to something which also enables you to talk about classical things. And you will see it works. It just works in very simple terms. So this is how it looks in Hilke space. So we have this Bell state, which I said is just a summation of, like, equal pairs.

1:00:00 And how do you refine it? Well, you take this vector, which assigns 1 to the sum of a base, and then this just duplicates the base vectors. So this is how I formally do this in Hilke space. And what is special about this is that if I look at this equation, So this equation is basically... So if I look at those vectors, so this guy is this thing. If I look at all my vectors in my open space, which satisfies this equation, which means the input will be mapped on the pair, and the only ones which satisfy these are actually these base vectors themselves. So specifying this operation is the same as specifying this base. So this diagonal... So it's a very different thing. The diagonal is not the natural operation. of space? Of course not. But it picks out a base for you. So instead of looking at the unnaturality as a bug, you look at it as a feature, which specifies a base for you, which gives you a base. So specification of this copying operation gives you a base, gives you the classical world. That's the physics of it.