 Rotates around the axis. And all of them just rotate around. In every basis, the modes rotate. OK. So just as a reminder, this lecture is going to be, I'm going to touch on a recent experiment we did where we sent this high dimensional entanglement that I've been talking about through a complex medium, in this case, multi-mode fiber. And then I will start to tell you a little bit about multi-photon entanglement in high dimensions. So more than two. And if time permitting, a few slides on free space quantum communication and classical communication. OK. So let's start with unscrambling entanglement through a complex medium. So with quantum comms, I like to think that we are at an early stage harking back to the days of classical communication when we struggled with things like storms and bad detectors and things. And so we have to overcome similar problems with quantum communications. For example, dealing with noisy channels is a huge issue. And a complex medium, essentially a scattering medium, acts as a noisy channel. So what do I mean by complex media? This is very trigger happy here. OK. So anything can be a complex scattering medium, from paint on a wall to your teeth to sugar cubes. Essentially, modes go in. They mix up, scatter in some ways, and come out in a different basis or with loss. A multi-mode fiber is a more controlled complex medium in that it's fairly unitary. There's no loss, little loss. But the modes go in and come out, all sorts of mixed. If you've ever seen light coupled into a multi-mode fiber, it comes out with a speckled pattern, laser light. And so there's a lot of effort being made in trying to overcome this scattering for things like classical communications. So SDM is space division multiplexing. We are at the stage where the classical data crunch is happening, and we want to push more and more information so that we can all watch YouTube videos of cats doing funny things. But one of the ways that people are going to do this is by our doing it currently is by putting lots of spatial modes into a fiber. And people have reached a terabits per second by space division multiplexing. And this was a work done by some people in Scotland, actually, at the time, where they figured out how to send images through a 30 centimeter section of a multi-mode fiber. So that's real data that you're seeing there. And you can see as the fiber is bent, the image stays the same pretty much. And so we wanted to do something similar in the quantum regime. So I want to talk a little bit about how people do this classically. So a complex medium is described by a complex transmission matrix, which is basically a set of complex coefficients that map a set of input modes to a set of output modes. So a lens has a very simple transmission matrix. A sugar cube would have millions, perhaps more complex coefficients that make up this transmission matrix. So using this device that I've been talking about, a spatial light modulator, people can map or measure the transmission matrix of a complex medium quite simply by sending in a basis of probe states. So they prepare a set of probe states, I, and measure a set of output states, J, which then allows them to retrieve the complex coefficients, Tij. Now, of course, there's some phase retrieval involved, because measuring J here, you would measure the intensity. So in order to get the phase of the Tij, you would have to do some additional work. But in the end, you get something like this. Of course, I'm showing you an absolute value here. And this is all very nicely described in this early paper by Sylvan Giga. Once you find the transmission matrix, you can construct a set of eigenmodes, which is essentially equivalent to a singular value decomposition or whatever you want to call it. But basically, you find the inverse of the transmission matrix, and you can either prepare a set of eigenmodes or you can rotate, prepare a unitary transformation after the complex medium to basically send information through this scattering medium. So let's move to quantum mechanics for a second. In quantum mechanics, we have this thing called channel state duality, which basically says that statements about a state can be mapped to statements about a channel. So if you imagine this scenario where you have a maximally entangled state, and now you should be fairly familiar with this ii1 over root d state, and if you put one particle of this maximally entangled state through a complex medium or a channel in this case, essentially the resulting state rho captures all the information about the t. So the maths looks like this. You have rho equals identity tensor t. Identity acts on one particle, and t acts on the other particle, and the resulting state rho then encodes t, essentially. This is called the Choy-Jamiolkowski isomorphism. I'm sure I'm still saying that wrong, because there's a slash across the L, and I know that's not an L. So proposed independently by these two brilliant theorists in the 70s, and so we're going to use this interesting feature to recover our entanglement. So I told you how one can use a set of probe states to measure t. What we do instead is we send one entangled state, or prepare one entangled state, and we send one particle from the state through our complex medium. And so the resulting state phi t, now I'm assuming it's a pure state here, encodes all the information about the channel. Now you can do a few things. If you want to recover your input state phi plus, you can prepare an inverse t inverse and apply it here, right? Or, which would then diagonalize your channel, and you would essentially recover your correlations. However, there's an interesting property of maximally entangled states that allows you to do something interesting, perhaps clever. If you take two operations on a maximally entangled state, you can move the operation on one particle onto the other particle or vice versa. So A tensor B can be written as B transpose. It's actually complex conjugate transpose. A tensor I, or I tensor B A transpose. What this allows us to do is instead of unscrambling Bob, you can scramble Alice and recover your state. So even though Bob's photon went through a scrambling channel t, you can put a scrambling operation on Alice and you will get back your entanglement. This is a unique feature of entanglement in that the entanglement doesn't actually go away. It's just rotated in funny ways. So if one photon is messed up, you mess up the other photon in a clever way and you get back your entanglement. Okay, so let's start with the initial state. As before, we have a crystal pumped with a carefully shaped pump laser and then we measure correlations between Alice and Bob. In this macro pixel basis, this is one basis. This is the mutual and bias basis. We measure very nice correlations and this allows us to certify a fidelity of 94% or so in this seven-dimensional mode space which says we are entangled in seven dimensions. Now, when we put one of the photons through multi-mode fiber, the following happens, right? The data looks terrible. You have no correlations. But if you think back to what I told you about the channel state duality, you remember, we realized this later actually, this data that doesn't have any entanglement actually encodes information about the fiber in two different bases. And so the correlations that are lost actually tell us what the fiber is doing to some extent, right, absolute value. So what we need to do is actually recover the entire transmission matrix from this scrambled data. And the way we do this is we prepare a reference mode that copropagates with the state going through the photon, going through the fiber. It doesn't matter on which photon you have the reference. It's a two-photon state. Whatever you do here can also be done here. So we use the space between our pixels as a reference mode and we phase-step the reference. And without going into too much detail, we recover what we call the S matrix which is the T matrix of the fiber times some diagonal matrix that we need to recover. And then we recover this diagonal matrix through another reference mode process. And with this information of the S and the E, we get our complex T. Okay, so the T looks like this, a real part, imaginary part, and if you remember, I guess I could go back a few slides. It's quite similar to this, right? Looking similar at least, that. And so what we do is we take this information of our measured T and we prepare a new basis on Alice's side. So now I'm essentially messing up Alice in a clever way such that the photon that's going through the fiber comes back to being maximally entangled. And this is what we see. We see this messed up data is now perfectly correlated again, which was really nice. And this fiber is actually just a fiber we bought from Thorlabs, right? Two meter long, just a commercial multimode fiber that supports a few hundred modes. So we are kind of only looking in this seven-mode subspace of the fiber. So this is not enough to certify entanglement. To certify entanglement, as I told you before, you need at least two bases. So we take the information about our transmission matrix and we rotate it using some mats to access the transmission matrix in all the other mutually and bias bases of our seven-dim mode space. And doing that, we find we can get correlations back in all the eight mobs of this pixel mode space. And with this, we can certify a fairly high fidelity that tells us we have six-dimensional entanglement. Oh, that was too quick. So I just wanted to point out that this work was done in collaboration with a colleague of mine in Glasgow, Hugo Defian, and then of course people from my group. And we had a summer student visiting who's actually starting his PhD with me in September. Okay, any questions about that? I know I've kind of rushed through that a little bit. Yeah, so the question was, did we have any problems of scattering or absorption in the fiber? We lose modes to other higher order modes, right? The fiber can support a few hundred modes, like I said. So light does scatter out. So we are basically only looking at the light, the photons that are left in our mode subspace. The idea eventually is to try and access the entire mode space of the fiber. In fact, it's not that big at telecom. It's about 45 modes for a standard like grin fiber. So we do lose photons, but it doesn't hurt us. As long as it's, as long as the loss is kind of flat across the spectrum. How bright is it? You mean how many counts do we have? Oh, it's two photon entanglement. Yes, yes. And what's the B3 for like countering? I think we had a few thousand counts per second in one mode. Okay, so I want to move on to three qubits. Essentially tell you about how do we entangle more than two particles. And I'm gonna start with a broad sort of introduction about multi photon entanglement and why it's important and how do we do it. So if we think back to like the first days of entanglement, you know, we've all heard about the EPR paradox and the EPR paper. But so Einstein, Podolski and Rosen proposed this idea which, funnily enough, didn't get much mileage. I think it had like four citations until the 60s. But they were not bad citations. One of them was by Bohr and the other one was by Schrodinger. So good citations to have. But then in the 60s, John Bell came about and proposed a test of entanglement, essentially, which we all know as Bell's inequalities. And this was a statistical test of quantum mechanics that allowed you to make a series of measurements on an entangled state that ruled out any local hidden variable theories. Essentially, that said that the correlations we are seeing cannot be nature playing tricks. It is inherently non-local or you have to give up either locality or reality. Not going to go into too much detail on that subject. The main point is that the main point is that it was a statistical test. So in order to measure Bell's parameter S, you had to make a series of measurements on both particles that allowed you to then reconstruct this parameter which then tests whether nature is acting in this manner. So in the 1980s, early 90s, Greenberger-Horn and Seilinger, seen here in order, Greenberger-Horn, Seilinger, proposed a new test of non-locality that is a non-statistical test. Essentially, it's one measurement that tells you, it's called an all or nothing test. So it rules out this idea of having to do a series of tests on entanglement that tells you whether nature behaves in this strange manner or not. And the main point is that it required three particles, three or more particles to be entangled. And so here I'm showing you a three-part type G at Z state. And then Merman proposed an equality just like Bell proposed a test of this two-photon or two-particle entanglement. Merman proposed an inequality which required you to measure joint observables on all three particles. These are essentially sigma X and sigma Y. And if you measured a Merman parameter between zero and two, then it was explained by local hidden variable theories, measuring it between two and four implied that nature behaves in a non-local manner. Okay, so what are the current records of multi-partite entanglement? This is possibly a little bit outdated because every year, Janhwe Pan's group adds a few photons and possibly a few students and postdocs to the experiments. And so 12 photons have been entangled in China. People have entangled up to 20 ions, although I say five in 20 because they only showed entanglement between any five in 20 ions in Innsbruck. And people have entangled again in China up to 20 superconducting qubits. Although Google may have done more, but we don't know. Right, so all of these describe a state such as this where you have zero tensor N, right? N times zero plus one N times one G at Z. What about high dimensional multi-partite entanglement? People haven't really explored this and I'll tell you why. So the simplest extension is of course, you take your two dimensional G at Z state and extend it to three dimensions, right? So you have a, I call this a 333 state and I'll tell you why. Or what happens is, well okay, I'm describing these states in terms of these sets of states. So here's my set of 222 states and here's my set of 333 states. When you go to higher dimensions than two and more particles than two, a funny thing happens and actually this is going back to a question somebody asked me earlier about Schmidt rank being different or Schmidt rank for multi-partite states. What is a Schmidt rank for multi-partite states? It's described by a Schmidt rank vector and this in fact is the Schmidt rank vector. So the Schmidt rank vector is the rank of every bipartition, so if I look at the bipartition of this particle A with respect to BC, the rank is three and so on and so forth. So this is an example of a state where the Schmidt rank vector is not symmetric. So it's a 332 state and quite simply put, one of the particles lives in a two-dimensional space while the other two live in a three-dimensional space. And this is in the middle of these two sets of states. That's basically explaining what I just said, the Schmidt rank vector essentially. Now there are many states possible, right? There's 332, there's 422, there's a vast family and they obey a certain relationship which again is all explained in the reference in the earlier slide. So how do you create states such as these? So this very nice paper from the mid-90s, in fact I asked Anton who came up with it, he's like, oh, who's Marek? Marek Zrakowski is a senior scientist in Poland and so this is a very nice idea of basically producing multi-partite entanglement from erasing information, right? So what you do is you have entangled photons produced in two different crystals and you combine them in such a way that you cannot in principle know which crystal they came from. I can show you how this works for polarization. So a PBS polarizing beam splitter, as we all know it transmits horizontal, it reflects vertical, right? If I come in with another photon from the other side, it will do the same, transmit horizontal, reflect vertical, right? So it mixes polarization components of two input photons. Now what happens when you send an entangled state through it? So particle two will go through and particle three will go through the PBS. There are few possible outcomes. Either horizontal from particle two is transmitted and transmitted from particle three, which then imposes the condition that particle four must be H and particle one must be H because of the entanglement, correlations currently, right? Or particle three is reflected and particle two is reflected and they're both vertical, which then imposes the condition that particle one must be vertical and particle four must be vertical. So post-selecting on these four detectors clicking together, these are the only two possible outcomes. You either have H, H, H, H or V, V, V, V, right? And this is your four photon G at z state. Now if you remember back to my last talk, I was talking about coherence, the importance of coherence in entanglement. And what I have described so far is just classical correlations, right? You either have all H or all V. This plus sign here is crucial. And in order to certify that you actually have coherence, you have to do a lot of things. You have to make sure, well, okay, I'll tell you on, but the test of that is you have to do a basis rotation and look for correlations in the other basis. Right, so this setup that I just showed you is the one that's basically been extended in these 12 massive photon experiments, 12 photon entanglement experiments in China. You can almost see one crystal cascaded onto two, three, four, five and so on and so forth. So why were people able to do this for polarization? It's because there are tools that exist for polarization like the PBS, right? And also half-wave plates, which can do basis rotations from horizontal vertical to anti-diagonal. So the challenge in doing this for other degrees of freedom or higher-dimensional degrees of freedom is to create beam splitters, essentially multi-port devices that can do the same thing for spatial modes or temporal modes for that matter. So such a device does exist. In fact, it was invented back in 2002 in Miles Paget's group and Ioannis Quartiel's group. It's a Mach-Zender interferometer and it consists of two dove prisms in each arm, one rotated with respect to the other. And a dove prism is a device that basically is an image rotator. So if you go in with an image, it rotates it by twice the angle at which the dove prism is oriented. Now, thinking back to your Quantum Mechanics 101, rotations and angular momentum are connected. So generator of rotations. So when you rotate a mode carrying angular momentum, you also impart a phase that depends on the angular momentum. So rotating this mode here gives you a phase of pi. Rotating this mode here by 90 degrees gives you a phase of three pi, which is pi modulus two pi. So you can almost see what happens. Odd modes interfere destructively. Even modes interfere constructively so that this device sorts spatial modes into even and odd values. And you can scale this up to many modes. It's just not very practical. So this is an example of a multi-outcome mode sorter for OAM, but not very practical because you need cascaded interferometers. So this is a picture of the device made in my lab. You can see the dove prisms on rotation stages, beam splitters, and this is data going in with these two modes gives us even, odd, and going in from this way gives us odd, even. So it was proposed as a mode sorter, but the key step to using it for entanglement experiments was to use the other input port, right? If you use the other input port, it's a mode combiner and a mode mixer, right? So let's see what happens if we put this device in the middle of our high-dimensional entanglement crystals, now two crystals, each producing a three-dimensional state. This is one minus one zero zero, minus one one, and the same thing here, right? There's a few possibilities. Oh, and I'm going to use the fourth photon as a trigger to herald the presence of my three photon entangled state. So one option is that photon C is transmitted through this, let's imagine this as a big beam splitter, and photon B is transmitted in mode zero zero, so even modes are transmitted, which then would herald photon D in being in mode zero, or I have odd photons reflected, so mode one B is reflected, mode one C is reflected, which then, or mode one B is reflected, and mode minus one C is reflected. Now remember, this is the herald, so there is no mode minus one B, right? And measuring these modes here imposes a condition that you have minus one and one over there. So these are only three possibilities. Combining them, and looking a bit more closely, you find that this is exactly this 3-3-2 state that actually, when we were not planning to create this, funnily enough, we were trying to create a three-dimensional GZ state, but this is one of those good cases of happenstance. So you'll notice that you have one, one minus one, zero, zero, zero, one minus one, one, photon B is in a two-dimensional space, zero and one, photon C and D are in a three-dimensional space. Now the key question is, is this in a coherent superposition, or is it just a classical mixture? The way you test for that is through two photon interference. In fact, it's four photon interference, but again, Benny showed this in his talk just to remind you. Hongo-manual interference, if two photons arrive at a beam splitter, there are four possibilities. They are either both transmitted, both reflected, and one reflected, one transmitted, and the same for the other case. And again, these two possibilities destructively interfere, they cancel out. So the only thing left is both photons going this way, or both photons going this way, this is known as bunching, photon bunching, and it's particular to bosons. Bosons like to stick together. And if you scan one of the paths here, you'll find this dip, right? And then the dip is not always perfect, and that's the key point here. The key point is that the dip not going down to zero depends on how distinguishable the two photons are. So if this photon looks a little bit different from this one in any particular degree of freedom, spectrum, polarization, spatial mode, you will not get very good interference, and this is essentially what then limits the coherence of your state. So we have to now look for Hongo-manual interference in a spatial mode basis. So what we do is we measure in a superposition basis using spatial light modulators, and we find that we, so this is again the basis rotation I was talking about, and we scan one of these path lengths, and we measure for a few hours, right? We see nothing. And then we run the experiment a bit longer. This takes a lot of patience by the way. After two days, you see a dip appearing, and it's like, oh my God, is this working? So after two days, we saw a sign that, okay, I've marked it as 15 now, but it was some arbitrary position. So that tells us that this is the point where we have the perfect sort of condition of alignment and distinguishability. It's not perfect perfect, but it's pretty close. So once we see this, we sort of have a sense that our state is in a coherent superposition. The setup, actual setup looks much bigger. It spans a big table, and if I can show you in some more detail, it consists of a pulsed pump laser, which is then frequency doubled to blue in a BVO crystal, and then the blue laser, UV laser, pumps two PPKTP crystals. The detection is done with spatial light modulators and intensity and single mode fibers. And then we have to be very careful about imaging and mode matching. So we have imaging systems everywhere. And this Mark Zender that I showed you earlier, the OEM beam splitter, had to be designed in such a way that it could be stable for several days. So we designed it in a folded Saniac configuration. So this is a Mark Zender. It doesn't look like one, but it is. And it's sort of a double path Saniac interferometer. So any changes in path length are sort of self-compensating. Lots of lenses, so anyway, you get the picture. It took us a year or so to get this to work. So now that we have this condition of perfect coherence, hopefully, it doesn't prove that our state is entangled in 332 dimensions. It just says that we have some coherence in some subspace. So to verify that the state is actually entangled, you can reconstruct the whole state. But this would take ages, especially since our count rate is so low. So we need an entanglement witness that can basically prove that the state that we measure cannot be decomposed into entangled states of a smaller dimensionality structure, essentially ruling out that our correlations could have come from 32222 or biseparable, bipartite states. So this is similar to the witness I told you before, but it's a little bit simpler and not simpler in the sense, but it's measured differently. It's a fidelity witness, excuse me. It's a fidelity witness, but it measures the fidelity not with mobs, but with two dim subspace measurements. So the way it works is you measure a state and you calculate, well, before you measure the state, you take an ideal state from the space of 322 states and you calculate the fidelity of the ideal state, 322 with the ideal state 332, and you find this to be maximal value of two thirds. And then you take your experimentally created state, which hopefully is here, and you calculate the fidelity of your experimentally created state with an ideal state. To do this, you need to measure certain number of density matrix elements. Ooh, right, 18, well, okay, I kind of went too quickly. Right, so you need to measure 18 diagonal elements and three off diagonal elements. The density matrix of this state has 171 elements. The three diagonals and the three off diagonals that we expect to be non-zero are the following, right? And in the lab, as I've told you before, you can only do projective measurements. So the diagonal elements are measured straightforwardly. You just put 0, 0, 0 on all three SLMs and you get this diagonal element. For the off diagonals, as I explained before, you have to break them down into measurable quantities. So I will not go into too much detail, but just an example of the real part of one of these off diagonals requires you to measure this monster, which can then be decomposed into projective measurements in two-dimensional subspaces and so on and so forth. So just the real part requires 32 measurements and the imaginary part requires another 32 measurements. So each off diagonal requires 64 projective measurements. If you multiply that by 171, you can see where this is going. So that's why tomography is prohibitive, but the witness only requires us to measure three, right? So let's go back to this witness. We took data for four days and we find that our expected density matrix elements look like this. The off diagonals are fairly sizable, which means that our state is coherent in these probability amplitudes and we get a fidelity of about 80%, right? Which means that our state is above this bound of two-thirds and hence it can only be explained by entanglement in three times three times two dimensions. Okay, so what do you do with a state like this, right? It's beautiful in my opinion. I don't think it needs to be used for anything necessarily, but we like to get money from the government. So one possibility is to do layered quantum cryptography. Essentially, the idea here is as follows, you can have secure communication where three parties share multiple layers of information. So Alice and Bob can share, have an extra key that is not known to Carol. So Alice and Bob could share a message, sorry, Alice and Bob and Carol could share a message and then Alice and Bob could share a second layer of message or key that is not known to Carol. That's just one sort of cute idea. So what about three-dimensional G at Z, right? That's still not, we're still not there yet. We searched for a while to get to this state. We tried on paper to figure out different setups and how do you actually produce a state that looks like this to no avail. Until we decided to try a different approach. This was a while ago, it was three, four years ago now, where we said, why don't we get a computer to do it, right? Clearly, our brains are not up to the task of combining lots of optical elements and seeing how modes mix. So we took a toolbox of elements that we programmed. This is a beam splitter, this is a mirror, this is a spiral phase plate, this is an OEM beam splitter and we told the computer to create random setups. Calculate the resulting quantum state, analyze its properties including things like its dimensionality of entanglement and we also use it to come up with ways to do unitary transformations. If it satisfied the criteria, if it didn't satisfy the criteria, start again with a random setup. If it did, try and simplify the setup and then finally learn the setup and go back to the toolbox. So basically save that setup in the toolbox and use it to create more random setups. So while looking for the three, three, three state, we found that not only did we find the three, three, three state, we found about 51 other high dimensional entangled states or ways to create them. So that's what the three, three, three setup looks like. It's really simple in the end. We're like, why couldn't we figure this out? And that's what a 10, 6, 5 setup looks like. So that's a spiral phase plate that shifts the mode by minus five and this is minus two. This is an OEM beam splitter, this is a regular beam splitter. These ones were states that were not found and these are states that are not allowed to exist. So this is what the setup looks like. We got it to work after another huge effort in the lab because now it involves hongomental interference not only here, but also here. So it's quite a challenge to actually make it work. And this is where actually going back to Benny's talk, you realize that you don't necessarily want to keep doing this with bulk optics, which is why we are trying to do other things. So again, I'm not going to explain every detail about this setup. If you're interested, you can look at the paper but just to give you the gist of it, we start with two entangled states and here I'm showing you kind of a graph that shows you that you start with a tensor product of these two entangled states where you have a connection between every possible probability amplitude and then this part of the setup removes certain connections and then finally this part of the setup results in this state which looks like a three dimensional G8Z state and actually this will tell you why we couldn't find it because we weren't looking for a state like this. We were looking for a symmetric state and the computer, the condition we gave the computer was find us a state that's three dimensional entangled but we didn't say it must be symmetric, right? So the computer found a state that was highly asymmetric but each particle lived in a three dimensional space. So minus three, minus two, one, minus one, zero, one and the same for the third. This experiment involved multi mode in harm interference between several different modes. I'm not going to go into detail here and the actual setup looked like this, right? It was two main components. We had to develop a much brighter source of four photon entanglement, four photon correlations and then this massive 27 dimensional multiport essentially. 27 because it was three OEM modes raised to the power of three paths. So we verified three, three, three entanglement in the same way as before except this time it took 21 days to measure and we also did these Merman non-locality tests between in each two dim subspace and we found that there was entanglement across in every two dimensional subspace. The people who did this was Manuel who recently finished his PhD or is almost finished with his PhD with Anton and Mario Cren who is in Canada now. Okay, so that brings me to the last few slides of my talk which is we're doing all this stuff with spatial modes. What about actually doing some real experiments outside the lab where we try to send spatial modes of light across real distance and see whether you can actually use them not only in fibers but actually to communicate with satellites let's say or earth links. So this is a photo of myself and a colleague. We're catching an OEM mode on the rooftop of our building in Vienna after it propagated for three kilometers over the city and we were trying to guide, at the time we were trying to guide it into the telescope and I'm on the phone actually telling my friends to move the mode a little bit to the left, a little bit to the right and this was the experiment you can see the beam going from this one old radio tower that was not being used where we set up the sender and we sent it across to the institute building, very simple setup and with this we were able to send basically information encoded in classical modes and then we took this experiment to the Canary Islands and to see how long could spatial modes of light survive over the ocean. So 143 kilometers, this is a long exposure. I took 25 seconds of Hermit Gauss mode. So it's a superposition of plus and minus one OEM or Hermit Gauss mode 01, propagating 143 kilometers to the island of Tenerife and actually you can even see the Milky Way. It was fun, I would say, but then a storm came and then everything was gone. This is what the mode looks like after propagating 143 kilometers, it's huge. So again, not necessarily practical, but this is the mode on the wall of the telescope, right? So we're beyond the point of trying to get it into the telescope but we have it on the telescope and we encoded a simple message and it was received with a slight grammatical error. So what are the open problems in this field of high dimensional quantum photonics? One is manipulation, we need to figure out how to do simple things like half wave plates, quarter wave plates in high dimensions, rotations. We've made some strides towards doing this but there's still a lot of room for improvement. How to generate them and measure them, both the theory and the experiment, transporting them. I showed you some things about transporting spatial entanglement through a fiber to couple them with quantum states of matter and finally the theory of how to measure entanglement, how to certify high dimensional entanglement both for non-locality as well as things like steering. So these are various things that my group works on right now. Okay, any questions that bore everybody in the end?