 Which unit does it matter? It doesn't matter because minus 40 is the same in both units. Yes, do the math. No, in Fahrenheit or Celsius, minus 40 is minus 40. Fun, do the math. Okay, so I hope that it was not tiring leisure. Now let's go back to our conversation. Okay, and the discussion that we had. So we learned how to generate the state, how to have the source, and we know how to detect them and perform the measurement, essentially. Okay, and we have a channel which I have no clue what's going on with the channel. Essentially, in the rest of my presentation, you will see a chart which is very common in the work of quantum cryptography or quantum key distribution. And right now, I want to just clarify that for you for the rest of the talk. So what is happening, you will create a state which we talk about that. It can be either in H, V, A, D, right? H, V, A, D. You will create them and I keep the color because they are in the same base, right? And then Bob performs the measurement on H, V, or A, D. True. What is the result? In theory, if Alice is sending an anti-diagonal, sorry, is that diagonal? Is diagonal, if sensing a diagonal and Bob measuring diagonal, you will see probability of one, right? For the rest will be one-half because it's equal superposition of the two. And the same story happens here. It will be one-half, here will be one, and etc. So you will get those numbers based on the measurement. We call it detection probability density. Detection probability density, right? And I write it in this way. So it's one on the diagonal term and the rest will be one-half. And that's the property of being mutually unbiased basis. In the n dimension or d dimension, that will be one divided by d for the rest, okay? Then usually I show it with this, okay? It's an avocado colliding and yellow means close to one and black means zero, essentially. And between these two, which is a greenish, it's some value between zero and one, okay? And this is what you can see. And maybe some of you may ask me, hey Ebrahim, maybe you're cheating or maybe your computer is not good. But look at those, they are not uniform, right? Are they? This experimental data is never perfect, okay? So those are the measurements that I perform. Now we talk about hacking, right? What is the easy way that you can perform a hacking? The easy and the most... Oh, camera is recording. TVL way is intercept, resent. What does Eve, Eve will take the information, project it on a certain state, and detect the state, the outcome, and generate the state that she performs the measurement and send it to Bob. He's the easy way. Okay, what do you all see as effects? Previously you had this, right? Of course it's not perfect because he's experiment. Now someone is intercepting your system. What is happening? The effect will be on the diagonal term. The diagonal term is not anymore detected with 100%. There is noise there, okay? That's the presence of Eve. If you recall, when they talk to each other, they say that sometimes they are not measuring the same quantities. They don't... Bob does not receive the same message that Bob sends. So these are those cases. So effect of Eve will be on the diagonal term. Essentially, what I have for the previous case, which is an experimental data in the laboratory for the dimension 2, I have about 0.1% error, which is coming from detectors because detectors, they have some dark cons or imperfection in generation. But when you do this sort of attack to your system, which is the most trivial one, you will increase the error to 37%. So the effect of Eve will be easily detected. Good? And this is higher than the previous lecture, Stefano? 11%. 11% is only true for dimension 2. Good. Fantastic. This is one of the case. Now I will go to the exciting part. Higher dimensions. I don't like to. I want to go with more. I want to have infinity times infinity because you can send more information as you wish. But before going to the quantum way, let's talk about classical picture of that. These are exactly the same number. Assuming that my computer is not in the base of 0 and 1, it's in the base of 3. This will be the result of that code. In the base of 7, that will be the result. In the base of 10, that will be the result. So if I want to send such a message, if you call it message, even in classical way, what is happening, you have to send 32 digits. But if your base is 10, you will send almost 9 digits. Exactly 9 digits. So 32 versus 9. Wonderful. Because you have to create 32 pulses, right? You have to send them and you have to detect them. Here, you will send only 9, and it contains exactly the same information, even good in some senses. And moreover, in the quantum mechanical language, if you work with two quantum bits, two independent quantum bits, they will never give you entanglement. They will never give you the four-dimensional hyperspace. Okay, there are two individual spaces. Okay? You should know about that. This is also important in some sense. Beautiful. Actually, Fripo and I, we were discussing about that today. So now, going back to classical way of communication. When I'm pressing letter M, what is happening in my computer? Depending on the CPU, it will be translated in 0 and 1. It either will be 32 or 64 bits, right? It's one of those two. So, asking code that I have it for M is this. So that will be the signal that you will be sent. Let's say, call it signal that you will be sent to somewhere else. So I have to send eight signals, which those signals, they can be encoded either in photon number or in multiplexing way, the classical way. So it means that it can be different color coding that you send through optical fibers. So it can be for the first part, orange, the other one, blue, orange, orange, blue, blue, orange, and et cetera. And you will send it to other person. Other person will detect them and record it down electronically. Of course, that was 1, 0, or et cetera. Right? Beautiful. But no channel is perfect. Sometimes you will not get the signal, which is what we call it losses. So if you have a loss and you lost one of those signals, you have to send the package again, entire package again. Well, what I can do, I can use an alphabet, which is those alphabet that I have is infinitely large. For example, minus 124 to plus 124 of orbital angular momentum. 124 helixes to minus 124 helixes. I can do this. By the way, I have not taken that from Wikipedia page. I made Wikipedia pages for orbital angular momentum in 2010. And you have the freedom to use it everywhere. And then when you have minus 124 to plus 124, you can pick up any of those and you can label it with a letter. Right now, what you have, you have only one of those modes associated to those digits that you are sending. Instead of eight signals, I'm sending only one signal and I'm detecting one signal. So it's green as well in terms of energy. And also in terms of noise because here, the error will not go with 8 eta, but it goes with eta. Fantastic. Wonderful. And this is a setup that I will talk about this and always I'm using this setup, but I will talk about the detection and generation system in higher dimension that Hamid, was Hamid that asked me. Yes, a gentleman asked me there. He asked about the higher dimension, how we can generate them and we can detect them because we only talk about polarization. If you want, that will be a part of tutorial for tomorrow. We can talk about, for example, orbital angular momentum or we can talk about time being and we can discuss about how to generate them and detect them. Okay? So now I'm not talking about the way that I'm generating but simply the way that I'm generating is special light modulator or tube plate or whatever, a device that does this sort of generation and a device that does the measurement. So this is Alice, this is Bob and this is a classical channel that they are looked at GPS clock. They can tell Alice and Bob when the signal was sent. Okay? They will never, nothing will be shared apart from the time coding and the time doesn't have any information there. Good. And the way that I'm doing that is with spontaneous parametric non-conversion. I already calculated the G2 function and I'm sure that I don't have more than one photon per time in this pulse, in this pass. There is no way that someone can attack by photon splitting attack to the system. So now my dimension is not anymore h and v but is this. What are those modes? One of those degrees of freedom. So the dimension is minus 3, minus 2, minus 1, 0, 1, 2, 3. If you prefer, we could start with minus 2, minus 1, 1, 2. It will be four dimension, for easy way. Right now I have done with 7 because 7 is a prime number. You will see later on. Not that I love 7, but there is a reason. Okay. Then what I need is not only the base but I need another base which is what? Fourier transform. Exactly. Discrete Fourier transform of that which we call it traditionally, mutually unbiased base. Okay? One of the mutually unbiased base is this one. So this is Fourier transform of that. So if I take one of those modes and projecting on those modes, I will get 1 divided by 7. So when I'm choosing one of those bases and I perform the measurement there which certainly I can define the state. So for example, Alice is choosing minus 1 and Bob is measuring in this space with 100 certainty Bob can understand if that was minus 1. But if Bob is performing the measurement in this space of 5, he has no idea what's going on. Exactly looked at that one. Sometimes this one will click, sometimes the other one will click. It's clear or should I write it on the board? Anyone has any question? Okay, good. How many mutually unbiased bases do I have? Definitely I know for prime dimensions. Certain values. So I have d plus 1 mutually unbiased bases which if d is prime. Since 7 is prime number, I have 8 mutually unbiased bases there. But when it's not prime, I have no idea. The easy dimension in dimension 6 which even we try experimentally in 2013, we couldn't get more. We got only 3, not more than that. And it's an open question. If you are solving it, that will be fantastic. People, they tackle this with group theory in different ways. Why we are caring about high dimensional QKD? I try to simplify in one slide. And believe me, this gives you the message. Why we care about high dimensional QKD? First of all, we know for sure that we are increasing the rate, right? If you go with 4 dimension, you have 2 bits of information. If you go with 8 dimension, I have 3 bits of information and etc. It's very clear. This is not the only part that you will be. If the error is 0, the bit will increase with the dimensionality, the secret key rate. Right? But always you have a curve based on the error, bit quantum bit error rate that you have it which will tell you if you have a positive information, positive secret key to be shared or not. For the dimension 2, what's that threshold? 11%. You got that. What is happening with dimension 4? Will be 19%. Huh. If I increase the dimensionality, what is happening? I have the possibility to handle more errors on the top of increasing the secret key rates. This is wonderful because we know that we have channels that they are really noisy and this is one way that you can overcome these difficulties. Beautiful. So high dimensionality will be useful for us in this sense. Good. So this is one of the case that has been performed in our laboratory. We have the dimension 7 as I showed to you. You have to rotate your head and you will see it. It's dimension 7. We have done that with two mutual embases of psi and phi. Essentially, I have to write them here but since it could get very, very complicated, I didn't write there. So if you are in the right base, you will perfectly detect them. If you are in the wrong base, you have unbiased situation. You have 1 divided by 7 as a probability of detecting them. Clear? Good. And look at the value. It's almost close to 100%. Apart from one of those bases that they were not perfect. Why? Because my detection skills and generating skills with special light medulator, they were not good. I have written articles about that, how to create modes with special light medulator. It's not an easy job. It's very, very complicated and mathematical is very, very, very difficult, I would say. But you can improve it. Good. Fantastic. And then we have performed this experiment in optical table of, I think it was roughly speaking less than 7 meters to perform the experiment of 10 meters. I don't recall well because it was long time ago, maybe 2017. And we had about 0.16% error and we were able to reach to 1.73 bit per photon, per shifted photon. Okay? Beautiful. How can you passly switch the states? We'll give you the rate. For example, if I can switch the state with the gigahertz, multiply that by gigahertz. That's a way that you can implement that. Good. Let's perform my stupid and naive way of hacking a system which is intercept, resent. What I have, I do, now I am doing in dimension two, dimension three, dimension four and et cetera. I will detect that photon in those states and I will generate them and I will send them back. This is the way. Dimension two was this, right? Do you remember? It was 34% error in the dimension two. For dimension three, make a guess. Will it be worse or will it be better? The errors will be worse because we already knew in classical way also the error will be enhanced there. In the dimension three, and I have not done that for only two muchalean based spaces, I have done for all muchalean based spaces. We call it tomographic protocols. Okay? For the dimension three, what we have, we have 49% error due to the intercept percent. Okay? So it seems that my security is increasing. It's more sensitive to hack. And dimension four, 61% error. And those are experimental data only for you. You have performed that only for you guys. It has never been published. The reference is not for that, it's for something else. It has never been published. Okay, so going to higher dimension also makes it more resistant. But this is a more trivial way, the most trivial way that someone can hack a system. There are more complicated way which I call it cloning attack. Is there a cloning machine? We just talk about it. There is no cloning machine. But I can ask myself that there is no cloning machine but can I build up an optimal quantum cloning machine? I cannot copy a quantum state, but can I make a good copy of that? Right? It's a question. And let's see what does it mean in terms of quantum mechanical language? It means that in the two-dimensional, I'm showing with the loch sphere or Poincare sphere, Alice is generating a state which is pure, which is on the surface of the Poincare sphere and sending to Bob. And Bob also performs a measurement on the surface of Poincare sphere. So Bob what he gets, he will get a state. Okay? But I can build up a machine for Eve, which what is the action of the Eve's machine? It will be an optimal quantum cloning machine. What it does? It reduce the purity, okay? And it generates a mixed state. But what it will give me? It will give me two copies of that. Okay? And we call that optimal quantum cloning machine. And let's do this sort of hacking on a system. And of course I know what's going to be the action of that. And it turned out that for the symmetric way, will be an easy effect which I call, I think people, they call it hongu mandal effect. If I want to be very careful, I have to also call it ferren and loaden effect. This is not in books. But ferren and loaden also, they had almost the same idea. Maybe it was published a little bit later after that. Okay. So what is hongu mandal effect? How many people they know hongu mandal effect? Oh, good. Very good. I love that. So what is happening in the hongu mandal effect? There's a bosonic nature of light. So you send of the state of light, of course. So you send two photons to a beam splitter, 50-50% beam splitter. If you send photons here, I label them with red and green. Red is passing, green is passing. It's one possibility. Red is reflected, green is reflected. Red is reflected, green is passed. Green is reflected, red is passed. Right? True? Guys? Fantastic. So I have the probability of detecting them from one gate will be, this gate will be 25%. So if I want to detect them, the probability of that will be 25%. What is happening if I take the color off? The color of them, those are green. You say the same probability, right? No. For light, what is happening? These two interference effect, they will cancel out each other because they have a negative sign between the two. What is happening? These two possibilities only will happen. Either the two photons will exit from this port, or they will exit from that port. Good? Now I'm asking you, what is the probability of having photon here? 50%, right? It will be one-half. In the previous case, when they are not identical, they are completely distinguishable. It's 25%. When they are going here, it will be 50%. So what is happening essentially, I label them with one and two. I call that one and the other one two. And when I do the action of beam speeder for the one and two, what is happening will be C in the port of, one in the port of C, two in the port of C, one in the port of D, two in the port of D, and et cetera. So that is port of C, port of D. Port of C, port of D, okay? What is happening that you have the possibility of the two which is here? But when you remove the label of one and two, these two, they will cancel out each other and you will get that state. Of course, you need a renormalization factor because I didn't do proper calculation. You have to do it with A and A dagger. If you do it that way, it will be normalized. There's a hongu mandal effect. Why I'm talking about hongu mandal effect because I was talking about optimal quantum cloning machine, right? I have to find the link between these two. Okay, good. So when the two photons are distinguishable, then what is happening? 25% of the time, both of them, they will exit from one port. If they are indistinguishable photons, that probability will go to 50% of the time. Agreed on that. This is an experimental data that I have taken many years ago, maybe 2010. So I created two photons. The two photons, they arrived in the same time at the beam splitter, but I had the trombone which I could control the time, arrival time of these two with respect to each other. So this axis is arrival time which I was able to control with the microstage which is what's controllable microstage in the precision of one micrometer. I could adjust them. That the two photons, they arrive in the same time or not. Okay, when the two photons arrive, and then after the beam splitter, I had two detectors and I look at the coincidence between these two detectors. When the two detectors, they are firing in the same time. And I record those cases. When the two detectors, they are firing in the same time and we call that coincidences. So what happens? Previously, you will see just what is happening. You will see that you have coincidence about, let's say, a unit of one. I don't know if it was about maybe 1,000 coincidences. And then suddenly you have a jump which I will explain to you if you want but it's beyond the discussion here. And then when the two photons, they arrive in the same time, they will exit from the same port. The two detectors, they will never fire together. It seems that both of them, they are going to the same direction. Right? Good. And then when they are becoming distinguishable again, you will get the coincidences. And some people, they ask me, ha, Ebrahim, maybe you have a misalignment in your setup that the detectors, they will not catch the light when you move. I say, fine. What I can do, what is the probability of these two photons coming from this port? In this case, it's 25%. In this case, how much is the probability? Exactly, it will be 50%. So when the two photons, they are indistinguishable, I will get 50% of the time the two photons coming out from the same time. So what I can do, I can place another beam splitter and I look at the detection, they click between these two detectors. When the two photons are coming here, they will be again same probability that either they will be reflected or transmitted. So half of the time I will have them here. But in the case that the two photons that are indistinguishable, this probability will increase by 50%, so it will be twice. Then if I look at the coalescence, I will see that there is a peak in that case. So it means the number of photons is increased by twice. Yes. Yes, this peak, it comes from the interference effects. I have interference filter at the front of that and that will give you bandwidth limits, okay? And what I am looking, frequency is the Fourier transform of that, right? I have a bandwidth of these and now I am looking at the time is a Fourier transform. So then a step function will be a sync function. That's a sync function. Okay. Good. Now let's look and use this technique for doing optimal quantum cloning attack to a system. What I do, I have my state which I have no idea what is the information there. It's a state of psi. Then I will generate a state of i which is totally mixed, is maximally mixed state. I will do the Hong-Gong-Mandal interference with that and then I will claim that you will get optimal cloning attack to a system. I can do the calculation. Sometimes this will be in the state of orthogonal, sometimes it will be the state of psi, then you will get the coalescence and the probability will be 1, then how many times it's happening to do 2 divided by d plus 1 and the other one, the fidelity will be 1 half. How many times this is happening d minus 1? Okay. If you do a little bit of calculation, you will find out the fidelity of the state there will be 1 half plus 1 divided by d plus 1. What is d is the dimensionality of the space. Dimensionality of QKD space that I am using it. Okay. And that's the experimental result. If you do that, you will see that for the dimension, for the dimension 7, which previously we had about 1.73 beats per second, sorry, beats per shifted photons, for the case of optimal cloning when you do these sort of optimal cloning, that diagonal term, the probability of finding the diagonal term will decrease to limit that you have it by optimal cloning attack, which is about 0.6%. So you are not detecting 100%. The effect of eave will be introducing noises to your system. Good. And that's a setup that you have it. So you need to create a maximally mixed state and making congo mandal effect with what signals is sent from Alice and then detecting on the other side and keep one for eave and doing the other one for Bob. And for those people that they want to see the experimental setup, this is the lab. It's not so complicated because it's done with a specialized modulator, but you will see Alice, I think, is this one and Bob is somewhere here and this is the cloning machine that is happening, of course, with a lot of detectors and electronics and etc. Good. We have done that for dimension 2, dimension 3, dimension 4, 5, to 6, just to show, and 7, just to show to the people that if you go to higher dimension, again, there are more resistance against this sort of attack. So, for example, for dimension 2, and by the way, this sort of cloning is universal. So it's independent of the basis that you choose. It does the cloning, I know it in a way. So, and what we have, these are the measurements that you expect. This is the threshold that you have it. This is the threshold that you have it for the optimal quantum cloning and that's the threshold that you have it for current attack. That's the one who talked about that today. And for the dimension 2, as I mentioned to you, the dimension 2, the threshold is about 11% and the cloning attack will give you 17% errors. So if you remember, the error went to be at the fidelity of the state depends on dimensionality. So, that was fidelity of the state was 1 half plus 1 divided by d plus 1, which d is the dimensionality of the space. For the case of d equal to 2, that will be 1 divided by 2, if d is equal to 2, plus 1 divided by 3, then that will be 5 divided by 6 and that will be 0.3. So, 17% will be error due to the optimal cloning attack. If you go to the dimension of infinity, this is my wish, then the fidelity will be 1 half, is what you expect. So, the presence of Eve will be easier revealed. Good. And of course, you can do the process matrix for this and finding out for different cases for optimal cloning and intercept percent and you will see what is going on with the process matrix. If you have no attack in the system, that will be the process matrix for dimensionality, different dimensions. And the process matrix, how these states are changing for different basis, of course, mutual end-by-spaces. And then for the case of optimal cloning, you will see that the action of these spaces is not anymore 1. It will be different. The other terms, they will be excited and for the case of intercept percentage, just off. Good. So, I suppose to stop here, I think. Let me see. Yes, I stop here for questions because that's four we have to stop because it will be a ceremony. I have two things to announce to you. Really. Okay, then I will leave it for the discussion because I prepared exactly based on that. So, the first announcement is this. They asked me to announce this. There will be a conference in Colombia, in Cali and Armenia. And the other one in Dubna, which is organized by Narine and Astrik. Where is Astrik? Yeah, she's there. They are organizing and usually, I mean, is a very exciting meeting. I mean, you can go there and usually, they have great speakers. Not that I was there, but usually they have very good speakers. And even I recall that Michael Berry, he acknowledged that meeting once because he found an idea. He got an idea during that meeting. So, this was the first announcement that they have. And the talk for tomorrow will be about doing QKD a different environment, fiber, underwater and free space. These are experimental results. And of course, I will cover some of the experiments and experiment by my colleagues, like Pino Balone and also by Jean-Marie Penn and Thomas Yerevan as well in Canada. So, they are doing the satellite and they are doing with the satellite. So, I will cover this. And moreover, I will maybe, I will cover the quantum network and quantum internet. So, what do you need for performing a quantum internet? So, that will be the subject of tomorrow and it will be open discussion. And also, I will talk about high dimensional state preparation and detection. If you ask me, I will describe that in maybe half an hour before going there. So, thank you very much. And I'm open to questions and let's have a discussion.