Symmetric Bimonoidal Categories & Quantum Systems — Part 2

0:00 So, delta H, and what you get is some I, alpha I, U, D. So, the only, you can check the only states for which this is not an genuinely entangled state, but can be written in some way as a tensor of two other things. The only for which this is true are actually these base vectors. So this is connected to this no-cloning argument which I gave in the beginning, where you saw that I made a sum of two base vectors and I got like this, that naturality was broken. So the only ones for which this diagonal behaves natural are the base vectors. So it's a very funny thing about this Carboni and Walder tables. It's a nice sort of observation about, I would say, the philosophy of category theory, find this sort of structure when they want to axiomatise a category of relations, they put in some uniqueness assumption to make it sound as if it was a property to have this structure. They really wanted to make it sound as if it was a property because it was in the spirit of a time, category theory should be about properties, not about structures. So they never use this anywhere in the paper or in anything where they do, this uniqueness assumption. And of course in the cases I'm interested in, I definitely don't want uniqueness, basis, which don't mess well together. I want incomparable observables, like typical things. So I definitely don't want it. So I actually asked a couple of months ago, Bob Walters, why they did it, and they said, well, it was just the spirit of the time. But it was wrong. We need to do it. We shouldn't have done it. It was just, people wouldn't have liked it, there wouldn't have been a uniqueness assumption or something like that, making it sound as a proper thing. But he agreed that it's wrong thinking. You won't think of it as a structure, this classical structure, like a refinement of your content. Would you say that when you introduce the cloning variation, you need doing that you introduce also the states which are able to become? To clone, yes. They will give you this. They will give you this. And this actually, this idea this idea, so maybe I shouldn't disclose this one So, assume that some state is clonable in that way. I will say that the set of states is orthonormal if, whenever I have done the following, that their even products are either 0 or 1.

2:30 It means it's the state multiplied with itself, or the state multiplied with an orthogonal. So this is a perfectly good definition of orthonormal. And you see, just by this isometric law, so okay, this top one should be different. I wrote it to be the same symbol, but this is a different one than this one. I think I have stated psi and phi, so I introduce this thing in the middle. also, by cloning I get two copies, so a thing is always equal to its square. So you get an orthonormal base, it is nothing, you basically have to assume nothing for that. So it's sort of very natural in the abstract that you get a base popping out of this sort of structure. Okay, now, so, and there is a result, I should mention, there is a result, there is a result which we recently proved, and it was actually very hard, surprisingly hard to get, that these things which I define as classical structures, if you look in the category of global spaces, they are exactly bases. We knew that each base was one of them. So the thing is, I just showed that you get an orthonormal set, but nobody says that it's complete, that the number of vectors would be the same as the dimension of your global space. So we are trying to figure that out, and it turns turns out, and you can show it, and we had to go to Gelf and Neymark and a bunch of complicated stuff. And we really needed each axiom of theory to be able to prove it, like all the axioms of a commonwealth structure, the Faubinus law, the existence of the Decker, we needed everything, just to see... No, this is just for a category of Hilverspace. Just to say that any structure in Hilverspace of that kind corresponds to a base. There are no exceptions. No very weird, staggered for being as common ones. Like if you, for example, drop, if you drop, we have counter-examples, for example, when you drop co-associativity, we have counter-examples. If you drop co-symmetry, we have counter-examples. So you really need everything, it's very tight. It took us half a year, basically, to get to approve, because I expected it to be simple, and it was not all simple. So, now, so, okay, so this conception is all very nice, of course, but the big attraction of this first common structure was this wire pulling, which was so simple to reason about,

5:00 and now I've introduced, I think, about five, six axioms instead of one. So you would imagine the beauty is lost. Okay, this is not, this turns out not to be the case, and in the same In the same way as in my big triangle I introduced one wire, now I introduce this sort of structure. I'm sort of going to open the box again and see what's inside. And can we find the graphical reasoning principle about these things. So then I introduce probably the most horrible mutational convention you've ever seen in your life. So whenever we'll encounter something of this form, we will rewrite it as this sort of spider. Whenever you see a spider, it actually means this. And then you can show the following theorem, that whenever you take any sort of composition, connected composition, of these copying operations, these deleting operations, and the red joints, and the natural isomorphism of this metagonal structure, it will always normalize in this way. Once you think about it, it's trivial. And conversely, all these actions which I assume are implied by this result. So the graphical reasoning here is completely trivial again. So you just, things just, if they are connected, connected is of course an essential assumption, things just depend on the number of inputs and the number of outputs. And, so, you find, so, if you've been looking at the book by Joachim Koch on Frobena's Osbrel and 2D topoesicocombic filter, So you find similar results there. The only thing which they don't have, the only thing which they don't have typically is this one. They don't use this one. So in our case here this means that if we have something of this form, we have a loop in the middle, we just stretch it away. Of course what they are thinking about are like these sort of manifolds. and so you connect with another one of course this is a hole in the middle so it's topologically very different from without a hole so they don't want something like that but you find similar results there for this sort of categories so yeah so basically this is just to show that all these actions are implied by this theorem so despite what we call spider normal form is the whole essence of what we call classical structure and just to summarize this sort of now is

7:30 so we start with this axiom then we refine these big triangles in these small triangles and trivially you just count the number of inputs this thing have to be equal it's just part of the the thing which has so it's really you refine the sort of wire pooling in the counting of inputs and outputs that's what it goes now to so we've got now okay how much time So, these are all nice concepts. So, to prove to, say, physicists that it's good of anything, you want to start doing actual new things, which they haven't done yet, and for which their tools or their formalism is just far too complicated to deal with. So, gradually, I want now to spend the rest of my time to get to actual results which would impress people who are doing quantum informatics now, quantum computing, that you can actually use this to, say, design new algorithms, stuff like that. So, so far, so we know what is quantum, we know it's classical, and these protocols will involve an important flow of information from classical to quantum. That's, that was my, I showed this in my teleportation protocol, so what do we need is an axiomatics of quantum measurement. Quantum measurement means I've got a quantum system and I want to extract classical data from it. Now, as you probably all know, in classical physics this just means looking at it and you know it, and quantum physics is a whole different piece. If you want to get to know something about a system, then you will disturb the system in some way or another, so it will change its state to some other state, and the sort of outcome you get only reflects to which state it went, not which state it was. there are some probabilistic correlations, but it's like a very involved process. There's no way you can just stare at it and know what it is. So basically now I want to define a measurement and I'm going to try to follow some operational path, like what is the, since we can't just observe what are reasonable assumptions, we can ask for a measurement. What are reasonable assumptions? So, okay, this will be my measurement, from quantum system input to quantum system plus classical data. And I claim that this is a reasonable assumption, that this is something I really want. If anyone, if there is anybody here who knows the actions of quantum mechanics by head, like you would open, it's typically five actions or something like that, does anybody recognize this?

10:00 So, I do a measurement, and then I copy the outcome. I do a measurement, I do a measurement again, and it turns out that my two outcomes are going to be the same as the one I got in the first one. It's called the normal projection posterior. So basically how do you do measurements in linear algebra? You have yourself a joint operator, you're observable, you're diagonalised, you look for your projectors, so they give you, the projectors give you the probabilistic probabilities of possible outcomes, and then you say, if you're immediate, then you go, what happens during the measurement is you go into an eigenstate, which means you have projector on one from one of these projectives since they're all orthogonal the probability to all the other ones become zero so if you measure again you get the same value that's what this says in a very short this is a general statement which says exactly the same thing without in all in absence of say linear output so this is normal projection to stress and do you prove that no no i'm defining i'm defining i'm i'm i will be proving something for some representation result. So at the moment I'm just saying, let's, let's, so what it actually shows is that there is at least some connection between whatever comes out here and this. So this basically is an assumption of the fact that the outcome here is in some way related to the outcome state of my measurement. Yeah? Because if I measure again, of this thing, I will get the same outcome. So it sort of asserts that there is... I'm doing this measurement, at least I know now that there is some connection between the outcome I got and what the system ended up in. So this is what it asserts for me. Now, when I'm projecting a constellation, okay, this doesn't matter. And I'm also going to assume this one. Now, I'm not going to explain what it means yet. This I will do later. But I don't know whether you recognize these two things together, this one or this one. Basically what I tell you is that you've got an Eilenberg-Morkel algebra for this co-monet. That's why I look here. See, this is my X tensor, this is my co-monet.

12:30 One, two, three. This is the other one. I'm specifying here that I've done an island where it's more close to the graph for the co-monet which just involves tensing with my classical structure. Just have this picture in mind, so what you take is you take a picture, I flip it down, and then to make the match I have to rewire this. This is some sort of self-adjointness condition and the result is that the self-adjoint Allen-Berrick-Moore co-algebras for this co-monet in the case of Hilbert spaces, are exactly your quantum measurements as you find in your quantum mechanics textbook. This is the very trivial proof of that fact. So basically, this Venoeum projection postulate, which is like your coals of a square, tells you that you've got a bunch of things which are identitots and mutually orthogonal. So you've got a set of mutually orthogonal identitots. And this triangle basically says that they all add up to the identity. So you get a full spectrum. And then this last condition basically asserts sort of the joiners of this thing which is an orthogonal project. So basically, there are two sides to this story. There was one part of the story which I didn't completely explain yet, which I'm going to be doing now. The fact that these measurements have clear operational significance. We want to make a reasonable definition here, like after measuring whatever we end up with is connected to the outcome. So this is the very reason, and we hope to recluse, and otherwise we have this nice categorical characterization, which is like that it's an Eilenberg-Morkel-Algebra or something. It's very conceptual, it's like adjoining, so you've got your category, you adjoin a classical context, you look at the being used Eilenberg-Morkel-Algebra, and then these are your measurements. You have to assume this additional thing. Actually, everything we do is like some sort of daggered. There is this dagger happening, so it's sort of an obvious assumption to ask this thing to be solved. So it's a very canonical notion, it turns out, quantum measurement. And in turn, it's all happening within my category now. It's not that I have to go out and start writing speck around a piece of paper. So, just to continue now the story, what does this mean? We know what this one means. We know what this one means and it turns out that this one also has very clear operational significance in the same way.

15:00 I'm going to write this side slide differently. So this is equivalent to this. So basically the bottom one is the same as this. So what this now asserts is like, you see I can exchange, so you think of this as having these cats and these bras sitting inside here, basically what I'm allowed to do is I'm allowed to flip this too. So I'm asserting that there is some connection with the thing which will identify the input and whatever it will spit out. It's again, what you could say, it's something we could hope for. That there is some connection to whatever comes in here and whatever goes out there. They are not completely unrelated. So that's basically what he asserts. So, well, now this one. This one really... So what does this mean conceptually? So, the first two were minimal requirements for the notion of measurement. This one very amazingly asserts that there is no faster than life communication. And this sounds very mysterious, but I'm going to show this now. So, you remember with the teleportation protocol that I set? You have no information whatsoever as long as you haven't set your two classical bits. There's no information in transfer whatsoever. And my claim is that this action sort of imposes this on your theory. So I have to prove this. Okay, so we go back to Alice and Bob. So you know there is this non-local state here. Like one side of Lake of Geneva, the other side of Lake of Geneva. And you know that the measurement, I wrote it down here, has like these internal projectors sitting. And so you would assume that maybe there is something in here and Alice can send this to Bob one way or another. Could use this mechanism to send information to Bob faster than life. This is something you might assume from as far as the story has been told so far.

17:30 She could use this mechanism to sort of perpetually send something to Bob. Now I'm going to assert that this is not possible. This is not possible. So how do I do this? So, formally I claim that this means the following. I claim that this exactly means the following. So, does anybody here know density matrices? You've got quantum mechanics, you've got pure state formalism and mixed state formalism. So, pure state formalism is what we've been talking about so far. Now, a mixed state you build as follows. You look at the projectors. You look at a bunch of them. Then you take a bunch of positive numbers and you make a big sum. So this is an operator. It's a trace. Typically you want this thing to be like 2F trace 1. And basically it's called the mixed state. You mix. This is the mixing. You mix a bunch of pure states into a mixed state. You don't know which mixed state you're in. Now, the maximally mixed state would be that I take all my base vectors and I just sum them up. And maybe I normalize if I want to. So this is called the maximally mixed state. Of course, if you look at it through, say, the closeness, then this is the identity, of course. So this is what I'm inserting here, the identity. So I took this one, then I doubled it up. It's basically the same thing as I do here, I take this one, I double it up, I make a big summation, which is actually this trace, this is my trace, and I say this is the maximum mixed state, this is the identity, basically asserting that God doesn't know anything. So, this is the process, and here we assert that God doesn't know anything. He hasn't been taught anything by all this stuff, he knows nothing. But if he looks at what is here, he just sees uniform probability distribution. He has no clue which theory he got. You can measure this and everything is equally probable. So, okay, let's now start computing with this. I'll go down. You see this? This is the same as this. Yeah, so we're looking at this now. So this I can do by my spider's theorem. I can just plug out the black part. Now, the top one, I flip back because I had these cell rejoinments or something. Now, the two after each other, I can use a normal projection postulate.

20:00 So, this is something you wouldn't think about when you're calculating with matrices. That this sort of assumption I had to impose categorically, this one just to get this nice representation theorem from it, that there was some assertion about, say, the nature of sending information. I must say that this is something people have been thinking about now for about 10 years, like reproducing quantum mechanics by information theoretic constraints. People have been getting the intuition that the structure of quantum mechanics is very much related to sort of limited abilities as an information theory. There are things you can do more and there are things you can't do. And this is, for example, a sort of instantiation of this thing. The problem with people doing this mostly is that they only have one model in which they work, which is Hilber's space. So it's very hard to make an axiomatic system if you're already working in the thing, you're trying to axiomatise. And this is your only model you're working with. So here category theory, of course, gives a really nice theatre to sort of try to analyze how things, say, are connected, how certain assumptions on information theory could, like, reproduce in some way. Here it's happening. You see this assumption on no information faster than light. It enables me to, like, reproduce the self-adjoint operators. So, there is a need to give you a better understanding about this. Like, this is what I said, actually, it means in Hilbert's space. that your projectors sum up to one. So it means in each measurement you've got a whole spectrum of possible things which can happen. And they can be very far apart. They're all mutually orthogonal. So it's an assertion of things are very non-deterministic. So it is an assertion of non-determinism. And it's exactly this non-determinism which sort of disallows you to send information faster than life, which forces you to also send this classical data to the teleportation protocol. So that's sort of the intuition about this. So here you showed, you proved the equivalence between low-person-like communication and this, the axiom of the right? Yes, in the presence of the two others. So in the presence of the two others, I used the two others too. So the thing is, if I already had, these two were reasonable assumptions of my measurements, where I assumed some connection between this and this and some connection between this and this. So this is how you can read these two. And then to get exactly the sort of measurements as you added, like exactly a representation theorem for your self-adjoint operator as the measurements of quantum mechanics, I needed to adjoin this one. And then I tried to read what it means.

22:30 And do you have any motivation? I mean, how would it read very clearly? Independently on his own. No clear, no clear. But in the presence of those two, I showed what it means. Well, this is all very new stuff. Like, I know some people are now thinking about these things, but this is the only thing I can say so far about this. It's all fairly new things, like from a month ago or a couple of months ago. So, okay, I should do this. So we got now a sort of very abstract, nice interpretation or analysis of whatever you want of measurement in a very abstract sort of scheme. So we got a good, so we have our notions of, and now we can start thinking about our teleportation protocol. So I want to have a black line here. So I have now my notion of classical data. So basically here will be a measurement, there will be some unitarity, so the way I'm going to fill this in is, so here there will be a certain box, which represents a family of these operations which I need to correct, which will now be labelled by my black line, by my classical data, and here, you remember how I've been setting up things, so this is how it was in my previous. this would be the projector in my previous teleportation protocol so now it will look like this because now it outputs some information and so you see this this this will represent this thing in my protocol this will represent this thing and this is the sort of this box i will sort of slide to the other side to correct because i did it before so the same mechanism works now so then think so i have to now assert i have to assert that this thing truly that this thing truly is a measurement, so I had my three actions before, and it turns out that you can just read that becomes this. This thing needs to be unitarity, the edge one should be equal to this inverse. It turns out that this is exactly this. And which is really nice is there is a paper which came out in 2001 by a guy called Werner, who said, what is the most general teleportation scheme we can ever do? And then he spent 25 pages of linear algebra to go up with these rules, which are right down here. And they are basically, they as I just write down without reason. They just pop out of the formula, exactly the same thing.

25:00 Of course, you need to be a bit smarter in the interpretation. For example, this is the map. Look here, you see, I've got unitary x, I've got a unitary labeled y, so this would be the unitary, this would be the unitary. One is upside down, so it's the edge one. x is the variable, this is the variable space, so these are these indices. Here I've got two variables and so it's an identity so right it is a direct delta and here is the trace this is the trace so it's you just read the linear algebra from it and then the other ones are similar okay so basically now we have our teleportation protocol included this is what it is don't these are just normalization scales and then the proof looks something like this which is just exactly the same. So here is another. So I think I should. So I'm just briefly going to, so this I'm not going to go in much detail. I just, these are very, very recent results and the reason I show this is because interesting new categorical structures again show up. So, so far, what have I been doing? So, we looked at the system, I asserted that there existed something like bell states, and start to reasoning that. And secondly, I say now assume that there is a classical context, which allows me to define classical information and all that. So, I asserted that as a factorization of my quantum structure, what I call quantum structure. Now quantum physics is interesting because there is more than one observable. It's like position and momentum, and they are incomparable. You cannot know the position perfectly, you cannot know and together momentum perfectly. That's like Heisenberg of certainty. So, so far, my theory allows for incompatible observables, and in fact, in the teleportation protocol, which I described implicitly, it actually asserts, but implicitly only, it asserts the existence of incompatible observables. So here, in this work, we have been trying to think, And we find the nice structure on, say, if you've got position and momentum. What are sort of nice categorical laws which those two observables together satisfy? And those observables are known as what is mutually unbiased. It means if you know position perfectly, you know nothing about momentum. Nothing. You've got complete ignorance about momentum. If you know momentum, you've got complete ignorance about position. So they're mutually unbiased.

27:30 They're very important, of course, in quantum mechanics to have these sort of things. Similarly, if you think about photons, x and y polarization. If you've got x polarized, you have no bias towards y polarization. This is another very important example of such two mutually unbiased observables. So wait, I'm going to skip this one. So here, this is an example of two mutually unbiased observables. So here, this is the one I've been talking about so far, and here is a different one. Like plus means that I take the sum of 0 and 1, and minus means I take the difference. So if I've got, so this base, 1, 0, and this would be plus, and this would be minus. And I can do exactly the same for this base. So I'm going to represent my initial base by green and I introduce new base red. And so what are the laws, other special natural categorical laws, or graphical laws, because that's what I'm asking, which relates these two things. So this is like a very clearly stated problem. And it turns out that it's satisfied following. Some people probably immediately recognize this. So this is almost what it's called, if you don't think of these two things, this is what it's called bio-jibang, which sort of appears when you're looking at off-jibang disorders things. So magically this turns out to be true for the situation I was just describing before. If you've got two mutually unbiased observables, they will satisfy this. The question is, is this useful for anything to know this? Can you actually compute with these things? yeah so what this is actually this this is actually a scalar for some weird reason you need so this thing is not equal to this thing there is some sort of scalar appearing which is actually the inner product between the green and the red so I'll just represent this like this because it makes it easier to read otherwise it would be some more balls here and sort of blur the picture so this is just a new product and for the rest of the talk I'm just going to ignore the So there is a quantitative, an essential quantitative difference between this and, say, a biosphere. Okay, let's compute with these things. So first of all, this is something you can prove from these two rules. How do you do this? It's a fair list.

30:00 So look again at this, the top one. So this one is very easy to remember. It means red copies green. We've got green. Red copy is green, and of course, green copy is red. This is sort of a more, they've got these two green ones that can sort of move through the red ones. So that's what this one is. So that's the way to remember them. So okay, I've got this side. So you see, I take this little thing and sort of adjoin it. This is my spider stealing, you see, now I've just got a piece of wire, I can adjoin something like this. And I can adjoin this. So now you see the shape of this action shows up here. I apply it, green copy is red, red copy is green, so I will get one green, one red, and then I need a product of the two, which I'm not going to bother right in. So I just proved this from the two three resections. And now, this is the sort of stuff you find in, say, quantum computing textbooks. This is called the CNOT gate. Now, how do you read? This is very weird, because I take a vertical line here, but you can basically show, you can show from, from, just from the Faberian structure, that this thing would be equal to this thing, in this case. So I can just write it like this. It doesn't really matter. And now it, so now it looks very much like something you find in one quantum computing textbook, where they're reasoning about logic gates and universal computation. and this is a very important one which they call CNOT gate. Why CNOT? If your first one is zero, then it doesn't alter the second one. If your first one is one, then it alters the second one. So it's some sort of, you can write down a matrix for these things. And then you open some computing textbook and then typically they ask, after a couple of pages, to compute what this is. So you've got CNOT gate, CNOT gate, where you exchange the input and CNOT gate. So I swapped the bottom thing. Now you see this is by algebra law. This configuration is sitting here. So I can replace it by this. I use my spider's theorem to merge down, to merge together degrees and the green and the red. Ah, I just computed what this was. And we know now that the result of these three signal gates is the swap gate. It's like something they ask you to calculate with matrices. So these are four by four matrices. Okay, this was like just a warm-up, so to say.

32:30 So, next we introduce what is called a color changer. This color changer is also a very important gate in quantum computing, which is called Hadamard Gate, which is, of course, for other reasons, like Fourier transforms. It's sort of a discrete version And basically, in concrete, it's your change of basis between the two bases which I defined. So now I abstractly assume that there is some sort of color change going on. And then, given this color change, I can define another gate. So you see, it looks like this now. And it's again a very important one in quantum computing literature, which is called CZ gate. And then one of the most recent developments in quantum computing is that people have been looking at something which is called measurement based quantum computing. Measurement based quantum computing means that you will be using the dynamics of measurement to compute. So the traditional model of quantum computing was as follows. You take an input, a big input state, you do a unitary operation on it, and then you measure, and you hope that you get something. So this is a very new model which is favorable for many different reasons. And the most important reason is actually practical implementation. Practical implementation as an actual computer. And what you do is the following. You assume some very big entangled state which is called the cluster state. And this is how they typically prepare it. You take a bunch of things in a certain state and then you apply a CZ gate there, CZ gate there, CZ gate there, CZ gate there. That's how you produce a gigantic entangled state. and then you're going to just start measuring and this measuring everywhere is going to introduce dynamics like we've been seeing this a lot now that measurement changes states like in teleportation and this is your form of computing it's very hard to reason about this in in matricial terms very hard so we people know this is very promising from from say implementation level but it's just extremely hard to compute with with matrices because measurements is always going out of your formalism of quantum mechanics all the time. So you would like to have a comprehensive way of formalism, which allows you to reason about this sort of very non-traditional quantum mechanical settings, where the measurement has always been seen as a bug of quantum mechanics almost, things become weird. Here, this takes this measurement as a feature, as a dynamical

35:00 phenomenon which you can exploit to compute. So this is how typically they prepare these Then, a couple of years later, some paper came out by two guys showing that there is a very different way to produce it, which looks like this. And they spend, of course, like a couple of pages computing a lot more than a couple of pages. They actually need to get the same thing here as you get there. Yes. So, it feels trivial. Completely. And so what is the sort of thing you want to show in such a model? You want, for example, you want to show that this is a universal computation of primitive meaning. Any arbitrary unitary can be realized in this way, in this sort of measurement-based procedure. Okay? So now, of course, if I want to say something like that, I need to introduce something like a continuous variable in my language. otherwise there's no way I can assert a statement about... I can do everything in this language which you can ever want to compute. So I'm going to introduce an alpha into my red and green dots, and I'm going to choose them in a very particular way in my yellow squares. This is called phase gate. And then this is just the color change on my phase gate. And then you can again prove the following generalized Spiner theorem, that if I define my alphas like that, like this, and then I do the following. So I should take the green one. No, the green one was bad. So I've got one of my dots. Then I define this one to be either a red dot with an alpha here, or with an alpha here, They turn around to be all the same. They turn around to be all the same anyway, so I put the alpha in the middle, and then you can prove that you get a generalized pie theorem, like everything just collapses again into one dot, but now you have to sum up the values. So you've got some additional monoid structure within your dots, which behaves extremely well with respect to your language. But still, so the whole goal is we're still in this graphical realm. The only thing we now have to know is how to sum all of us. We have to compute sums. So there's some by-hand computation now. But, okay, so now we want to know how to realize an arbitrary unitary gate.

37:30 So here was my initial cluster state. I've got my inputs. I've got these jet-set gates which I apply. And I claim, I claim that in this way I can realize any arbitrary unitary gate. Okay, let's see. Let's see. These are the Euler angles. All the green, just all the green collapses together. These two H's, Euler angles. Euler angles represent your whole block sphere. So, the people who have been proved, so this particular scheme is in a paper by people you might know, Van Son Danos from here, and Elam Kachepi and Prakas Panagarden, they sort of have like a 30-40-page paper in the Journal of the ACM to establish this sort of way of realizing universal quantum computing. So this is how long it takes in this language. Okay, so I think I can... So let's just... So, because this was a question at the beginning of this... before my talk, for my answer. This is just a note for Chlor, that if you take any symmetric monoyal category and you look at a category of community of co-monoyants in it, there will always be a Cartesian category. So, of course, these classical structures, which I defined as a community, a little bit of additional structure. So, obviously, if you look at them, and then you look at the operations which behave well with respect to it, which are natural relative to it, then you get an easy category. So, the classical world pops up as a limit. Very awful. Now, the really nice thing is, there is a lot more which pops up. Namely, you've got the whole zoo, and this part of this is already in the Carbone and Walter's paper. Like, if you look at a category of real spaces, just with this language of delta and epsilon, this copying and deleting, I can define all the permutations, what a bi-stochastic map is, what a stochastic map is, what a total map is, which I just did, it was the theorem, a partial map. And actually, by classical and stochastic maps. So, classical probability theory shows up. I'm not going to do this in detail. And in fact, you can extract the category of relations. And this is something very weird, because to construct the category of relations, you have to define...

40:00 There is something very funny there. The composition of relations doesn't match with actually how you define it within these categories. So you have to define a new composition, given your language of your category, and then you exactly get your composition of relations, which is something very weird. Anyway, I don't really understand, so this is like something, like hard for categorically efficient markers, so to try to understand, there's something very weird about category of relations which I don't understand. So, anyway, so, just so you get basically classical probabilistic theory and all these sort of things popping out of this very simple language we started with. And that's sort of where we are now. That's it. Thank you. Inside your boat, the little boat, you showed us that there was one vertex. And the vertex, the boat, was the same as in space. Yeah. But inside the heart, the heart works. You didn't show another vertex. There are two kinds of vertices in the heart. So, I think you're alluding to the right, this and this, yeah? Yes. This was the first vertex. So initially I had like dots in which there arrived three things and then in which there arrived one thing. and something which I didn't say so this basically will mean this and then I define this this spiral representation where I thought when you make a classical measurement yeah make a measurement your class for the time yeah can you also give the vertex representation so everything which is happening here is it is happening within that language Not in general. When you throw the hard fork, you know this cappella. Yeah. So the vertex showed up in the implicit definition. So they are playing a role here. Yes, this is an algebraic text. When you erase classical information, you have a dot, a dot which makes it disappear. But inside, you have it.

42:30 Inside? Ah! Could you find it? Well, okay, yes, yes. So, this is defining arbitrary observables, which can be degenerated and all that. You get all sort of a journal. If you want to find a canonical one, you can basically just take this copying operation. It does the right job for you. This is an example. You see, like, this is co-associativity. If you think of this one as being the dot itself, this is co-associativity. What is co-associativity? This is, well, this is just trivial because... But this follows trivially by the spider's theorem, and this, of course, is also trivially. But this is another kind of vertex, because one of the outputs is classical. But you have to think of, but this is the quantum world and the classical world embedded in the quantum world. So, if I think of this as a measurement, then this will be in my category a linear operator. Now here, I've been slightly cheating. I've been slightly cheating. So this is something I wasn't, it's a lot to say in a very little time, so let's go through. In fact, my question is how many kinds of different vertices do you have? As in QED, in QED we have one vertex of interaction between electrons. Yeah, so the number of vertices like, okay, so here I had two families. labelled by designers. So here I have quite a number of them. Usually in my teleportation protocol I have one. Only one. In general, there are two kinds of things. So it depends, like in this case where I was calculating, these are the two. So this mutually unbiased basis has two and the relation which relates, and they are related by this biological structure. So it depends how much you want to do. So this is not really a reconstruction of quantum mechanics program which really wants to say this set of actions is exactly the same as working in other spaces, although it's an instance of something you could do. So you could dream of finding, and actually I think we're pretty close with the alphas and stuff to do we have an equation, a complete set of equations which allows you to prove everything you can prove in the other spaces? Of course, a hard moral theory question. Very hard question.

45:00 So actually, the guy with which I'm working with is actually looking to which extent we can say something about it. Not really solve it, but how close are we, or can we say that it's impossible, or at least something of that kind. So, I don't know, so I don't, so basically there's a lot of different motivations going on, like this thing with this color was about doing things which people in quantum computing actually care about now in which the paper topics on which the papers are now today written and doing it simpler than they do it and better than they do it so this is what this is about so there is of course a more philosophical side to this story like this operational reconstruction side so so i can't give an absolute answer to how many vertices have you. It depends what you want to use it for. At least two vertices. So for doing this sort of stuff in two vertices, if you want to really then start reasoning about universal computation and simulating any classical computation, you need to put by hand this alt pass in which I had some erratic monoid structure. Concrete erratic monoid. But if you keep it discreet? So you want to keep it, yes, for purposes of automation you want to stay as discreet as possible, of course. And for example, for this additive monoid structure it doesn't really matter how many alphas that I have. You know, I just say I have some alphas and I don't have to be too specific about whether they are continuum or not. same but you introduce the alpha yeah just to get a continuous to to yes to get something which like looks like a phase information it's like fundamentally you don't need it if you are approximating the continuous yeah so that's what i was just saying like the idea that there are some because i didn't specify too many rules for them the only rule which i had was the spider just summing them so you can just think there is a finite number of them and then they will by taking them arbitrary and you basically prove everything about the continuum too so you can still implement it it stays two different yeah yeah yeah so it stays discreet essentially but yeah that's the main the main thing it's it has still a discrete feel to it

47:30 although it's not really the actual in the actual model of quantum mechanics it wouldn't be but but it doesn't seem to matter like it does the actual structure of these alphas doesn't come in at all in this computer here just because there are some it's the only thing i need actually there's a you're a remark that the classic sizing structure amounts to choice of bases right yeah it's actually makes think about choice in stat theory and proposes There are philosophical questions there in physics, like, is there a very special observable in nature, like, is physical space a special one? And mathematical physicists will disagree on that, some will say, oh, the theory should be uniform, independent of which choice of base, you don't want a special base. Other people say, obviously, physical space is a very special one, because any experiment we do is basically a localization. If you measure momentum, you're actually doing a localization experiment still. Any sort of experiment boils down to localizing something somewhere, a dot on screen or something. So there are questions there too. So you spoke about the Compact Dagger categories for rel and hilt. And I was wondering if you have any other examples that you played with? So, so, so what, so, so what, there are the code word, these are categories which are also compact closed. So, so actually what I, my, my, my past couple of days here in Paris I was basically talking to people who have their own sort of axiomatization programs which have nothing to do with category theory, which have a feel of being good examples for other categories. And so there is one guy, Rob Speckens from Cambridge, and he has a It's a very discreet model, I think John Bies won't discuss the whole model of this in a category, if you might have read it. And it's very discreet and it reproduces a lot of phenomena of quantum mechanics, not all. So there I've been now to structure the category for it. It's a very pretty category which I don't understand myself. And that's a compact-close category which has Frobenius structure. It's a sub-category of the category of relations, it's a sub-category of the, so like, the category of relations has one canonical Frobenius object on each, Frobenius structure on each object, yes? Well, that's not what he did, but that's how I recast stuff. He restricts the objects, so he only takes the one-element set, the four-element set, the sixteen-element set, and so on.

50:00 And then he restricts the morphisms in some way, and then you get three Frobenius structures on each object, corresponding to, like I said, three X, Y, and Z polarizations of photons. For example, there are tons of categories of this kind which are byproducts. You take any involutive semi-ring, and then you build the matrix calculus around them, and then you get one of these kinds. Or categories of spans, which is almost the same thing. But you always get byproduct structure. And the interesting examples for me are those where you don't have byproducts. The byproducts are still too much of a vector space field, of course. And you want to go to models which are relatively different. still have to write the stuff down so there are there is actually i think there is there is something to be done in producing new models also michelle fiori is doing things with with distributors where he builds categories of this kind which are very very different in spirit than say the byproduct academy so but it's good to have a lot of models of this sort of thing But actually, if you just take, say, two categories in the sense that you have two compositions right there, is it how it relates to, say, monoidal tensor categories? Some things get, of course, a lot more complicated there. I'm sure there will be some use for these sort of things. The nice thing about the symmetric neural categories is that you don't have to tell a story or a justification there, because you just get them immediately on your plate. So that's, definitely, if you're thinking about like, so of course it's like proof theory, like here you've got these protocols, I'm just rewriting these things, of course it's are living one dimension higher. So if you want to make a rewrite theory of these pictures, if you want to make a theory about rewriting, then of course you have to go one dimension up in your category. So there's obviously use for these things. More questions?

52:30 You mean to use this for like classical data flow? Yeah, so the fact that you've got a commonal instruction. The fact that you have a collection, you've got a commonal. Because I've seen it in some books, so I don't know about it. So I know there is some work by Dusko who sort of... So you've got the lumbic stuff on categorical logic by assigning variables, which is very sort of similar in spirit, that's where these things show up. So in the Lombik-Scott, the way they assign variables and polynomials, and then build polynomials, it's very much in the same way, but it's all Cartesian there. So here it's Monois. I know there's one paper by Dushkovich called Names and Abstraction in Action Calculus. So that's very much, I think, names and abstraction in... I assume it's a structure in the quantum quantum system. Oh, sure, like quantum, in this sort of, and all these biomes and stuff, it shows up, like if you read Ross Street, book on quantum groups, all these mathematical structures are sitting there. I mean, besides it, the fact that you may have a common way to work in between this, that you have. Yeah, yeah. But you're aware of it? I mean, I'm just curious about connecting with the questions. So the operational connections I have no clue about. These things, I see. quantum groups is typically this domain of mathematical physics where people are not really sure what is the physical like topological quantum field theory what do they physically mean nobody really knows although these structures are very much the same thing so it's it's the operational meaning of these mathematical structures this is mathematical physics these are things people in foundation quantum mechanics until now never cared about because it didn't mean anything for People in foundations of quantum mechanics would have never cared about these things. These quantum groups and all that. But it's not at all clear what it would mean operationally. It's like a path of mathematical physics. I'll give you one. So...

55:00 What if a topological quantum filtering, can you predict anything with that? No. You can't. Okay, thank you again, Zanbob. Can we have your slides? Thank you.