 OK, everyone, we're going to do something really exciting now. I think this is always the most. So basically, so far, he signed a picture with teleportation that I had to represent the telephone by some curly little thingy. And that's, of course, not very elegant. We really want to have the whole story in pictures. The whole story has been also the classical communication. And for example, measurement-based quantum computing, there's a lot of classical computation going on. You want to get these only pictures, everything in pictures. And that's the goal now. So we have to come up with a paradigm on how we distinguish something that is classical from something that is quantum, like classical computation, like data versus quantum systems. And in a way, von Neumann already knew how to do that. Von Neumann already knew this in a way. So basically, what we're going to do, and this may seem ad hoc, like if I just draw a diagram like this, I will always think of it as classical. So what we've been drawing so far, these diagrams, I'm now going to think of all of them as classical. Of course, there can't be any phases, because that doesn't mean anything phases classically. And this will be automatic, as you will see. Now, if this is classical, what is quantum? Well, quantum simple is everything doubled. You just draw everything twice. That's quantum. So if you see single wires, it's classical. If you see double wires, it's quantum. And this is something you actually already know, but you didn't realize. So, oh yeah, I should say, if you double, what you have to do is then, if I've got an alpha phase here, then the other one should be the conjugate. In the doubled picture, everything should be conjugate. So you've got something, a picture, and it's conjugate. And that's quantum. And so you can write it like this. It's a bit more elegant, takes less space. Let's now see why this is the case, why double corresponds to quantum and single corresponds to classical. I mean, intuitively, if you think of density matrices, if you think going from pure states density matrices, what do you do? You take psi, psi ket and psi bra. So you double it. Now, taking psi ket and psi bra is actually the same as taking psi ket and ket conjugate. It's the same thing. And that's really what we're doing here. So we move from a ket to a ket and it's conjugate, just like you do in density matrices. Okay. And so, okay, here is a spider. Two inputs, one output. What is this? This is a measurement. This is a measurement. This is what philosophers have written 100,000 papers about in 100 years. Measurement problem. This is a measurement. It's just a spider. Two inputs, one output. What is this? One input, two outputs. Okay, this is encoding classical data as a quantum state. This is encoding classical data as a quantum state. So going from quantum to classical and back is actually very simple now. It's also just using spiders. So okay, so how would this look? Suppose I've got a density matrix here. This is a density matrix. So I've got all the diagonal P and one minus P and then I've got some off diagonal Z and maybe Z bar. If you apply this, you remember this is an operation which now you have to read this backwardly. It's take zero, zero in, spits zero out. It takes one, one in, spits one out, which means it only takes in the diagonal elements. It gets rid of the off diagonal elements. And so, you see some density matrix like this would actually turn into this probability distribution. So we've got a probability distribution here. So if you've got a density matrix, if you take this density matrix and you measure it with the Z observable, then this would indeed be the probabilities for your two possible outcomes. So it does what you expected to do. It does what you expected to do. The encoding, the encoding takes a probability distribution and just lays it off on the diagonal. So it's what you would expect it to do. So this is just plain quantum representation of this. Now we can do stuff again. Okay, so this is, I take classical data. I encode this as a quantum state and then I measure it in the same basis. It's not a very useful thing to do because what comes in goes out. That is a lot of work for nothing. So this is, so again, encoding, measuring does nothing. Okay, so the measurement and coding, blah, blah, blah. Now, I encode in one basis and I measure in the other basis. So I encode in Z and I measure in X. Now Z and X are mutually unbiased. They're as far apart as possible as they can and the probabilities for any outcome in red will be equal for any basis vector in green. So we get this. This actually is randomness. This represents the state of randomness. So nothing that goes in comes out. Everything is lost. This is deletion. You delete your input and you get randomness. See this simple equation which we saw before tells you now something about what happens if you encode and then measure in different colors. Okay, I said that. So these are basically just, this is the state. This is the uniform probability distribution associated with the, let's go back. This is the uniform probability distribution associated with the green measurement. This is uniform probability distribution associated with the red measurement. So we discover something new. So these are classical things. I'm gonna remind you that this measurement is actually measurement against the red basis and this measurement is actually measuring against the green base. I said this before. So that's the only thing you have to be careful if you don't get this wrong. Don't get this wrong. Okay, this is a non-demolition measurement. So sometimes in the measurements and then the system still exists. Like in our I&Trap computers, you can actually do that. You can measure something and the thing is still there. It just collapsed, just collapsed. And so this is a measurement with collapse. Let's disentangle it. So here you got quantum in, you got quantum out and classical out and really what this is is you got a measure. This is as if you do a measurement, then you copy the outcome and then you encode again. And this actually corresponds to the collapse. So this really is the collapse which is happening here. And now you can see this internal spy. This collapse of the wave package, like people say, okay? So I got two measurements, non-demolition measurements, one in Z, one in X. And let's see what happens if we compose them. So suppose we do first a Z measurement, then an X measurement and then a Z measurement. These are sort of the question philosophers asked very early on in quantum mechanics. Say okay, if I observe this and then I do another measurement and I observe it again, of course, we would expect to get twice the same. That's what a philosopher said. But if we do the calculation, we see two wires in the middle there, they just vanish. So this one is completely disconnected from this one. There's not gonna be any connection between them. And this is pretty much gonna be randomness and this is gonna measure whatever comes in. So you see, and this is also the technique who knows about quantum key distribution. Yeah, yeah. So this is also the reason quantum key distribution works. So Alice encodes the certain bases and Bob measures in the corresponding bases. And then if Eve is messing in the middle, it's gonna break down the correlation between Alice and Bob's measurements. And they can then detect that there has been an intruder. That's really how quantum key distribution works. It's because of this phenomenon. Okay, you've seen that. Okay, this is also an interesting one. So these are all very simple calculations. So suppose I've got a phase, a state with a phase. State with a phase, a non-trivial state. And I'm gonna see what happens if I measure it. So I said these two have to be conjugate. These two have to be conjugate. So they're gonna cancel out. So this is kind of saying that if I go from quantum to classical and I measure, all the phase is vanish. So phases is something that has no counterpart in the classical world. They get eliminated. Okay, so now you understand. So you can actually buy these t-shirts with this dodo if you want to. So you have to look on Etsy for Dave, the dodo t-shirt. And I can advertise this because I'm not making money with it because my co-author, Alex Kissinger, takes all the money. These day, many quantum scientists go work for companies and he says, okay, I'm also gonna do a company for t-shirts, dodo t-shirts. Oh, you can also get MUX and stuff like that. But what I'm showing, this is really teleportation, this little picture. So you see, you've got this cup state. You've got this cup state. This is what's coming in. That's the incoming state at Alice's end. This really is the Bell measurement. This is the classical communication of your two measurement outcomes. And this is the correction. So this is now how it looks teleportation just as a picture and nothing more than a picture. So that's kind of cool. I mean, here it's, so this is what we had before. These two possible measurement outcomes, these two possible corrections. What we do now is we use this representation in terms of circuits. We use this representation in terms of circuit of these Bell states. So we turn it upside down. We get this, we get this. And now we just need to do a measurement here and a measurement there to get actually these outcomes there. That's what we do. And this is full teleportation with classical and quantum wires. Everything inside. Okay, I said that again. Okay, I think this is more or less the last thing I'm gonna do before questions. Let me check, let me check. What else do I have? Oh, so I'm going backward. I think do I have anything? No, that's the last thing I have. So this is from the kids book. This is in here. And what is this? This is the proof of non-locality. This is a proof of quantum non-locality for kids. And it involves both classical and quantum wires. Let's, let's, does anybody who knows sort of the mermin architecture, the mermin proof for quantum non-locality using a GHZ state? Nobody? That's the most, it's much more elegant than Bellini qualities, but okay. So, so basically what you got is like, so what you got here on the bottom are three, four GHZ state. It's actually one GHZ state, but in four different scenarios. So you got a GHZ state here. And then you got some rotation here. You see, you got some rotation. This is doing nothing. There's no, and this is like doing a 90 degree rotation. This is rotation around the, around the X axis. Rotation around the X axis, 90 degree, 90 degree. So, so basically you got the GHZ state. Here I do nothing. There I do two rotations to the two last particles. Here I do rotation to the first and the last. Here I do rotation to the two first ones. And then I do a measurement. So basically I'm actually taking this, this, these are, these are Z measurements. And I'm actually rotating them, the Z measurements to a Y measurement. So these are, the 90 ones are actually Y measurements. And the zero ones are actually Z measurements. So I'm doing four different measurement scenarios on this GHZ state. That's what I'm doing. Got a GHZ state. I'm considering four different measurement scenarios with these, using these different rotations. And what I do then is I take all the classical data together and apply red spider to it. So what does a red spider do to classical data? It's a parity test. It's, it's, it's like an XOR, but it's like a generalized XOR. So it's like a parity test. It sees whether the outcomes are either even or odd. So, I mean, this is something you can do in the laboratory. As I did, this is in the laboratory. So I've got these four scenarios, four measurement scenarios. And then I basically measure the parity of the whole thing. Now, what? No, I'm gonna, I'm not, I'm not, I'm gonna do the calculation just like that in my head. So basically, do you remember, do you remember that we had this rule for the squares? And I said it's like commutation, green and red sort of gets exchanged. Remember, I said this commutation? So, so you can actually use this rule here to actually move these red dots above the green dots. And then the green dots are gonna vanish. And so then basically what you have is I've got 90 degree, 90 degree, 90 degree, 90 degree, and 90 degree, and 90 degree. So I end up with 180 degrees. So I will, and then ultimately, you just end up with this thing, no wires anymore with 180 degrees. That's really the computation you do here. I mean, I didn't put the slides in, but it's quite the easy thing to do. Okay, now, so this is what quantum mechanics tells you. This is what quantum mechanics tells you. If I do these four scenarios, I get under 180 degrees there, which is actually one, which is the classical state one. With zero, it would be the classical bit zero. This is the classical bit one. You get the classical bit one here. Okay. Now, we're gonna assume that there is a local hidden variable model. That's what monocality is about. Local hidden variable model means that whatever these measurements are, whatever these measurements are that we are here, we've got one, two, three measurements. We've got this measurement on the first system and these measurements on the first system. So one with zero and one with 90. One with zero and one with 90 on the second system. One with zero and one with 90 on the third systems. So there are six different measurements we need to consider. Six different measurements we need to consider. And so this is the first measurement that's getting the zero. And this is the 90. Second measurement, second system measurement of the 0 and of the 19, 30 system measurements of the 19, and we assume that there are variables determining the measurement outcome. That's what the local hidden variable model is. So for each of these measurements, there's a variable, and so we copy these variables out, we copy these variables out to these four different measurement scenarios we have. So this is really just copying them out to the four different measurement scenarios live somewhere here. And then we do the parity test. So this is what the local hidden variable tells us, and what do you see? Anybody knows how to simplify this? We got red dots, we got green dots, we got wires between red and green dots. Yeah, you see, between every one of these green dots, and this is because of the choice of the scenarios, the way it was made, there's exactly two wires to the red, so they vanish. So basically what we get is a red dot with nothing inside, which is the bit zero. So we got bit zero here, we got bit one there, so quantum mechanics gives you bit one, local hidden variable theory gives you bit zero, and zero is not equal to one. So we have a contradiction. So whatever quantum mechanics predicts in this particular scenario is in contradiction with whatever a local hidden variable model could ever give you. So quantum mechanics is not local, and this is now entirely just in picture, so that this is in honor of this year's Nobel Prize. Zeilinger came to ask me two weeks ago to sign his copy of quantum in pictures. So that was pretty cool. So he's the one who came up with this idea and did it in the lab, and that's why he got his Nobel Prize pretty much for this experiment. I think I can stop here. It's been long. I've got nothing more to say, so I'll leave it to questions now. So let me just end up with this. Any questions? If you have questions, raise your hand. I'll bring you a microphone, actually. Yeah, sorry. I would like to go back to the question I asked before because I didn't really get the whole answer. So I understand that, I mean, when you have a real quantum device, for instance, the one of continuum, it's not like as simple as just applying a synod gate. You have to implement it physically, and this makes sense. So what you understand is if you have this complicated synod stack and you want to translate it into the phase gadget, that's really cool. But did you say that the phase gadget is what you can actually apply physically in this quantum device? Yes, I mean, I'm not a specialist there on the hardware implementation, but specifically for iron traps, they're in a way more native than your usual quantum gates. I mean, if a tonic's people use completely different, I mean, like the measurement-based thing, which I explained, that's what people are trying to build that psych quantum, not gates, because it's much more natural to do that sort of stuff with light, than the usual gates. So different architectures will have different limitations. It's not just all gate-based. Thank you. And like I said in the beginning of the talk, with photonics, it's sort of encoding of qubits. They do something like dual rail encoding and stuff like that. It becomes really complicated to describe this in Hilbert space, not the same possible. So they're all using these diagrams now to basically, yeah, reason about their program, reason about the error correction and stuff like that. And in Quandela in France, they're doing the same. They're also just using diagrams for everything now. That this is the place where this is now the number one language, quantum photonics. So I have another question here. Oh, okay. So thank you for the wonderful lecture. Yeah, the sound comes from here for me. Yeah, I never know where. Okay, sorry. So I have two questions, actually. The first one is, so would you say that ZX calculus or like what you showed us today is more natural for optical photonics, right? Than gate-based architectures? So I can tell, so basically, Rosdank and I, we came up with ZX calculus, trying to look at a problem in measurement-based quantum computing specifically to do with what's called generalized flow because it doesn't look like this measurement-based quantum pattern. So you've got a very complicated state and then you start doing measurements all over the place. And there are some criteria which tell you when you can actually translate this into a circuit and not and stuff like that. And it was looking in a problem like that that we actually came up with ZX calculus. So ZX calculus actually came out of measurement-based quantum computing so it's not a surprise, it's quite useful for it. On the other hand, when you actually want to work with beam splitters, beam splitters and stuff like that, then it turns out that there's a different calculus than ZX calculus that's quite useful which is called ZW calculus, which is something Alex Kissinger and I did in 2010. And in the way we now are reasoning about photonics, we actually mix the two up. Z, because for some things having the W is more useful around and for other things having the Z and the X is and Richie who's here, you can talk to Richie about it. He's not asleep anymore. So you can talk to him, he's doing a lot of work on that sort of stuff. So for the second part of the question, you mentioned about beam splitters. So any other quantum optics or photonics experiments or basic stuff can be explained by ZW calculus, right? Could you give a small illustrative example? So any other useful practical that we see in photonic quantum computing? So just compilation, just compiling circuits to optical, that's something we are now developing at the moment, specifically for Quandela. Kragi is here, he's involved in that sort of collaboration. And we're going to do similar stuff for quantum now, where they use these diagrams. It's really in their calculation of error correction stuff of a photon and it's super complicated. You should look at this paper from Naomi. You should go and look in the sort of diagrams you find there. And you should look at this paper from Naomi. You should look at this paper from Naomi. What you find there, they're just too, they're so big and it would be completely impossible, let me go back. So if you go in that one then you find tons of applications and you should just imagine trying to do these things with Hilbert spaces from the pictures you're gonna see in there. And I think they have a new paper out too. Like I think a few weeks ago where they even do more complicated stuff. So my slides a bit outdated. Thank you so much. Sorry, one last quick question for me. What is the relationship between tensor, networks, and the x-dent conclusion? Okay, I just answered a question, it's a very good question. So on the one hand say tensor, networks, and Feynman diagram they're in the same space. They are a graphical representation of stuff you do in Hilbert space. But it's just like you have a calculation in Hilbert space and they will help you to do something there with that calculation. They are no substitute for Hilbert space. You use them to guide your calculation in Hilbert space. This stuff lives on its own. This stuff lives on its own. You don't need Hilbert space anymore. You can completely forget about it. I mean I'll tell this afternoon a little bit history like the sort of historical development of these sort of ideas. I'll talk about that this afternoon. But so we've got a genuine substitute of a Hilbert space here. And you do everything in the calculus. You never go back to Hilbert space. So that's different. It's a complete logical maximization of Hilbert space while a tensor network network is like a calculation of eight when you're working in Hilbert space. I mean these are tensor networks. You can think of them, but they are sort of very precise, well-defined tensor networks. In many cases tensor networks have a lot of variables inside, say okay a matrix product state or something like that. So they're not fully defined. Here we work with exactly defined entities and all you need is the rules I showed you. There is no such thing as the tensor network calculus that lets you compute everything you want. But these are tensor networks, very special example of tensor networks. I don't think anybody can hear that. There's been some work out there where people use ZX calculus to rewrite ZX diagrams to make them as small as possible and then use tensor networks to contract it leading to faster tensor network contractions. You can combine ZX techniques and tensor network techniques and also stabilize the compositions. So if I understood it, can you also do some expectation value calculation if you have density matrix? Oh yeah, yeah, yeah. So I mean in one calculation I said I throw away the numbers. But basically the numbers in the doubled world, so you got a number, multiply it with a complex number. With this conjugate, so it's a positive number. So basically you would just drop, I mean this, the example of G and Z which I gave is not like actually a probabilistic process. It's an exact thing. But a close diagram like that would usually give you a probability. The diagram would become a number and that's the probability. Oh okay, and so like if I imagine quantum chemistry, so you have molecular basis and like certain basis to compute. So in this case what's the input and output basis would look like? I mean I'm not involved in that but I know that some of our people are talking to Nate in there and they're starting to do some quantum chemistry using CXW. Oh, actually you're involved in that, right? Yeah, so we only just start to apply but this is basically full blown quantum mechanics. The entire quantum formula is everything is there, probabilities, everything. So whatever you can do in quantum you can do here. But it's all just as a diagram. Okay, thanks. Yeah, this is just for two by two intense products essentially whereas quantum chemistry would require more general basis states. But I think that's what Hani's working on. Yeah, yeah, yeah. With this CXW in higher dimensions, yeah. Okay, if we're done I got one more thing to say. Are we done? So who has a copy of this one? That's bad luck because everybody's gonna get one for free now, ha ha. They're there. Can I ask one more question? Yeah? Can you explain again this deformation of the rectangle? Like how do you break down the rectangle again? Oh, what happened? Oh, there. Yeah, this one? Yeah, so it's my microphone's alone, yeah. So basically, did I take this? I don't know. So you got four loose wires. You number them. They are like alternating colors. You take whatever's connected to green and you connect to a red dot. You take whatever's connected to red and you connect to a green dot and then you connect those two. So in this case, if we take like this red and the green on the other side, that would be an equivalent representation. Would this be? So if we take that green and the red on the right side, so that would be an equivalent diagram. Oh yeah, this inequality. Oh, sorry, like in the right plot, if we take this green and the red on the right side. Yeah, that's all the same. Okay, okay. That's all the same. Like this crossing doesn't mean anything. Oh, okay, I think. Crossing doesn't mean anything. You have to think of these wires lying on the table. So there is no crossing. The only thing that would matter is what is connected to what? Like in electricity. Okay, good, thanks.