 OK, so welcome to the second Salam lecture of this year. The lecture, as you certainly know, is Professor Sandu Popescu from the University of Bristol. And so the title of the second lecture is going to be Quantum Non-Locality and Beyond. And I understand that Professor Popescu is going to give the lecture on the Blackboard. Finally, thank you very much for that. And let me also remind you that the Blue Salam Distinguished Lecture Series is generally supported by the Kuwait Foundation for the Advancement of Science. So please. Thank you. Well, thank you very much. Thank you again for inviting me. As I said yesterday, it's great to be in this place that contributes so much to the development of science in the larger world. So as I told you today, all my lectures, I decided that I will give a little piece out of different subjects. So today, I will talk about quantum non-locality and beyond. So you see, this story starts, in fact, long time ago. It starts with Einstein, who, as you know, is one of the fathers of quantum mechanics because he gave his explanation to the photoelectric effect, which was one of the first effects where it was understood that something is interesting in nature and cannot be explained by the usual classical laws. But as it happens many times with parents that at a certain moment, they get to be a little bit displeased by what their children are doing. That is what happened also to Einstein. So he didn't like what quantum mechanics turned into. And the reason that he didn't like is that quantum mechanics turned out to be a probabilistic theory of nature. So you may know his famous sentence that God does not play dice. So he could not envisage that nature is probabilistic at a fundamental level. So let me tell you more or less what the issue is. We, in everyday life, we are used with the fact that if we repeat the same experiments in exactly the same condition, then exactly the same thing will happen. I throw something. If I know exactly with what speed I throw, I know what the air is around. I know where it will end. I will do it again. It will happen the same thing. Of course, you can never do identical initial conditions. You know, I cannot throw it perfectly the same velocity every time. But I can make the experiment better and better. I can replace myself with some robotic machinery that throws one can have better devices and so on. So then you make it more and more reproducible. And there is no limit how well you can reproduce the experiment. And philosophically, this is very important because that says that the same causes, the same experiment leads to same effects. On the other hand, basically what quantum mechanics says is do something, which means you make the same preparations in the lab. And look what the effect is. Then do it again, and you will find that something completely different happens. So as an example for that, you just take this piece of chalk and you let it fall. Now you do it, it took a second. Well, less, half a second. You do it again, it may take an hour. That is what quantum mechanics says. Or let me put it in quantum mechanical terms. Quantum mechanics describes everything by a wave function psi. And let me denote here psi absolute value square. Now what does it mean that I prepare a particle like this? I have all kind of devices in the laboratory and I put all the dials in the same place. And then I look where the particle is. So sometimes I find a particle here. Now I repeat the experiment again. I do exactly the same measuring devices. I put the dials in the same place. I look where the particle is and the particle comes out there. So Einstein said, look, this is impossible. If you found a particle here, it meant that it actually was there. If you've done exactly the same thing and you find a particle here, the particle must have been there before you found it. It is impossible that you do exactly the same thing and sometimes it's here, sometimes it's there. But the theory of quantum mechanics said no. This is what it is. We just have a wave function and that's the end of it. So Einstein for a long time wanted to, he was so sure that quantum mechanics is wrong, he tried to find all kind of example to show that the theory is factually wrong. And in one famous event, there was a big conference, the Solvay Conference, where all the great physicists of the era were there. And Einstein came with an example to really show that quantum mechanics is wrong. Everybody was shocked. Nobody understood what is going on. And next morning, Niels Bohr came and showed that Einstein was in fact wrong because Einstein forgot his own theory of generalized relativity. So from that moment on, Einstein decided that, OK, quantum mechanics is probably right, but probably it is incomplete. Meaning we've already known about theories that are probabilistic. All the statistical mechanics is like this. But that is because we do not know all the facts about the system. We have many molecules. We don't know the exact positions, whatever. And we can only discuss about probabilities. So Einstein said, perhaps this is all it is. It is a statistical theory. And the fact that sometimes is here, sometimes is there is because although you've been in the lab and you put all your dials, you didn't actually control all the parameters. There must be some parameters that you missed. And that's why this all happens. So you would think that all these ideas of Einstein made a big impact on the physics of the day. But while science is, well, while nature is something stable and objective, the way in which science works is very subjective. So scientists, believe it or not, they are also human. So they just did what they thought is interesting. And they had quantum mechanics at that time was something new. And using it, you are able to explain a lot of things about nature. Quantum mechanics allowed you to explain why there are atoms in the first place, what are the levels of atoms, what are the properties of matter, what are nuclei, and so on. People really didn't have time to think of, is quantum mechanics right? Is it not right? What about the philosophical implication, and so on? They just had things to do, and whatever they were doing with quantum mechanics was good. So Einstein was sort of left alone, but being Einstein, he never let go of the idea. So in the end, this was quite early on. But in 1935, Einstein, together with Podolsky, with Boris Podolsky, and Nathan Rosen, published a paper in which they wanted to show that, indeed, quantum mechanics is incomplete. So as we will say today, they show that there are hidden variables. I will put here local hidden variables. So they presented the following experiment. So they say, look, imagine that I have here a source. Yes. And what this source does, it sends one quantum particle that way and one quantum particle this way. And it sends them far apart. It sends them from the Earth to one galaxy and to another galaxy, long, long, long distance away. So let's suppose that here I have a measuring device. It has an opening. The particle may come and enter there. And there is another measuring device here, another measuring device here. And this particle comes there. And by prearrangement, we arrange that the particles, say, in the laboratory frame of the source, they arrive at the same time. Now, the entire argument of Einstein, Podolsky, and Rosen can be made in very abstract terms. So all I tell you about these measuring devices is that they have two light bulbs, a green one and a red one. And here, just two light bulbs, a green one and a red one. So what happens? The particle comes here. And as it enters, one of the lights comes out, either the red or the green. That's all. So Einstein, Podolsky, and Rosen, they said, look, we can arrange a situation in quantum mechanics where the following thing is the case. When the particles come, there are only two possible answers. Either both measuring devices show green or both measuring devices show red. This situation occurs. In 50% of the cases, this occurs in 50% of the cases. Importantly, green red never occurs. And the red-green never occurs. OK? You can do that in the lab. You cannot be in two different galaxies, but it doesn't matter. Just imagine this. So then Einstein said, look, I always prepare the same wave function here, but sometimes the answers are green-green. Sometimes the answers are red-red. How is this possible? See, quantum mechanics said that until the particle arrives here and the measurement takes place, you don't know whether the answer will be red or green. So then they said, look, this poor particle comes here. And imagine that you are the particle. This is the best way to imagine. Imagine you are the particle, and you will need to make a decision. If at this moment you have to decide whether to say green or red, how do you make sure that your partner, which arrives here, will say the same thing? So there are two ways. One way is that before you tell green or red, you just take out your mobile phone, and you call the other particle and said, look, this time I decided I will say green. You will say green as well. But here is a problem. And here is what was the genius of Einstein, Podolsky and Rosen. They put this measuring device is far apart. So that even if this particle now takes and tries to call the other, it takes a long time for the message to come, because that is limited by speed of light. And the message cannot reach its partner by the time the partner needs to give the result. So the experiments take place what we scientifically call in space like separated regions. There is no way for anything to do. So you don't have to be afraid of cheating. These days mobile phones are very small. You can make them even smaller. You can hide them in a filling in your tooth or whatever. So particles may want to cheat, but relativity prevents them. Incidentally, Einstein, Podolsky and Rosen thought of very large distances. Today, in fact, you can do it in the lab within two meters or something, because experiments can take place so quickly that even for this very short time of propagation of light, you can have experiments outside of light corner of each other. OK, so particles cannot communicate. So if this particle arrives here and has to make a decision on the spot here, it cannot ensure that the partner will know. So then they may give the wrong answers. Red, green, or green, red sometimes. And then Einstein asked, how is nevertheless possible than the particles give correlated answers? They said, look, the only way it is if they know in advance what they are going to say. Because if they have to make the decision, only here is too late. But they have been here together at the source. They know what game they are playing. So they decide, look, this is what it has in its mind. I will say green. You will say green. So you remember, and you go from here, tack, tack, tack, all the way, you arrive here, and you say green. Very nice. Next time, or the next pair of particles that go, when they are at the source here, they flip a coin. And if it comes red, they say, OK, this time we will say red. So they go slowly, slowly, slowly here, and they say red. And everything is fine. But what does this mean? This means that the particles are not only characterized by the wave function psi, but they also need, in one of the cases, they need to remember that they said red. They need to hold that information in their mind by the time they come here. So there must be some other variable. In one case, that variable says red. In other cases, that variable says green. So in addition, quantum mechanics thought that you are preparing exactly the same thing every time. But there is this hidden variable, maybe something in the brain of the particles that remembers their decision. Hence, quantum mechanics is incomplete. It's very important to understand this point. Everything relies on this. Any question about this? OK. It's a simple game. It's easy to understand. In fact, Heinchland-Podorsky and Rosen said, look, let us show you even a different example. Yes? So the velocities of the particle should be larger than half of the speed of light, right? No. No, they could be whatever. They could go even very, very slow. But they have to take the decision only when they arrive. Yes, but then, OK. And then they are very far from each other, may take a year for the light to propagate. So they cannot take the decision on the spot. That is the important thing. OK, so Heinchland-Podorsky and Rosen said, look, let us show you something even more interesting. Suppose that this measuring device, in addition, has a switch, which has two positions, up and down. Putting the switch up means some experiment takes place here. Putting the switch down means some other experiment takes place here. Similarly, here. There is a switch with two positions, up and down. And there is an experimentalist here. Perhaps we call this Bob. And there is another experimentalist here. Perhaps we call this Alice. So what happens? What happens now, if I get my things, is the following thing. Particles come. And just before the particle, it arrives here. Alice decides whether to flip the switch to up or down. And similarly, Bob, who is here, just at the very last moment, decides to put the switch up or down. So the particles, when they were here, they don't know what position the switch will be. Now it is a tougher experiment. Previously, it was trivial. OK, so now what Einstein Podolsky and Rosen did, they showed, look, there is a possibility of preparing the following experiment. In the case when both switches were up or when both switches worked down, the possible answers were red, red, or green, green. This happening with 50%, that happening with 50%. However, in the case when Alice's switch was up and Bob's was down or the opposite, only the answer is red, green, or green, red were possible. And again, 50%, 50% of the time. So again, you have these correlations. But now it's a much tougher life for the particles, because the particles, when they are at the source, they don't know where the switches will be. And furthermore, when this particle is here, all this particle can see is whether the switch on its measuring device is up or down, but has no idea what happens there far apart. So the decision has to be taken only based on what they decided here and what this single switch is for Alice's particle. And Bob can only rely on what his own switch is. It's a far more challenging thing. But Einstein Podolsky and Rosen said, look, there is not a big deal about it. Let's see. So this is the strategy. Imagine that you and your friend are the particles. And you play a game, play this game against me and my friend. My friend Alice is here. I take the place of Bob. And we want to play this game. Perhaps if you win, you get a prize, which means that we pay. Would you? Should I play this game against you? What will be your strategy? Take a few moments and think if you have an idea how you can win. Any suggestions? You will all lose? OK. So then I should invite you to the game. All right. Look, it's not very bad. So since they don't know what will happen, they have to be prepared for whatever the switch will be. OK. Right. You and down. So let's say that this pair of particles, so in the first round of the experiment, they decided, look, if I, the particle going to Alice, see that this measuring device has the switch up, I will say green. OK. Then what do you have to say if you see the switch get up? Well, these are the rules of the game. For up, up, they are winning only if they give the same things. OK, these are the correct answers. So it's very clear. This particle may say, must say green. Otherwise, they lose. OK. But now when this particle arrives here and sees the switch on up, it is clear it will say green, regardless on what happens there. Because it never knows. It doesn't look to it. It doesn't know. That's far away. So whenever this switch is up, this particle will say green. But it is very possible that this switch is actually down. So then this particle has to be prepared. What to say for this down, given the fact that up there says green. But for up down, the answers must be opposite. So if it is clear that this said green, this one must say red. So then we already arranged for this, and we arrange for that. But now, if I say down, it may be possible that the switch there was down, in which case the particle says red. But for down down, the answers must be the same. So I must say red, OK? So by that pre-arrangement, this one is done as well, OK? Which leaves me the last one. For down up, they must be different. So let's see here. For down up, they are different. So in effect, here they say, if I see it up, I say green. If I see it down, I say red. And then they can do this sort of correlations. How can it be that is in 50% this, in 50% that? Because next time, when they play the game, they may say, no, no, let's start in a different way. I should say red, red, green, green, which is also a solution to the game. So you play like this, and you are able to do even though you don't know what happens far apart. So this idea of hidden variables is a very strong thing. So again, we are now in 1935, and you would think that, wow. Now, this time, Einstein, Podolski, and Rosen really showed it that quantum mechanics is incomplete. So we should start seeing what happens about quantum mechanics, modify quantum mechanics. It should make a great impact. As a matter of fact, it did not make any great impact. It made almost no impact at all. And the reasons are multiple. First of all, I presented for you what I called the Einstein, Podolski, Rosen model. Well, whenever you give a talk, you never present it exactly as it was. Every time when you discover something, things are far more complicated. You never get the simplest solution and simplest statement. This is how we see what they've done today. They did it in a more complicated way, with position and measurement and momentum, and all their arguments were very complicated. And then Bohr wrote a paper which was even more incomprehensible, and everything just fizzled out. OK. All right. But this is what Einstein, Podolski, and Rosen said. So it took some more time. And it took almost 30 years in which some philosophers discussed a little bit about this thing, of Einstein, Podolski, Rosen, and so on. And it comes now to 64. There were actually a few things that happened in between, but I will talk about those later. And in 1964, John Bell reanalyzed this idea, this setup. He, in fact, used the setup more or less, how it's written here. And that was due to some advances. I told you, Einstein, Podolski, Rosen wrote it in position and momentum, but there was an essential paper by David Bohm and Yakir Aharonov that simplified this situation from position and momentum to something with just two variables. And then what John Bell said, Einstein, Podolski, and Rosen, EPR, they claim to have seen that quantum mechanics has hidden variables. That is wrong. That was the decision of John Bell. And in fact, he discovered something that we call today non-locality. So what John Bell did, he considered exactly the same experiment. But he said, look, here there is another possible experiment with some other quantum states, which does the following thing. Whenever the switches are up, up, up, down, or down, up, the answers are either red, red, or green, green. And they occur 50%, 50%. When they are down, down, I would like red, green, or green, red. And I would like to do this again. So he asked, is it possible to find that? So the question is, is this possible? OK, so let's see if that is possible. So let's get back here and try to apply the same thing. Remember, if the particles do not agree at the beginning what they will do, if they wait till the very last moment, there is impossible to keep correlation. Let's say, look, without any loss of generality, let me say, I will say green if I encounter the switch up. What do you, my friend, Particle, coming here, should say if you see your switch up? Well, the game, in order to be won, requires that for up, up, the outcomes should be the same. So if here I said green, this must say green. But what happens if here the switch is up and Bob's switch is down? For up, down, they must again say the same thing. So there must be a green here. So this one is down, is done, and this one is down. But it is possible that the switch of Alice is down and the switch of Bob is up. Bob Particle sees the switch up, says green. For down, up, in order to gain the game, must say green as well. But see what happens for down, down. So this one is OK. But for down, down, the game that John Bell asked whether you can win requires the answers to be different. But there, for down, down, the answers are the same. So it means this cannot work, which means that in this case, the probability for success is equal to 3 out of 4 cases. That is all you can do by prearrangement. So any two particles, if they would have hidden variables, they would only win this game in 3 out of 4 cases. Sometimes they will give the wrong answer. But what John Bell found, he found an example that the probability of success in quantum mechanics, in the example that he found, is 2 plus root 2 over 4. Do you know how much is root 2? Bigger than 1. So this is bigger than 3. So the particles can win more. It's indeed 1.41. So the particles cannot do this, quantum particles. They win more than it is possible. How is that possible? It is impossible via pre-decision, OK? There is no way. If they try to have a strategy, a prearrange strategy, they can only win in 3 out of 4 cases. But quantum mechanics gives more. How can one do more? Only if the particles communicate with each other. That is the main message. So if the particles can communicate, obviously the particle of Alice will take out its mobile phone, will tell the other, look, I saw this switch up. I said, green, you take care of what you do. But remember, what is the catch here? These mobile phones have to communicate superluminally. No message with speed of light could arrive in time from Alice to Bob. So Alice's particle must communicate to Bob's particle faster than light. That is the conclusion from here. So this hidden variable, the prearrangement, doesn't work. And they really need to communicate. But if they communicate faster than light, does this mean that Einstein's theory is wrong? So this is now a very subtle question. Because we are here in a probabilistic theory. We are not in the standard theory of Einstein, which was classical mechanics, which was a deterministic theory. Here, one of the answers may occur, or the other answer. The other correct answer may occur. So one has to be a bit more careful of what we want to demand from relativity. So here is the thing. The particles, when they arrive, this one is here, arrived here, this one arrived here, they have their own tiny mobile devices, tiny cellular phones in their pocket. And they are able to talk to each other. Look, look, I got my spin up. I said, green, take care. The question is, however, can Alice and Bob hack these phones? Can Alice transmit a message to Bob using the particles? If they could, then that would be a violation of relativity. And then Alice and Bob could enter in all kind of paradoxes. You know, if you can communicate faster than light, you can make a rearrangement, various frames of reference. You can send actually messages back in the past. And then they always want to kill their grandfather. I'm in a bad position here, according to the standard thing and so on. So the question now arises, are Alice and Bob able to use these correlations between the particles in order to communicate from each other? So there are two points here. Number one is all Alice can do is to put a switch either up or down. It cannot order this particle, now you say green or now you say red. The second thing that is important to bear in mind is that although I said what the correlations should be, Alice, they do not see these correlations. All Alice sees, she sees whether her result is green or red. To find the correlation, she must compare the results with the results of Bob. But that takes time. If they wait until they can compare, there is nothing that goes faster than light. So let's see what happens. How could Alice, putting the switch on up or down, send a message to Bob? Well, there's really no way. Because let's see what happens in this particular instance. The answers of Bob in all the situations are sometimes red, sometimes green, 50%. So no matter where Alice's switch is, Bob will not see a change in the local probabilities and similarly Alice. So in this sense, here's the dramatic thing. Einstein started with the idea that he's very bad to have a probabilistic theory. The theory should be well-defined, should be deterministic. But the fact that the quantum mechanics is probabilistic allows you to have no locality. Just the fact that the answer is probabilistic allows for this phenomena, whether particles talk to each other superluminally, to exist without leading to contradictions and to parallelism. So the fact that the nature is fundamentally probabilistic is not, in this sense, a limitation. But in fact, it's something that gives you much more, gives you a lot more freedom than what you would have in a fully deterministic probability. OK. In fact, all the effects used in quantum information today, they all are based on this thing. And what people that received the Nobel Prize this year did, they just verified this thing. More or less, they verified that particles could get like that. How the experiment takes place, well, that's pretty easy. As a matter of fact, here is one experiment. You have, let me get some space here. There are different ways of doing it. But the way that is standardly doing today, you just shine an ultraviolet laser to a tiny crystal, which has some nonlinearity. Every crystal is nonlinear. So more or less what it does, it splits the ultraviolet into two optical photons. You put some mirrors here. And with mirrors, you send one one way, one the other way. You have some polarized beam splitters at this end. And you have two detectors, a detector here, a detector there, a detector here, a detector there. And you can slightly turn this polarization, or you can put something that is called a halfway plate in the middle to sort of rotate polarization. You may put this or take it away. And that is like the switch up or down. And then you look. This is what I call green, and that is what I call red, and so on. Before it was discovered that you can just put this with a crystal, people have to use a more complicated thing to create the photons. And the photons you create in some state psi that is up to 1 over root 2. Say horizontal, horizontal plus vertical, vertical. So it's, you know, when these experiments were first done, the way to produce that state like this was not known. It was produced in a more complicated ways. To do all the electronics, to move these things, was also very, very challenging. Imagine we are not talking about two galaxies. We are talking about a tiny lab where we want to do the experiments. So they were very, very demanding. And to see that quantum mechanics allows for this thing, which is something that was never tested in this regime, was dramatic. So these were the two people that received the Nobel Prize for the Bell inequality. The third one, Anton, received for further developments in quantum information. But what Clauser and then Alena Spedid was basically to test this. Today every student of quantum optics does that in the lab just to align the equipment. So that is the level to which we got now. OK, but let me leave the experiments on the side and go back on the importance for the foundations of physics. You see, I told you that the fact that quantum mechanics is probabilistic suddenly allows for a phenomenology that is impossible classically in a deterministic theory. Because in a deterministic theory, if when you move something here, something else wiggles there, you could see that thing wiggling. So then you could signal faster than light. It is only because whatever changes here, you could have obtained all possible results just by randomness. You don't know whether they are due to some changes here or just by what happens there. It is only later when you see them. OK, now comes something else more interesting. We understood that the fact that nature is probabilistic allows for non-locality. But is it necessary? So let me take a step back. You see, when you learn about special relativity, you find that special relativity can be derived out of two basic axioms. One of them is that all frames of reference or inertial frames of reference are equivalent. So if you are in a train that moves in a straight line without you just move smoothly, you don't know whether it goes or not if the windows are closed. And the other is that the maximum speed of signals is finite. And out of this, too, you derive the entire thing. And there are two natural things. All inertial frames are equivalent. The maximum speed of signal is finite. When you go to quantum mechanics, the situation is more complicated as far as the axioms go. The axioms are every state of a microscopic particle is given by a vector in a complex Hilbert space. Every physical variable is given by a Hermitian operating acting on this Hilbert space, which are very far from all frames of reference are equivalent. It's nothing natural. And when people try to say something, there were always negative ideas. If things are random, if I try to measure this, I disturb that. Well, there's not too much that you can advance from such axioms. On the other hand, and this was the idea of Aharonov, he realized that I can turn things upside down. Suppose that I take the axiom that nature is indeed described by special relativity. So all frames of inertial frames are equivalent. Speed of maximum speed of signaling is finite. And I also say that nature is non-local. From this follows up that the theory must be probabilistic. So although I could imagine a theory that is probabilistic but does not have no locality, if I want a theory that is non-local, the fact that it is probabilistic follows. That is the main thing. So suddenly one of the properties of quantum mechanics that was so strange that nature is fundamentally probabilistic immediately comes out from taking non-locality as a bonus. So that is, to my mind, one of the great questions to ask. And then the question is, could we actually derive quantum mechanics out of the axioms that special relativity plus the existence of non-locality? That is, is the only theory that you can imagine hypothetically, the one that contains non-locality is quantum mechanics or there are other theories. So then what is interesting, if you come here and you see this is how much the particles succeed in the experiment shown by John Beth. Somebody else called Boyce Chirson, different spellings, found out is that actually quantum mechanics gives this thing, this is the maximum, this is the maximum value in quantum mechanics for such games, OK? Which is interesting. He says quantum mechanics can play this game, succeeds more than classical, but it doesn't succeed all the time. It only succeeds 3.4 cases out of four. Why not always? Why not always? The question is, does special relativity prevent you doing that? And the answer you can see immediately and the answer is no. In fact, there is nothing wrong of succeeding all the time here. So you could, in principle, envisage a theory where this happens all the time, OK? So that is, in fact, what Daniel Rohrlich and myself found many years ago. You see, once you put everything in this language, it is trivial, but it was not in this language. It was put in very obscure terms. The game was not clear by that time. The meaning of it, like probability of success, was not. Everything was done just in pure quantum mechanical terms. So one had to rephrase it in abstract terms in order to be able to compare quantum mechanics to something that is beyond quantum mechanics. And once you do that, you understand that there is this hierarchy of successes, which is the following. Is that, in principle, you have, if you put here the probability of success, classical mechanics, classical mechanics can take you up to three out of four cases. Quantum mechanics can take you up to two plus root two over four cases. But you, in principle, can get to four out of four and still be what we call non-signaling. That is, Alice cannot signal faster than light to Bob. Relativity doesn't prevent this. So the question is, what happens there? So here there are two different ways in which you can go. One of them is to say, look, we've seen until today only correlations coming up to here. It is possible that there exists some setup that would give stronger correlation in this game. Now, if such a setup exists, it means that quantum mechanics is wrong. And quantum mechanics is not the good theory of the world, because quantum mechanics doesn't allow you to go more than this. Classical mechanics takes you up to here. Quantum mechanics takes you only up to there. So perhaps nature is not quantum mechanical, and you have more stronger correlations at a distance. On the other hand, perhaps such correlations do not exist. Then the question is, why don't they exist? What is the other basic principle of nature, apart from relativity and existence of non-locality, that prevents you to go more than that? OK. So many, many things were proposed. And let me just, I think that my time is gone, let me take five minutes and give you just one example for that thing. You see, when we invented, we looked at this game and found out that, in principle, you can get beyond it. All we proposed, we proposed a very simple thing, some particles that can really only give these probabilities without mistake. But that is not a full theory. A full theory, if we go beyond quantum mechanics, should explain the hydrogen atom, should explain nuclei, should do a lot, a lot of things. So this is just a very, very small beginning. However, Van Damme, a computer scientist, came with a very clever example. And that is the example in which Alice here and Bob here and Bob here, they want to talk they would like to meet. And they have very busy, they have very busy schedules, Monday, Tuesday, and so on. So almost all of their days, they are very, very busy. And they would like to find, is there a day when both of them are free and they could meet? And they talk by telephone. This is a telephone line. And suppose the entire game is, Alice will tell things to Bob, and Bob would have to decide when they can meet, okay? Case are more interesting game. Let me make the game more interesting. And the game is like this. Bob just has to answer one simple question, is the number of days when we can meet even or odd? That's all. That is the question Bob needs to answer. Not when, on Monday, the 7th of May. No, only if the total number of days when they could meet is even or odd. So there is just one single yes or no answer. So how much did, does Alice need to tell to Bob? Alice thinks the following way. Look, it is conceivable that the only day in which Bob is free is 1st of January. So he really must know whether I'm free on 1st of January because that's the only possibility. So I must tell Bob if I'm free or busy on 1st of January. But it is conceivable that Bob is actually free only on the 2nd of January. So then I must tell Bob also whether I'm free or busy on 2nd of January. And in fact, I have to tell him about every single day. So that means that although Bob only needs to find out the answer to a single yes or no question, Alice must tell Bob, her entire calendar. Like this, Bob finds out much more than what he wanted to need, than what he needed to say. He knows exactly all the calendar of Alice. He can tell which days they can meet, how many the days are, which are the days. He's not supposed to know that. He doesn't need to know that. So the communication in this situation is redundant. And it is not redundant because people generally like to talk too much. Like I like to talk too much now, you know? Sometimes the mathematics of the problem forces the redundancy. And this is what happened to poor Alice in this case. All right, so when I said look, but here it's another possibility. Suppose that in addition to this telephone, let me replace telephone by saying a classical, a classical communication channel. Suppose that in addition to that, they have some particles that were prearranged here. And here they have these measuring devices. And they have many particles like this. Like the classical line, which was put there in advance by the company, these particles were shared between them. But they were just kept outside the measuring device. They were not measured. So then perhaps Alice looks at her answers, at her days. And she would put the spin up or down according to whether she's busy or not. Because we know that these devices are correlated. Can they do something more? Aha. So let me come here. And you see, here is what he proposed. So he said, let me say that up means busy and down means free. Okay? So in all these situations, either both of them are busy or Alice is busy or Bob is busy. So all of these are busy. They cannot meet in any of these situations. The only situation where they can meet is this one. When both of them are free. So this is, so let me put here. So here is no, we cannot meet. Here is yes, because both of them are free. So what they do, they put this thing in their devices. So in cases when both of them are free, the answers come, let's not forget what were the answers here. The answers in all these situations were green, green or red, red. And the answers here were green, red or red, green. In all these three situations here, in all these three situations here, when they cannot meet, the number of green outputs is even. Okay? Either zero or two. So for every day when they cannot meet, the number of green outputs just increases even. The only thing that is relevant is here because for every single day when they can meet, you add just one green. Either because Alice got the green or because Bob ran the green. But whatever it is, you just add here. So if the total number of greens is even, these really don't count, they always add even. It means that there were even number of yeses. If the total number of greens is odd, it is because the number of yeses, each yes as a single green is odd. So all they need to know is the total number of greens. Okay, Bob can count his greens. Alice can count his greens. But actually, all Bob needs to know is whether Alice has an even or odd number of greens. Because then he can add to his. So all that Alice needs to do is to send a yes or no, or a zero or a one, telling zero if it is odd, one if it is even, or the other way around. So now the entire redundancy in communication goes away. She needs to only send a single yes-no answer for Bob to find the result. So already we find something very interesting. If it would be possible, why did I erase this? If it would be possible to go three over four, two plus root two over four, if I would be able to go up to here, all redundancy in communication for this problem will be gone. Now, this is not a silly game. This is in fact the most difficult communication problem that you can imagine. So if this can be done, all other communication problems can be solved without redundancy. Okay, the technical name for this is the inner product problem. So that's an interesting thing, something else. You could imagine playing that game with quantum particles, so with something here, okay? So here, this is up to where quantum mechanic can tell you, can take you. None of this will help. That is a more complicated proof. But if you have quantum particles, they only succeed with this probability, and that is not enough to reduce the redundancy in this problem. There are other problems of communication where quantum particles can also reduce it. But for the strongest one, quantum mechanics is no good, although it has no local correlations. More interesting, what happens if you are not perfect, whether you are imperfect? So this is 0.87. This is, of course, 0.75. This is one. Well, it is known that if you go up to 0.91, you are still okay, so no redundancy. What is happening here, it's not yet known. It would be astonishing if something happens at a quantum level. In this problem, we don't know. But there are some other problems. There are at least four other similar problems in which precisely when you hit quantum mechanics, something happens. Something that you could do, you can no longer do. So quantum mechanics appears as a limit to some informational problems that have nothing to do a priority with quantum mechanics. Not for this problem, for a few other. So first quantum mechanics was discovered by thinking of spectra atoms or why atoms exist in the first place and things like that, photoelectric effect. But now there appears the possibility. We already see glimpse that quantum mechanics is important without starting from physics, starting from purely informational tasks. And something happens the moment when no local correlation hit the quantum limit. So it is perhaps conceivable that we can see some deeper reason why nature is quantum mechanical. But this is just a story at the beginning and perhaps you could contribute to it. Thank you very much. Thank you very much. It was really a fascinating lecture. Is there any question? Maybe I can start with the question myself. Yes. I mean, it's a very probably trivial one. But I mean, in all these models, there is another fundamental ingredient which is the fact that you have a person, Alice and Bob, who can take decisions, right? Okay. So is this ingredient relevant for what are these considerations you made? Yes and no. That is obviously it is important that you can have people that can take decisions. But if you would not have people to take decision, then why should I answer your thing? Why do you answer? Why anything happens? So from this point of view, none of the physics would exist if we could not choose what measurements to do ourselves. Because then we would be limited and nature can play a trick on us. You know, whenever nature knows that we want to perform a collision, we'll give the results that we want. So if we are not free to ask, then nothing works. But the question is not, I try to make it silly, but the question is actually profound because there might be cases using this thing for cryptography. So you may use this sort of correlation to transmit secret messages. And perhaps there you need many, many decisions to be taken. So it's not Alice and Bob that take their decisions because they could not make 100,000 decisions per second. So then they will use some devices to decide where to put. And if devices are even a little bit predictable, maybe the adversary, the eavesdropper, could make use of that. And the question is how much pure randomness versus predictability can still ensure that, which is not nature in this situation, but is the eavesdropper, cannot use your own predictability to do something. So in fact, it is a very relevant question when you think like this. And it is another basic fundamental question, how robust this thing is versus the predictability. So there are some things that are known here. Once you phrase it like that, there is something you can calculate. Thank you for this lecture and another silly question. So if quantum mechanics can bring you, also classical mechanics can bring you to three fourths. Quantum mechanics can bring you to say 0.97. So what are the mathematical structure? Can you give an example? What are the mathematical, can bring you beyond the, so what should you replace the wave function with? This is actually a very good question. So how can I show that quantum mechanics? Well, that classical mechanics can come up to here. So quantum mechanics is of course bringing up to here and classical mechanics only up to here. That classical mechanics can bring you up to here, I just proved. Whatever you do in a deterministic things can bring you up to here. How can I prove first of all that quantum mechanics is limited? Because there you would think that I need to take all kind of models, what particles to use, whether to use spin or polarization or momentum, you know, or some molecules. The fact that quantum mechanics is limited up to here, that only depends on the Hilbert space structure of the space of states of quantum mechanics. And the fact that results of measurements are done by operators and the fact that operators at a distance commute, these are the ingredients that come in. So they just based on them, we know this. If you want to change, for example, that the space of states is not a Hilbert space or perhaps that the space of states at two locations is not a tensor product of the two separate Hilbert spaces. All even tiny changes like this can allow you to go more, okay? What is interesting, however, if you try like people try to make some nonlinear changes of quantum of Schrodinger equations, those immediately will allow you to signal. They will make correlations that when Bob changes where he puts the switch, you know, let me put in a different way. I presented here just one particular situation, but generally instead of these results, I can describe everything in a black box, a black box scenario with some input x, output a, input y, output b. In my case, x had only two possibilities, up or down and y, and a and b were only binary, but you may imagine more complex things. And the entire physics you describe by a probability of the answers a, b, given the inputs x, y. What Alice sees on her side, she only sees some of b of probability of a, b given x, y. She doesn't know what b is. And what is important, so this is the probability of Alice to c, a, when the inputs were x, y. And the non signaling, the non signaling condition is that this would be probability of Alice of a only of x. If the variable y would affect the probability of the output a here, you would see. Most things that started, people longer go thought the Schrodinger equation is linear, add some tiny nonlinearity, perhaps that would help you understand collapse or something like that. They would immediately lead to probabilities that are signaling, are not of this form. But if you start thinking of the structure of the mathematics as discussed, like taking the Hilbert space not to be a tensor product, but some other mathematical formulas, then you may ensure that things are non signaling. So there are a few papers along that line. And in fact, there is an entire hierarchy of mathematical formulations that go towards quantum mechanics, but they violate quantum mechanics a little bit. We can discuss more about that later. More questions? Yes? Yes. How do we actually know about this 0.91 on the probability axis? How do we know about that? Yes. Well, because there were some clever people that calculated it. Okay, that is basically the answer. They wanted to look at that particular problem. They imposed various probability distributions here with X and Y being binary, A and B being binary, and then they tried to look for various protocols to solve that problem and tried to use some clever methods. When you have noise, you may want to clean up via redundancy or all kinds of things. In information, for example, if you talk to my mother, she's old and I tell her something and she says, could you please repeat? I didn't hear well. I say it again, could you please repeat? I didn't hear well. So in the end, you do error correction by redundancy. In the end, she would listen and know very well. And there are better methods invented by Shannon and others how to correct things. So trying to do that, if they were not getting perfect correlations but slightly unperfect, then they tried to use some standard ways of encoding information to try to bring it down like a majority counting or things like this. And with the tools they have at their disposals, they did this. Now to go lower, perhaps you cannot or perhaps there are better methods. And incidentally, all these questions, they go back and forth between information theory and physics because once you raise this question, then you can see where are the limitation in the present knowledge of information theory. So for example, this game that succeeds always does something mathematically very interesting. And that's the following thing. If I call up to be zero and down to be one and this to be zero and that to be one, the answer is the type of correlation, say x dot y equal a plus b. With that sign being modulo two. Because when either x or y are up, so when either of them is zero, this is zero. And the only way in which they can be zero is if a is zero plus zero or one plus one. On the other hand, if both of them are one, then it is one times one is one, which can be done when the answer is zero one or one plus zero. So that machine that always succeeds does a fantastic mathematical transformation, transforms a nonlinear function of the inputs into a linear function of the outputs. So you already see that this game that started with Einstein, Podolski and Rosen and with hidden variables has a dramatic sort of representation. If you push it to the limit in what the mathematics of it is. And the one in which you get the vertex in which you always succeed is if you wish is the most powerful zero rate of communication channel. It doesn't allow you to communicate but you can do the most. Nobody imagined things like that. Okay. There's a question down there. Yes. Hello. And this might sound more like a philosophical question since you don't change, if you don't change the postulates of quantum mechanics, all the discussion follows. But do you have any opinion about the interpretations of quantum mechanics to explain the measurement problem and all sorts of philosophical open questions? Whether do I have? Or whether do I have it related to this? Let me... Okay, so first of all, why is the question? The question is because you may know that although we can use quantum mechanics to tell us things in the lab and one of the basic element is superposition, we are made out of microscopic particles. So then you may ask why we are not in a superposition of being in different places. And we've never seen that. So the question is, will we see if we were actually, could we distinguish and so on? And since these questions are not experimentally available, then they are just philosophically available. So that's why that's the background of the question. Now there are a few things to be said here. Number one is that the measurement problem is not the only important problem in quantum mechanics. So this is another type of question that people just thinking of the collapse were not able to, well, did not address. Once you know this, and this comes with its own issues. Now, there is a connection between what I said here and some of the interpretation. Here I said it is non-local. What you do here influences something here. This of course implies that Alice will see a result here and Bob will see a result there. Some interpretation of quantum mechanics like many worlds interpretation, there's not a single Alice. The moment she does a measurement and there are a few possible answers, there will be many Alice's and there will be many Bob's there. And then when they come together, the question is which Alice couples with which Bob? And if you do it correctly, you can get rid of non-locality. And everything is local, but you pay the penalty that suddenly you may have a million Alice's instead of one and many worlds and things like this. Some people like that, some people don't. Ask Alice, well, in this case, which of them? Ask my colleague Levite Mann, he will say yes. Ask Aharonov, he would say no. Ask me, I would say no. So there are all kinds of things, but clearly thinking of non-locality adds an element to the philosophical discussion. To advertise my talk tomorrow, that I think will add much more. Okay, yes. Thank you. I think we have to stop now, it's already one hour and a half. So before I close, then the final lecture will be tomorrow morning at 11, so not in the afternoon, but it will be in the morning and it will be precisely on the floor of time, another fundamental property of quantum mechanics. And you can understand the students will have the right to stay here. Yes, and ask me questions. You mean now? Not necessarily, actually that was only for the first lecture. Only for the first. If they want, I will be here for the next few weeks. Otherwise we have refreshments for everybody outside, so perhaps you can also continue the conversation outside with the class or something. Okay, thank you very much.