 people that just put up a poster that will be added to our list for the three minute talks. So at any rate, that'll happen later. You already did it. I say a few people did. All right, so that's the only announcement I want to really make right now. Mike just went up. So we have Professor Malick from, now you're Harriet Watt, from Harriet Watt University in Edinburgh. And he's going to explain to you from all over the world why that's pronounced Edinburgh, not Edinburgh. Okay, thank you. Thank you for that introduction. It's a great pleasure to be here. So my name is Mayhul Malick. I work at Harriet Watt University right now, where I've recently started a group, but some of the work I'm going to be talking to you about today was done also in Vienna, which is not too far from here in Austria. Okay, so I thought I'd start by telling you a little bit about my academic journey and looking at the list of participants. This is probably one of the most diverse audience that I've ever spoken to, which is really great. It's great to see scientists from every part of the world. And my journey has been pretty international so far. So I was born in India. I started my PhD in Rochester in the U.S. in 2006. I had a brief stint in Glasgow in 2011 when I visited a research group. I did a couple of post-docs, I guess one would say, in Vienna starting in 2013. And as of last year, I started my own group in 2018 in Edinburgh. I don't know why it's called Edinburgh. Scottish, I don't know. So my group is called the BBQ Lab. We do like BBQ, but mainly because we work on beyond qubits. It's beyond binary quantum information. It's the official name. We do everything sort of in higher-dimensional systems. Benny talked to you a lot about temporal modes. I will be talking to you more about spatial modes and entanglement in spatial modes. So we are located up, let me see if I can get this to work, up there in Edinburgh. And it's quite beautiful. Harriet Watt campus, it's outside the city of Edinburgh. It's very picturesque. It doesn't look like a college campus, but that's a field that gives you different colour flowers at different times of the year. And my group is fairly small compared to Benny's group. It's four of us right now. That's Will, he's a post-doc, Wattscholl, my second PhD student and Natalia, my first PhD student. And we have two more people joining us this year. And that's part of my lab. Okay, so a rough outline of my lectures today. First, I will give you sort of a brief introduction, talk about qubits. Second, I will introduce just kind of recap what is superposition, coherence and entanglement. Then I will talk about a theoretical tool that we developed to measure high-dimensional entanglement. And time permitting, probably not. I will talk about noise resistance of high-dimensional entanglement. In the second lecture after lunch, I will talk about a recent experiment where we sent high-dimensional entanglement through a complex medium. And then I will talk about the first experiments on multi-photon entanglement, more than two photons entangled in high dimensions. And finally just some fun slides on some free space experiments that we've been doing in the Canaries and in Austria. Okay, so let me just start off by reminding you that the first quantum revolution, as people call it, refers to technologies that we're all familiar with and we use every day that are based on the coherent control of many quantum systems together. Right, and these form the basis of modern society, things like the transistor, the laser, the GPS that is in all our phones now and MRI. It really is based on the control of many quantum systems. The second quantum revolution, as the media and us now like to call it, is based on the control and engineering of individual quantum systems and uses uniquely quantum effects such as superposition and coherence and entanglement. So some examples of recent sort of landmark achievements in this second quantum revolution, the Quantum Satellite that recently beamed down the Chinese Quantum Satellite, Michius, not sure how to pronounce it, sent entangled pairs of photons across 1200 kilometers. Google recently developed a chip that has achieved what is called quantum supremacy, which is not the best term for it, but essentially it was able to perform a computational task that would take a classical computer very, very long. And LIGO is currently being upgraded with squeezed light and I don't know if you can read the fine print there, but it says quantum squeezing light could lead to daily gravitational wave detections, which is quite exciting. And alongside, you know, there's been big national programs, funding programs, so the UK launched this National Quantum Technologies Initiative a few years ago now and then the EU Quantum Flagship launched last year, I believe, and then the US has recently, I think, started to have its own quantum funding program. So there's a lot of excitement in this field for good reason. But what is the role of photons in all of this quantum computing and quantum communication? So photons are really the ideal carriers of quantum information, right? There are many platforms that are in the running for quantum computing from NV centers in diamond to superconducting qubits, but photons are the only ones that can really transmit quantum information very efficiently and the reasons for these are they are fast, right? They travel at the speed of light, they are cheaply produced. They don't really interact with the environment, which is pro-anicon, con in the sense that they're not good for quantum computing. They don't interact with each other either very easily, but they're not affected by noise, so they're great for communication and they're efficiently transported in fibers and integrated circuits, as Benny showed you earlier. And we're building on this massive classical photonics industry when we do quantum photonics. So today, sources of quantum light and detectors are reaching a stage where we are really getting close to real-world applications. We have very bright sources of entanglement, very pure sources like quantum dots of single photons and high-efficiency superconducting detectors. These are some nanowire detectors. So quantum optical interconnects are envisioned as a link in a global quantum network for people called the quantum internet. Essentially, quantum processors interconnected by quantum optical links. This was laid out in this early, not too early paper by Jeff Kimball in 2008. Okay, so let's move on to introduction to Q-dits. This is basically a rehash of a slide that Benny already showed you that talks about, you're all familiar with polarization, polarization qubits. This is a state of left circular polarization. It's really formed a workhorse for many quantum optical experiments from fundamentals to applications in quantum communication. However, as you already know and have seen, you can encode a lot of information on a photon, not just zeros and ones. You can encode information on the temporal and the spatial structure of a photon. These are some examples of spatial modes that you've already seen. So I'm just going to move quickly through this slide. So something new, the amount of information that you can encode on a photon is limited by a few things, the size of your physical system. So for spatial modes, it's the size of your aperture. Benny talked about single mode waveguides. If you have an optical system with finite apertures, you can encode a certain finite number of spatial modes given by the Fresnel number. And for temporal mode, it's the bandwidth, for example, that limits you. So it's the uncertainty principle, in other words. Why go to high dimensions? Again, there's a massive increase in information capacity. This is a small calculation. It's actually showing you that the amount of information contained in 14 qubits is informationally equivalent to the amount of information in two qubits of 100 dimensions. So instead of having to control 14 qubits, 14 ions, if you could exercise the same amount of control on two qubits of 100 dimensions each, that would simplify things quite a bit. And then you have this increase in noise resistance. That high dimensional systems offer you. I don't think I'm going to have time to talk about this today, but in a recent paper we showed that you can add a lot of noise to high dimensionally entangled states without losing the entanglement as compared to qubit entangled states. So the challenge for high dimensional quantum photonics is developing ways to efficiently transport, manipulate and measure these states, which is a fairly mature thing in qubit photonics. So let me start by introducing an example of a spatial degree of freedom that we have used in the past has been quite popular. If you solve the Helmholtz equation in cylindrical coordinates, which is great for optical waveguides and optical communication systems because they're cylinders, you get this big equation, which is a Laguer-Gauss mode. You can divide this equation up into an amplitude part and a phase part and further simplify it down to a simply a times an azimuthal phase e to the iL phi. This describes light beams with an azimuthal phase dependence shown here by very appropriately placed pasta fusilli. This has three intertwined spirals, essentially. And so if you look at these beams head on, you get a phase winding going from 0 to 2 pi here, 0 to 4 pi here, and this is 0 to 2 pi in the other direction. And the intensity of these beams, because of destructive interference in the middle, you get these donuts, donut beams. In quantum mechanics, a single photon is described as carrying an orbital angular momentum of L times h bar, so it's a very small amount of OAM. And the OAM state space can be huge, of course, limited by the aperture of your system. So there are clear applications for quantum information. You can encode a lot of information. What's interesting is you can also interact with quantum states of matter. So this is a video showing you a bunch of micron-sized particles trapped in a beam carrying OAM. You see they were spinning around. And OAM has been transferred to a BEC and retrieved from it. So there's some applications in terms of transduction of quantum states of light from light to matter. How do you make, generate spatial modes, arbitrary spatial modes? So the standard method is you put a phase profile, a transverse phase distribution on them. And you do it with a device called a spatial light modulator, which is basically just a liquid crystal display, like the one on your TVs. And so if you put a helical phase profile on an SLM, as it's called, and send in a Gaussian beam or a plane wave, essentially, you get this nice spiral phase beam out. If you add this helical phase profile to a plane wave, to a blaze grating, right? So that's a plane wave modulus 2 pi. You get what looks like a forked hologram, which is something that you'll see in many papers on OAM photonics, essentially. And what this grating does is it produces an OAM mode in the first diffraction order. So a blaze grating puts all the light into one diffraction order, and this will contain a spiral phase mode. This is an optical system that we would use. We would use a 4F system and pick out the first order of diffraction using a pinhole in the Fourier plane, and then the OAM mode is nicely produced in the image plane of the SLM. This is an example of a hologram phase and amplitude hologram that produces a nice Laguer-Gaussian mode of order 5 minus 5. And what's nice is you can use this same technique to generate complex superpositions of OAM modes, Laguer-Gaussian modes, rather, or any spatial, a complex spatial mode, in this case. And now I'll explain what these mobs mean in a bit. You use the reverse process for detecting the same modes. And how does that work? If you generate a mode and you put the SLM spiral phase conjugate of the generating phase onto your beam, it would convert it back to a Gaussian beam in theory. So the way this works is if you go in with an OAM mode of plus 3, you put in a spiral phase of minus 3, it converts it to a plane wave, Gaussian, and you focus that Gaussian down to a, and couple it efficiently into a single mode fiber. Any other mode will not couple into the single mode fiber. What does this look like in terms of maths? If you went in with mode A and you have the conjugate of the mode A star, the result is non-zero. If you went in with mode B multiplied by A conjugate, since modes A and B are orthogonal, you get zero, so no light will couple into a single mode fiber. Now, in practice, you have to add this Laguer-Gaussian term into your integral, which actually makes this work not as well. So this previously zero term becomes non-zero. And in a recent result, we showed how to take care of this with using some clever optics. Essentially, you magnify the backpropagating Gaussian beam from the single mode fiber onto the SLM such that the Laguer-Gaussian component becomes flat. And at the expense of some loss, you can have near-perfect mode measurements. And so this is without using our technique. When you try to measure radial modes, which are notoriously hard to projectively measure, you get very bad visibilities. So this is mode in, this is mode out, red out. And with our technique, we were able to increase this visibility almost to unity. So this was very useful and we've recently applied this in entanglement experiments and had great success. Okay, so moving on, yes? I'm sorry? So it's a good question, exactly the same process, right? So the mode that I'm describing here, this A, is any complex spatial mode, right? And that's a A star is simply the complex conjugate of that complex mode. So it doesn't matter what mode is going in. It could be any superposition of any mode basis. Does that answer your question? Yes, so this is... Exactly, so what this is is basically like a polarizer, right? And a polarizer, it picks out the mode you want to measure and then you can do a basis rotation with a polarizer, at least in linear. You can do the same thing here. And it's still, you can choose the basis you want to measure in, right? And it does the same thing. So I can measure 0 plus 2 or 0 plus 1 plus 2. But how do you do the rotation? It's exactly the same. So like I said, you can generate any spatial mode with the SLM. You can also measure any spatial mode in the exact reverse process. Okay, thanks. Right, so a quick recap on superposition, coherence and entanglement. You're all familiar with this diabolical thought experiment that was proposed by Schrodinger which puts a cat into a superposition of being dead and alive and opening Schrodinger's box constitutes a measurement of this state. Now, what's key here... Ah, right, and then this collapse model is... I want to point out it's just one interpretation of quantum mechanics. The fact that opening or the measurement collapses the state to either being dead or alive. There's many other interpretations, but that's a topic for another day. What's crucial to point out is the idea of coherence in superposition. So here I'm showing you superpositions of electrons going through two slits interfering or each electron interferes with itself. And the fact that the state of an electron going through a left and a right slit is described by this equation which implies that there's coherence between the left and the right probability amplitudes and not this, which would be a classical mixture. So let me just flesh that out a little bit more by the concept of a density matrix. I'm sure many of you have seen this before, but I'm going to go over it in any case. That's a state of polarization describing amplitude H plus amplitude V with probabilities A and B. Now the density matrix for this state looks like that where the first row refers to the diagonal elements of the density matrix and the second row refers to the off-diagonal elements. Now the off-diagonal elements contain information about the coherence. They tell you how coherent this state is. If the off-diagonal elements were zero, essentially drop these down to zero, this would be a purely mixed state. Well, that's a terrible term for it, a mixed state, which means that you have a mixture of H and V. So you have equal probability of getting H and V and there is no coherence between the two. How do you measure a density matrix? So the diagonals of a density matrix are measured quite straightforwardly. You simply measure in the basis that you are interested in. For example, if I want to measure this diagonal element HH, I put a polarizing beam splitter in front of my state row and the transmitted output tells me the size of my density matrix element HH, which should be modulus A squared. Now the off-diagonal density matrix element HV is a complex quantity and to measure this, you need to play a few tricks. You need to do a basis rotation. So the real part of this density matrix element is given by a measurement in sigma x and the imaginary part is given by a measurement in sigma y. So this is an experiment. It looks like this. You put a half-wave plate, you do a basis rotation and you measure in the DA basis, the diagonal and anti-diagonal basis. When you extend this to multiple qubits, two or three qubits, it becomes much more complicated and I'll touch on that in subsequent slides. So this is called quantum state tomography essentially. I just want to point out that we detailed this a little bit in a recent review that I published with some colleagues. Okay, going back to my favorite analogy of quantum states, which is dead and alive cats. So extend your imagination a little bit and imagine the case of conjoined kittens born in this beautiful city here that are separated by skilled veterinarians and put in identical Schrodinger boxes. You keep one in Trieste, send one to Edinburgh. Now the cats are of course in a superposition of being dead and alive and the state of these entangled cats would be described in the following way. You have alive, alive and plus dead, dead. This is what we're all familiar with, entanglement. What I want to point out is that this is not a causal connection. You cannot shoot one cat and expect the other one to die. The outcomes are completely random. You can open the box. So if you find this cat to be alive, the other one is alive. The story doesn't end there. If you open the box in the daytime, you can see whether the cats are dead or alive. If you open the box at night, you find out whether the cats are zombies or vampires. What's a zombie and vampire cat, you may ask? Well, of course, a zombie is dead plus alive and a vampire is dead minus alive, right? So this is a basis rotation. The whole point of this complicated story is to tell you that no matter how you measure an entangled state, it is perfectly correlated, right? So if you measure a state in one basis, you get perfect correlations. You measure the state in another basis, you get perfect correlations. So if you take a polarization entangled state, such as this, HH plus VEV, and you measure in the HV basis, you get perfect correlations. If you rotate to the DA basis, and that's my zombie vampire, right? If you plug these equations into the first one, you'll find that the state looks perfectly the same. So as Benny pointed out, we produce these types of states through spontaneous parametric down conversion. To produce entanglement, you need to play some more tricks besides just SPDC, at least in polarization. You need to produce two indistinguishable polarization pairs. And I'm just going to skip through this part. Okay, applications of entanglement, of course, are measurement-based quantum computation to do quantum error correction. And an exciting application that is more in the realm of theory currently is device-independent quantum cryptography where you can have security independent of any of the devices used. So you can literally give your entire communication system to an eavesdropper, and you will still have guaranteed security, which is the ultimate limit of paranoid communication. So you can have entanglement in high dimensions, of course. The earliest works on this are from my postdoc advisors group in 2001 where they showed entanglement of orbital angular momentum modes. So if you go in with a pump that is a Gaussian, you get out pairs of photons that are carrying equal but opposite amounts of OAM. So the OAM is conserved just like energy and momentum. You can have timebin entanglement where essentially pairs of photons are produced in correlated timebins. And of course, you can have combinations of these different degrees of freedom, which is called hyperentanglement, which is an example would be polarization, tensor, three dimensions of OAM, tensor, two dimensions of time. The corresponding density matrix starts looking like a small city. Okay, so pausing point. Are there any questions about the introduction? Yes. I'm sorry? OAM. Oh, we don't actually do macroscope. Do you mean like how do we... So was that theory, like the stuff you showed on dead and alive and zombie and vampire? Yes, that's just a metaphor for polarization, for example. Okay, I thought you implemented that. I mean, I do have colleagues who are doing this thing with producing superposition states of viruses going through slits. It's getting close to dead and alive, but is a virus alive? Not sure. Yeah. Yes? Are all of them have the same probability in the first line? No, that's a good point. So when you pump it... Well, it all comes down to phase matching conditions, right? And you saw all of that in Benny's talk. It's the same idea. In this case, you get something like a Laurentian due to the phase matching conditions, but people have played around with engineering the process such that you can have near-flat SPDC, essentially spatial modes, equally distributed. This is just... I've even skipped the normalization here. It's just to give you an idea of a maximally entangled state, but no state is actually maximally entangled. Okay, moving on. So how do you measure high-dimensional entanglement? So I talked a little bit about tomography, and if you can reconstruct the state perfectly, you have the density matrix, and the Schmidt rank, K, tells you how large your entanglement dimensionality is. What is the Schmidt rank? Again, Benny talked about the Schmidt decomposition. The Schmidt rank is a number of terms, non-zero terms that appear in a Schmidt decomposition of your state. And in other words, it's the minimum number of levels that you need to represent any given state and its correlations in any local basis. So if you do a Schmidt decomposition of a state, it's diagonal. It's expressed as a product of two terms. The number of non-zero coefficients, lambda m that appear in this decomposition, tell you its Schmidt rank. The Schmidt rank is also given by the... essentially the rank of the reduced density matrix of any bipartite state. So in this case, this is a bipartite state. It gets a little bit more tricky with multi-partite states, but again, we discuss this in more detail in this review. A quick toy example of entanglement versus classical correlations. So this is a perfectly classically correlated state in three dimensions. So you either get a 0, 0 or a 1, 1 or 2, 2 with no coherence. If you did the calculation on this, the Schmidt rank of the reduced density matrix of this state would be 1. If you construct a state made up of essentially three bell pairs, where each of these individually would have a Schmidt rank of 2, so they are two-dimensional, if you constructed a state that is a classical mixture of all three of these bell pairs, this would also have a Schmidt rank of 2, even though it looks like there are three levels involved. It's a typo there, that should be 0, 3. Oh, sorry, 1, 2. And this is a maximally entangled three-dimensional state, and again, would give you a Schmidt rank of 3. So how do you certify entanglement dimensionality? Is there a question? Yes, exactly. Actually, that's a very relevant question. What you do is, it's called, well, so there are many bipartitions involved. Let's take a three-party system, for example. So a three-party system, ABC, has three possible bipartitions. You can look at the partition A with respect to BC, B with respect to AC, and C with respect to AB. And in general, these can have different Schmidt ranks. So one way to describe this is with what's called a Schmidt rank vector. So you need three numbers. No, well, you could, but why would you? Why not just use a vector? So I'll talk about this later. We actually created a state with different Schmidt ranks across different cuts. And as the parties grow, then the vector becomes bigger. And there are certain relationships that exist between what is allowed, between the Schmidt ranks. The generalization of the Schmidt rank to mixed states is called the Schmidt number, where you have to sort of maximize over all possible pure state decompositions, but I'm not going to go into that. So to certify dimension, entanglement dimensionality, you can either do full state tomography, and this takes ages. So for example, in Bob Boyd's group, they did full state tomography of an eight-by-eight dimensional state, and this scales as d to the four, so many, many measurements required, and it took them about two days to measure this state and two days to just do the computational analysis. So when you do tomography, you have to do sort of state reconstruction, and that takes a lot of time when the state is very big. You can do a bell-like test of non-locality. This was done in 2011 on an 11-dimensional state of OAM. Now, this is a very strict criterion. You need to have an almost perfect state, and then in this case, they made some assumptions on the state in order to certify the entanglement. There are other techniques developed recently, including one by myself, where we essentially calculate the fidelity of a state, but we do this by looking in two-dimensional subspaces of a high-dimensional space, and this scales as d to the cube. This similar witness also scales as d to the cube. So they scale really badly. It takes a lot of measurements at best d to the cube for high-dimensional states. So what we were really looking for was a way to certify entanglement dimensionality that was fast and did not make any assumptions on the state. So this is very important for quantum communication, because if you want to do secure communication, you cannot make any assumptions on your state, and you don't have a lot of time. So with that, let me introduce the concept of mutually unbiased bases. For polarization, you're familiar with the different mutually unbiased bases. Essentially, a mutually unbiased basis states from any two mutually unbiased bases have the same overlap. So for polarization, you have three possible mutually unbiased. I'm just going to call them MUBS, three possible MUBS. You have the HV basis, the DA basis, and the LR basis. This can be generalized in high dimensions and has been done in this paper by Bill Wooters and Brian Fields in 1989. Here I'm showing you MUBS for 11-dimensional OAM mode. So if I take my 11 OAM modes from minus 5 to plus 5, I add them together. I get one possible MUB is this, another one is this, another one is this. Now, you can almost see how this would make sense. This is OAM, and this is its Fourier basis of angular position. So if you go from angular momentum to angular position, you get modes that are localized. And because I have a finite bandwidth of 11 modes, my angular position is not infinitesimally small but has some side lobes. This is what the construction of MUBS looks like. The jth mode in the kth basis, where omega is the dth root of unity. And of course you can have an infinite number of constructions of MUBS. This is just one nice one that works. So we developed a witness that uses MUBS to certify entanglement and I'm going to try and go through it quickly. This is what the experimental setup looks like. I'll just describe it really quickly. We produce a two-photon state in a crystal, a nonlinear crystal. And we separate it with a polarizing beam splitter. We use spatial light modulators to do projective measurements of the state. So any matrix that you see here, this is not a density matrix. This is in fact a matrix of data, which is telling you how many counts do I measure. Ooh, seems to be stuck. I'm still not used to this fancy laser pointer, a laser-free pointer. So any element here is telling you how many counts do I measure in a joint measurement of mode M on one side and mode N on the other. And this is coincidence counts, two-photon counts. So how this witness works is first we measure two-photon counts in our Schmitt basis. So in this case we know our Schmitt basis is the OAM basis. The state is expected to be diagonal because of conservation of momentum. And with this we can recover the diagonal density matrix elements, Mn, Rho, Mn. And this allows us to sort of construct a target state. So we calculate the weights of all of these elements, and we nominate a given target state phi. So this target state then looks like sum over mm with these weights that we've calculated from our non-zero diagonal density matrix elements. So every element in this matrix is a diagonal density matrix element. So this density matrix is huge. In this case it's 11 by 11. Oh, 13 by 13, actually. That's the number of coefficients. In an ideal case is your Schmitt number or Schmitt rank. So the way this witness works is you want to calculate the... So you measure a state row, experimentally measured state row. You want to calculate the fidelity of your state row to your target state phi, which is given simply by the trace of that term there. And the key point is this. For a state of rank k less than or equal to d, the fidelity to the target state is upper bounded by this term Bk, which is given by the kth largest Schmitt coefficients squared summed. I know that's a lot of information, but I'll give you an example. So if your state fidelity is greater than B3, which is basically the first, the three largest Schmitt coefficients squared summed, and is smaller than the four largest Schmitt coefficients squared and summed, you can say that your entanglement dimensionality is four. So all you have to do is somehow find a way to calculate this fidelity if you want to find out what the entanglement dimensionality is. So how do you calculate the fidelity of a state? You can split up the fidelity into two terms, f1 and f2. f1 contains all the diagonal elements of the state, and f2 contains all the off diagonals. I already showed you how to measure the diagonal elements in the Schmitt basis, but to measure the off diagonals, as I showed you in a previous slide, you need another basis. Now, this is a challenge in high dimensions because all efforts before have done this by kind of going back to two dimensions, looking at two dim subspaces and measuring things in two dimensions and rotating bases in two dimensions. However, it should be obvious at this point that you can do this with mutually unbiased bases. So what we do is we take our lambda m's that we've measured before, which are the Schmitt coefficients of our target state, and we construct a new basis, which we call the tilted basis. It's kind of like the mubs that I talked about earlier, except they take into account the fact that the state may not be maximally entangled. It may have some kind of, the question you asked me earlier, it may have some kind of distribution that is not flat. And so what that does is it introduces this lambda m here, right there, and then this normalization, instead of having one over root d, you have one over this term. And then you simply measure in this new basis, using the spatial light modulator method that I showed you before, and then you get a whole bunch of counts. This is actual data measured in this tilted mob-like basis of OAM. And then I will not go through this expression to save you, spare you the misery of explaining all the details, but what I will point out is that essentially it contains measurements in the Schmitt basis, it contains measurements in the tilted basis, and then the crosstalk in the Schmitt basis is subtracted from this term to lower bound the fidelity of the state here. So that's the crucial point here. So with measurements in these two bases, you can lower bound the fidelity to your target state, yes. Basis, how you made the experimental setup to... My question is, when you have defined this new basis, how you experimentally made the setup in order to define experimentally that basis? You mean to measure this basis? Yes, indeed. Right, actually that's related to the question by the gentleman earlier. It's the same thing. So you can use the spatial light modulator to construct any mode, including any mode in this basis. So in an ideal case, you can produce Laguerre Gaussian mode, but you can also produce superpositions of Laguerre Gaussian mode with whatever weight you want. So we use the exact same hologram to measure this basis. Right, so it's exactly the same as before. Right, so using measurements in two bases, I can lower bound the fidelity of my experimentally measured state to a target state. And with this, I am able to then calculate the bounds and calculate how large my tanglement dimensionality is. So we did an experiment on this, and we were able to certify entanglement dimensionalities of 9 in 11 modes, which required a fidelity of about 75%. And at the time, it was the largest assumption-free entanglement dimensionality measure. And you see, when we assume a maximally entangled target state, we get slightly lower fidelity than the tilted state, because our state is not perfectly flat. It has these slight drop in the diagonal elements as you go to the high-order modes. Now, what I want to point out is that how this method scales when compared to tomography and previous techniques. So for example, if you use single outcome measurements like the kinds we do with spatial light modulators, you see that the number of measurements falls from D to the 4 to D squared. So for our state, we needed 242 measurements as compared to 17,500 for tomography. But when this method really shines is when you have access to a multi-outcome device, like a polarizing beam splitter, which does exist for spatial modes, but it's still sort of getting to the point where we can use it in experiment. It's a multi-outcome sorter that takes OEM modes and decomposes them to plane wave modes. But the point is, the main point is that the number of measurements becomes independent of dimension. So you can essentially measure any size state with only two measurements. This is a picture of the people involved in this paper. So that's Marcus Huber, whose group I was working in at the time, and there's a whole bunch of guys here who are theorists pretending to be experimentalists. And we built a box around the setup. The primary purpose of the box was to stabilize the experiment in temperature, but we would joke that it's a theorist-proof box because the theorists would come into the lab and try to touch the things, and it would mess up the alignment. But it was really great fun actually working closely with the theory group. So recently, my group has extended this in other spatial mode bases. We work in these macro pixel bases, and this is showing you data in 20 different mobs of a 19-dimensional pixel mode space. And with this, we've been able to certify, including other methods that I haven't really talked about, 94% fidelity in this 19-dimensional space. We've gone up to 51 dimensions, and we have about 40 dimensional entanglement and 50 dimensions, and we're working on the paper. Okay. Right. I should point out that this was work done by my PhD student, Natalia. So I think what I'm going to do is I'll stop here, and I will take questions if you have any. Five minutes.