 No, there is no roadblock. Good. There was to rush through at least about a few of my favorite arguments why photons are a nice system. But now we're jumping to stuff that we like to talk about more details about recent results. So now we go back to the original quantum physics. No computing, just curiosity. And the interesting testing is the mathematics, the white one that we use for describing quantum physics. So what do we mean by that? As you might remember, back in the days, quantum physics was developed hand in hand with the development of mathematics. Heisenberg developed also matrix formalism and so on. And remarkably, the smart mathematicians at that time realized, even though we have not developed this new beautiful mathematical framework, we still have problems and issues of quantum physics. Nowadays, we all know there's this randomness which is still hard, but now we accept it. But at that time, it was something people didn't like. Measurement, problem, and so on. So as the new mathematics showed up, people like a phenomenon realized that even though we have not this beautiful new mathematics that allows to describe the entire quantum physics, we still have an issue that we can still not predict the result of a measurement outcome, for example. So here is the fair question. Guys, maybe you used the wrong mathematics. Maybe quantum physics works nicely and you just have the wrong toolbox in your hand. So it's basically the summary that he defined. He said, well, if you work in the Hilbert space, you have the freedom to choose what kind of numbers you take. Real numbers. In fact, you can use real numbers, but then you have issues to describe some kind of properties. With the complex numbers, the things we use, too, since high school, they work nicely. But there are higher classes, so-called fraternions, before elements, or octonions, if eight elements, they also allow a full framework of so-called rays in the Hilbert space. So he asked the question, what about if you take, basically, a more complex mathematical framework, can we solve problems then? So what are the next level of mathematics there, the so-called fraternions? In principle, they're the same as complex numbers, but just they have three imaginary parts. You see, that's a fraternion number, one real, and here the three imaginary, and they also have been defined by Hamilton back in the days as a full mathematical framework. What's the difference to complex numbers? The difference is that they do not commute. So we are used with complex numbers, 5 times 3 is the same as 3 times 5, fraternions, it's not the case. So mathematics slightly has different issues there. Well, of course, this means when you think about making experiments and trying to investigate if mathematics is right, then it was pointed out by Pérez that, in fact, you can do experiments where you might see different results if mathematics is based on fraternions who do not commute or complex numbers who commute. So you basically thought about experiments where C basically phase shifts from two elements where A and B is not the same as B times A. And that's basically his pioneering work. Now comes an important issue. Let me talk about commuting behavior. It's not the phase shift of 5 and 7 pi. It's basically the quaternion number, which is given by the mechanism of the material how it interacts with the light. So in other words, to have different complex or quaternion numbers, you won't have different physical mechanisms that basically make light, for example, interactive something. So the key challenge was here to define basically different quaternion numbers that originate from different interactions of phase shifters with light. So if you have different mechanisms, different quaternion numbers, then you can make a test if A times B is the same as B times A and compare complex numbers versus quaternion numbers. The first people who were interested in that time were neutron guys. And they built precisely this kind of experiment. So they've done a mach 10er. They built a mach 10er, which is like entrance beam splitter, outer beam splitter, mirrors here. If the neutron comes, it's superimposed to go either the upper path or the lower path recombined, and then it goes out to this port and that port. As you remember, by choosing the phase shift in there, you can choose which outer port has been taken, all the neutrons here or there. The two elements that were taken to have different to test these mathematics right or wrong are basically very required to be very different in interaction. And indeed, they found two materials which have very different properties, one for so-called positive scattering amplitude, the other one for negative scattering amplitude. Very different mechanism how neutrons see this material. And they've done the experiment and said, OK, let's see if they commute or not commute. Nothing in there. So they've changed some mirrors, the fringes. They've seen fringes with nothing, fringes with just one element, the other element. Then they compared A, B, aluminum, and titanium, and B, A. And they said, is there any difference if there are no difference? Complex numbers, difference, continuous. They've seen no difference up to this little noise level in there. What's the problem? Good for us. The problem was they couldn't see anything. So when I take a closer look at these mathematics, it turns out that for non relativistic, i.e., massive particles, regular scattering equations, the amplitudes dumps exponentially. In other words, you cannot see an effect if you have massive particles. So you have to go to a framework of relativistic things and massless things, like photons, our business, where we actually could say, well, now we test properly if there's something out there that we can see. Is there some noise that might come from cotrionic contributions? So what we have done was precisely that. We took different phase shifts, A, B, and compared this A, B the same as B, A. And that's basically the sketch of such an intermediate we have done there. Light comes in, one element and the other. But I have a few more slides to show in more detail what we have done out there. So we have taken, for stability reason, a folded Marzenda, which is so-called Sanya Klub, because the arms compensate themselves if there's noise. So how does it go? Light comes in there. It goes 50-50 beams per liter, either this path or that path, and everything is aligned properly. Then you have interference that all the light must go out there, and nothing comes out of this port. It's so-called dark port. So what defines the light that comes out there? Very simple. So there's no light coming out if this term is one. That means the visibility is perfect, and you have perfect interference. And you have no contribution from this gamma, which is the quaternionic contribution. So the test, of course, you have to make the visibility as good as possible that we have no noise. Otherwise, you would see no signal from the noise floor. And then we could really test if this term is basically existing or not. So just sitting there and comparing either any clicks and do the clicks higher or lower, it's not the best way to check something. You won't have a tunable parameter. So what have we done? So we added here another Martinda. You see this one here. Where the change of the mirror leads to fringes down there. Very simple to understand. If nothing comes out there, best case, then light just comes here. You tune the mirror, it stays flat. It's basically as if there is no light. If some light comes out there, then you see little wiggles coming up, and the little wiggles get bigger and bigger, the more light sneaks out there. And you can really have a controlled fringed visibility telling you what's happening in there. So we compared them with different elements here and there, similar to neutron guys. What's the challenge? And we're a little proud about that. The challenge was to find materials that interact in a different way with light. And that's not that easy. Because the normal thing where you get a phase shift from light is by refringes. It gets slower in some thicker material. So that's one mechanism. Then we looked for some other mechanism, how we can basically have a very different interaction of light, similar to the neutron guys, like for example, positive and negative scattering length. The thing that in my time was to use a so-called negative-effective index material or metamaterial. The way our light interactive metamaterials is totally different at the fish antennas as with the liquid crystals. Therefore, we assume there must be different quaternions. And here we might see a strong effect if something is out there. Good. Well, let me not go into details here. Metamaterials are those things which allow this cloaking, which give this negative phase shift by having negative permittivity and permeability by using these kind of antenna structures. If you would have a liquid out of metamaterials, it would look like this, whereas normal liquid bends the straw like that. We have very interesting properties. We haven't done those ourselves, but we got a very lucky situation that we got in touch with the Tsang group in Berkeley, who at that time made the first metamaterials working for our visible light. Normally, they are far off in the wavelength. They did something for us. And it was also the best quality so far because they worked with single photons. They wanted to test quantum physics. And they were not used to have very low losses in there. So we said, please, we fight for every photon. Make the best. And we got good devices that were in the order of 50% at the beginning. Experiment took so long, or we took so long, that they oxidized a bit. And they went down to effectively 10%, 13% at the end. Good. Test experiment, you see we have here, these are the metamaterials. We see a negative phase shift with respect to different wavelengths. We stayed normally at 800. And you see a positive phase shift for regular liquid crystals, which is the same as a piece of glass. Very, very basic interactions. Good. Now we need quantum systems. So we took a single photon from a down conversion source. Two photons come out. One is a trigger. Other photons send into here this Sun-Yacan diffimeter. One element is metamaterial. Other element is the liquid crystal. With that, you really could investigate the story. We got different fringes for nothing in there, A in there, B in there, and both. We zoom in a bit. So you see basically, these are the real wiggles when you zoom in, you see this kind of better resolution. The summary of the story is that you observed that up to this 0.03 degrees, you don't see anything. So at least it was the fact of 10 better than neutrons, plus based on relativistic particles, which have a chance to see the effect. So I personally believe there is no reason to build something up with proterions. There was a hype many years ago to use those because they're very handy. As you can imagine, these four elements is something immediately to jump into your eye when you think about tensors or generativity, free for space, one for time. So people tend to use those as very handy mathematics. But here it seems that complex numbers are still very fine. And there's no, at least up to this little noise level, no strong indication that we have the wrong mathematics. But at the end, we have to test. We never know. And sometimes this little epsilon of uncertainty might open a new door. So here, but that was our approach. Good, oops, how much time do we have? Good, now we change gears. The only thing is the same as the setup, more or less, in terms of concept. But now we jump to interesting computations. And I would like to show you how by changing the framework, you can even boost regular quantum computers. So that's the story here. By changing what is superimposed, you might be surprised what can be done with respect to standard quantum architectures. Good. The story starts basically with the investigation. And people said quantum mechanics allows to superimpose everything, OK? Not only we're used to superposition of states and so on, but in principle, we can also superimpose orders, or causal causality, or basically causal orders. So what are causal orders? We're used to them from the very early days of our life. A comes before B, or A affects B in one way, or other around B affects A in one way, and there's a time flow, these errors. Sometimes they could be also common cause, that twin berth gives A and B, something like this, OK? We're used to that. And we go to quantum mechanics, where we like to superimpose things, because that's the interesting stuff that's not out there for classical physics, the normally superimposed states. We're used to them, no 0 and 1 spin up, spin down, polarization states, and so on. But in principle, we're also allowed to superimpose here, so causal orders, OK? There's nothing that's against that. You can see that basically you don't know who comes first, for example, in this case. When it be like a computer guy, you say, wait a second, causal actions or causalities are also in gates, not one gate before the other. So you can think about superimposing this kind of transformation or gate actions in the same way. In the bigger picture, you can feel the thing of a standard of a quantum computer, where in principle quantum physics allows to superimpose the order of gates, states, but also causal relations. Is the history of the state, or is the history of the problem? Yes. Is it the state and quantum? Yes, but with the causal relation, it goes beyond the regular state, so there's time dependence. You can superimpose time orders. There's time for your state, depends how you define it. OK? You want to test time order? Yes, you can do that, yeah? Superimpose, at least you can have superposition at the end, you do not know who was first, OK? So that's a rather new field where a few groups jump on that, because it's interesting to think about the time flow, which always considered as one particular trajectory can be also superimposed. At the very end, you can say, I don't know who was first, as in quantum physics, at just the particular interference that tells me there was this superposition of causal orders, OK? So mapping it down to applications, OK? Which is the case, which I would like to talk today, but it goes far beyond that. You can think about, wait a second, we can superimpose gate orders of computers. That is different to all the standard quantum computer architectures, because as you remember, the speed up for computers come from having superimposed zero and ones, input states, your massive parallelism. Then you basically have to fix gate orders that do the job you like to do, a shore, a grover, who are smart architectures who basically knock down the wrong results and give you only the right one, OK? But the gate order is fixed. You don't change the circuit, basically. Now you're allowed to superimpose different circuits in a way that you have at the same time different architectures used at the same time, in addition to having here this massive parallelism, because you can still use zero and ones at the same time, OK? That's the new idea out there. So let's take the simplest case, and you will see what's suddenly interesting, how you can basically do something better. So the simplest cases, you have a control bit and a target bit, and the control bit, the control bit in the state zero. You have the order of first this operation, and then this one, you see, blooms. Or you have the state one, then you have the other operations first, this one and the other one. Two different cases, very distinguishable. You can write it down like this, of course. So when you're not superimposed the control bit, then, of course, you have superposition of U1, U2, or U2 and U1 as written there, OK? And this is something that was developed a couple of years ago, and it says that mathematics and quantum physics allows to do so. The question is, why the heck should you care, OK? One reason is, well, you can do things faster. It was shown that with such a superposition of gate orders, you can, for example, by asking only once, going through the gate only once, tell them, do the commuter, anticommuter, OK? That was the beginning. And here, it was shown already, there is a speedup. People tried to find other applications that go beyond it. So by having one shot only, you know, do the commuter, anticommuter, by seeing either this port or that port gives a click. Here's the summary page. So if you don't need to listen to all the stuff there, but the summary page, why these causal superpositions are interesting. First of all, they show a linear advantage with respect to standard quantum computer architectures. In other words, they are faster than what you have seen so far from standard circuit model or measurement-based models and so on. And they are super interesting in investigating causal orders and the relation to that, because they might shine new twists or new insights how time, quantum physics, up to even general relativity might interact because you're allowed to superimpose time flows in causal relations, OK? Active field is not clear picture yet, that is will end. But you will see, I guess, every month interesting papers are using this kind of concept out there. So these are the two driving forces why this is interesting at the moment. Good, let's go to experiments. We like to see it in reality. We want to see it on the optics table. This is really happening. So what we need to do, we have to build something that allows to do that. So photons, again, very nice. Mobility, key issue. We can do something nicely. So the simplest way to do it is to take two operations. You want you to. And you take photons which take polarization as the degree of freedom to tell them where to go. So if the horizontal polarized and if this polarizing beam speed, they go just straight. So let's show it like this, OK, very lack of time. If the H polarized, they go straight first here. H goes straight through, up here. H goes straight through, then this and out. If the vertical polarized, then this device reflects the light. So you start with this one first, up here, reflect again, here, and out. So you see the polarization defines the order. And if in a superimposed polarization, which is very easy, just turn a wave plane, then you have superposition of u1, u2, and u1 as written here. In the computational language, people call this a switch, OK? So there's two operations, the two switch gate. And people would draw it like this. The control bit defines basically how the switch acts on the operations, OK? That's basically the computer language to that. So we have done this. So we have control and target was put into one photon. Again, nice advantage of photons. You can use more degrees of freedom. So the path set, one direction is 0. Other one is 1 for the control bit. And the polarization, the intrinsic spin of the photon was used as basically where the computation takes place. What do I mean? So that's a beam splitter who says, if the photon comes here, it's in the one state. If it comes there in a zero state, or it's superimposed, it goes basically superimposed in a superimposed way. This kind of paths. Then we basically folded them like this and folded them like that. And then we recombined the beam splitter in the same way as shown before with the PBS. It's really identical to that. So I put in the two operations. As the first case, we just use some transformations, or polarization rotations. And by basically building that, you could really have one-shot only supposition of this for that or that for this operation, which allowed by one shot only to check to the commute or not commute. Similar set as before. Down conversion, two photons, one is trigger. Other photon goes in there. And there's the beam splitter. And then defining on the ratio here, you've got this path and out, or that path and out. So my fingers are not that good. So I use this pointer here. So here comes the photon, superimposed. It goes either this order or that order. And then, again, recombines here at the end and comes out. By one shot only. In other words, by going through each gate only once, you can check the commutator, if the commute or not commute. Here comes the, we have tested this in the experiment. If you choose unitarists that are commuting or not commuting, again, the question to you guys, how good do you remember one-on-one in quantum physics. So if you choose in the beginning, very simple operations. Polyoperations, you know, x, y, z. So question to you, when you have now, for example, x and y polyoperators after each other. Do they commute or not commute? Please. Not commute. OK, so they go out to port blue. You see? Right. If you have now, for example, x and x, they commute red. So you see, at least, from the basic knowledge that we have, the experiment works. You can see, yes, commuting here, green, not commuting blue. Then we took other operations, basically very weird unitarists who also busy commuting with respect to each other. We could see that we get all these very nice one-shot only result being commuting or not commuting over there. It cannot be done by a standard quantum computer. That was the story here. Good. Last three minutes, I see, probably. I make it short, so I will rush for this slide. So the next step that was interesting to us was, are we really sure that we have this causal supposition? Because so far, just entertain you with that. But people could say, wait a second, Philip. You see a speedup, fine. That's something a standard computer cannot do. But this is really convincing that you have supposition of causal structures. Because at the end, you might have found a better algorithm and still have this kind of classical world scenario that doesn't supposition of orders. Well, fair enough. So the way to convince ourselves and others, we do have the supposition of causal orders was to take a witness. And basically, the witness violates if it's above zero, it's really basically supposition of causal orders or non-separable case. If it's below zero, bad for us, then this is not the case. Identical to entanglement witnesses years ago, we also have basically a fingerprint that says, yes, that's a belt pair or not. Or that's a cheat cistern or not. It was the same concept here. Just takes a few measurements out there. Good. So the story is now done in a process matrix, in a matrix. But it's in the process. It requires process matrix where you choose different input states, different settings for the operations, and output here. The nice thing is you can implement any operation here. You can even make measurements or you can superimpose measurement results. Quantum physics allows to do everything. So there's details. There's a process matrix. I don't need to tell this to you. The question was, can we basically define case? Can we prove that it's either separable in this way? That basically it really says, oh, I'm A before B or B before A. There's no coherence between them. Can we basically prove that? Yes, we found an operator, similar to entanglement witness, that applied to the process matrix is the following. If it's bigger than zero, we have a separate case, boring case. If it's smaller than zero, it's a yes fingerprint you are in superposition of causal relations. Now I rush through because of time. The point is now, this slide, so we've done many measurements, many input states, and many settings for these operations here. The challenging part of this experiment was really a beast to do so, OK, was the fact that he replaced the unity operation with a measurement device. Really put in something brutal. Measurements are brutal in quantum physics. Now, random, they give you zero and one. He put this inside. The tricky part was that it doesn't destroy everything is that the result of the measurement operation here stays inside the box. Nobody knows the answer until the very end, OK? So let me phrase it mathematically. So basically, we have basically here this kind of probabilities to get, which allow us to reconstruct the witness, which defined on the input state here, that says superposition of the control bit go both ways, or just one or the other. Measurement settings here, of measurement A, measurement settings here, and re-preparation of the outcome. And here we just took the unity operations as before. So the beast, of course, was to put in here this measurement operation inside the interferometer. So it'd be an interferometer inside the interferometer, which was not that straightforward. That's the picture. So previously, we had just this unity operations acting on the polarization. And now we put in this kind of beam splitter, a processing beam splitter where photon is either H comes out here, or B comes out there with re-preparation operations afterwards. But then you see, it doesn't leave the circuit. It stays in here. So this result is recombined here also, and goes out here. And the other result stays inside the interferometer stable, and goes out there, OK? So the point is now we have two more outputs, OK? Fair, as you can do that. But the point is the measurement result is not known to the outside until here. But at the end of the day, we're able to superimpose all the measurement results. Good. That's the result of our witness. That's the quality of the setup. If the visibility is perfect, you've done a good job. As you really stay in a good regime. Then you can see your violates. So we flip the plus and minus sign. But basically it would be above 0. We have this proof, the yarn superposition of causal relations. If it would be at 1, then it'd be up to this value of 0.2. We never have 1 in reality, so we're more like 1.9 something. So that was the best value we got with this number here. And then to show that it's really like the case that we see, I wanted to, again, we like to touch knobs. And we want to touch a knob and see how things change. So what can you change here? Your purpose misaligned the setup to get basically to allow nature more and more to understand what's happening inside. So when you decrease the quality of your interferometer, which is basically this axis, you go from 1 to 0. Then of course, the witness gets smaller and smaller. And at this point here, being 70% quality visibility in your setup, you get results that could be also explained by just regular classical causal relations, no supposition. We like to have this kind of plot, because it really shows that it makes sense what we are doing. You want to really have not only one short measurement, but you want to see basically correlations or basically how one thing affects the other one. The highest number was 0.2 with up to, I don't know, 5, 6 and the standard deviations for our case. Good before I get kicked out here, the last slide. So besides the personal view about photons, I think they are nice and we'll have a very good future for applications. I showed you recent results that we have done, and we like to, first of all, investigate this kind of mathematics of quantum physics comparing complex number versus quaternions. And then I jumped over to interesting new twists in quantum physics that allow to superimpose gate orders. I've seen we can have speed ups with respect to standard architectures. And also I've shown that basically we could find a witness operation, really prove uniquely yes. It's the case that you have this kind of supposition in the box. They're also superimposed and measurement operation in there. I would like to go a lot within this framework. There are many things to do. It turns out there are more applications. It was shown communication can be done better. There's new ideas for restricting computational speed ups. And of course, we like to think about foundations. Basically, can we really have weird suppositions of course in relations where with active switches and so on that might twist our brain even more what quantum physics allows to do so. Good. I'm just showing off with results that I haven't achieved myself. That's the team who stands behind the experiment. Particularly I want to mention Lorenzo who has done the metamaterial business. And Julia and Lee who are still working on the supposition of course relations. And they basically dig into there and have done the great job. If they want to stop here, thank you very much. Thank you very much.