Thank you very much for the introduction. Can you can hear the book? Is it okay? Yes. Great. So, yeah. So this talk will be about story mostly. So it's an interesting story. So a story of quantum information, a small thread that went happen at some point. But we start with a very basic point, which is thanks to quantum physics, we all have learned at some point this statement in a way or another that the outcomes of a measurement do not exist.
The measurement. This is one of the. Things that we learn to believe when we start studying quantum mechanics. And if you think, where's this been proved? Where does this come from? Probably in the traditional way of teaching quantum mechanics. The answer would be Heisenberg saying that the relations right, because there is the first case where explicitly it was noticed that not all physical observables can have sharp values.
If I know the value of the if I if the state is such that the value of the valuable position is well known, it turns out that the variable momentum is not so well-defined. And which means that if I measure momentum, a negative result where this outcome did not exist. My measurement. Now. In order to introduce inequalities, I want to somehow criticise and sovereignty relations. My criticise no. But Bush explained that they don't have some features that we would like to have.
Especially you think of trying to convince someone who doesn't believe in that statement. That is, after all, a pretty strong statement. So what happens with 71 nations? First of all, they could be interpreted. It's not the orthodox way. I hope your teachers have told you not to do that, but they could be interpreted as imperfections of our measurement devices. And it turns out, as you probably know, Heisenberg himself made that mistake.
So after driving correctly, mathematically, he's on 70 relations. When asked in a Sydney popular book to explain what happens. He came up with this idea of the microscope where, you know, there is the particle coming there and then in order to measure it, they must interact with a photon. And a photon is almost each barber calculation. Something comes out and then a semi classical model for that, which is not wrong, but it doesn't capture the whole yet the whole idea.
Besides, I can easily falsify an uncertainty relation with crappy measurements. Imagine that my measurement of position I don't plug it in. And so the needle is stuck there at zero. I did. You know, I get that x equals zero. Violating the uncertainty relations. They always say, Oh, but this is a bad measurement of how you check it. So you need to have perfect measurements, even perfectly well-characterized in order to test and uncertainty relation.
And I can again. I can easily fool you by selling you and calibrate the device so badly. Can you buy the device that would convince you to do something which you actually do not hold? And I can measure position and momentum perfectly. So in fact, the conclusive test that the outcomes not exist is these violation of inequalities. I'm sorry if I don't even write down in the whole talk a single inequality. To understand this talk, you don't need to have the equation and the inequality.
It's an inequality, something smaller than something else. But I would not spend time describing what it is. You have to believe me in these. Most of you probably have heard these things before anyway. So what happens with inequality is, first of all, you cannot fake it experimentally. So if you have a bad measurement, you may end up not violating the inequality. But if you do violate somehow is a simplification that things are going pretty well.
Contractually on seventh in relation. Now it's a is a statement that these things are independent. So the inequality itself does not rely on quantum theory. I'm not gonna tell you again to yourself that the relations for which I need the formalism of quantum theory in order to then verify that indeed this to operate does not compute. And they have all these various relationships. It's something that you really write down. What does it mean that the results pre exist?
And this implies some conditions and that's what you write down. And then you test that in the lab or in quantum theory, these conditions may not be satisfied. That can be violated. A consequence of these on which I will come back a lot is that this device independent? This means that it does not rely on a description of what is being measured. So it doesn't matter if I'm measuring photons, I'm measuring atoms and measuring Bose-Einstein condensates, electrons in solids, whatever system.
The inequality does not depend on that. So the the assessment that the benefit what is being violated does not depend on my knowledge of the experiences. Needless to say, the experimentalist always doing the thing as better know their job. So the experimentally should know that they are dealing with photons and measuring polarisation. They should do it very, very well. But myself, as a verifier, I don't need any of this knowledge.
I just can look at statistics and say, Oh yeah, this phenomenon, whatever it is, by all these bad inequalities. It got very exciting because I've be first noticed by John Bell in 1964. So more than 50 years ago and have been many experiments with what John Bell noticed is that the predictions of quantum theory are incompatible with that.
Of course, at the end of the day, the jury is out and the jury's nature, you have to ask nature whether quantum theory is correct or is close to correct, gives the right prediction in these cases, or maybe not. And if both had been made and all of them easily came very close to the predictions of quantum theory. The first one that is considered conclusive is 1982 by Alan Spain near Paris.
And you might know that two years ago, one year and a half ago, three groups reported three allegedly loophole free bell tests, whatever that means. We can enter into these technical details later. There were some loopholes. I mean, if you are really a crackpot, you could think that there were some loopholes in the experiments of us being other people.
You really need to think carefully. But yeah. So anyway, this loophole free had been made in Delft in Vienna and well with a source and optics based in Neath and then with plenty of collaborators around the world. Now, as I said, measurements do not reveal a pre-existing property is a pretty strong statement. In fact, as a student, at least you and I should have questioned it. Well, normally we don't we just accept we have to pass the exam, right? So you better repeat it.
But it has been. It is to get some. Debates some confusions. Some people have been speaking of local variables. It's not the way he done is not the way local. What does he mean local? Some people get very, very angry if you say that quantum theory is non-local because quantum theory is local. Of course, these different definitions and they just like to entertain the discussion
instead of clarifying the definition and knowing that there's no discussion. But, you know, things happen anyway from the youngest among us. Probably this is a very constructive kind of discussion compared to what we might have had, which something of this type. But I made up these ones. But nobody cares at this point. Right? So whether it's authentic or not is no longer a question.
So anyway, so that's the situation of quantum theory around the 1980s that yeah, there are plenty of, let's call them conceptual problems or things that are a bit troubling. The measurements modify the state. The realism is no longer there. Yeah, whatever. And then, well, you know, this quantum theory is fantastic. So you can just go ahead by not listening to all these problems and calculating things and you get always the right predictions.
And somehow I like to think that quantum information is like the guy in the middle of you recognise who he is, who decided to not to give in to the song of the sirens. Not by not listening, but by listening, but being forced not to follow them. And because otherwise, you know, what's called the the mythology goes that if you listen to them and you are free to go, then you would automatically go and jump in the sea.
So the question that the quantum information community started asking is, can all these we have stuff be actually useful for something given that it's there? And. Well, I'll give you the first example where some of these real quantum stuff as it is itself was identified as useful is quantum key distribution of quantum cryptography. Let me remind you what that is. That's the scheme of Benetton Brussel in 1984. The idea is to distribute between two partners, Alice and Bob.
All good people. Secret key at a distance so that later they can exchange secret messages by encoding these messages to the secret code. So the idea that is, of course, everybody's been exchanging secret codes for many years. The idea here is to do it at a distance, because otherwise, if I'm lucky, I can also go to LA to give him my piece of paper. He keeps it and we have done it. But the idea is that I don't want to come closer to him at a distance connected by some channel.
And then one of them got some notice that if Alice prepares a speech in half in four possible states, plus z minus that plus x minus six. And Bob measures the Z based on the X basis. What happens? Well, it's pretty clear what that means, right. So if if we are placid and Bob measures said he would get plus minus Z minus. But if Bob measures Z but at least we build something like X in the complementary bases. The results are completely random because again, they don't exist.
So the if the state specifies the direct, the speeding the direction X in the direction Z is completely randomised. And similarly, of course symmetrically, I'm not going to tell you how the scheme goes about in creating. Well, essentially you see that the idea here is that now if they've measuring the same basis, if the measurement and the point based on the same basis, then Alison both share the same at least has kept the plus. And Bob has discovered a plus or minus and a minus.
And if they measure in different bases, they can describe them later. And the idea here is that measurement modifies the state is put to the fruition by thinking that it's an annoyance. It's an annoyance, yes, but for the enemy. So the eavesdropper theory called evil is there in the middle trying to learn because she would like to share the secret in order to decrypt the secret communication symbol. But if Eve tries to measure what happens here, she modifies the state.
What does it mean? It means that this that these things will no longer be seen. And I stumbled by checking some of their results. They could see that there was a problem in the middle. We don't have the perfect statistics. We have some noise there. Someone may be on the line. Let's stop the communication. So that's the idea how the idea of measurement modifies the state. One of these quantum we of NASA's whatever to call it is put to fruition in order to guarantee security against sort of.
Now here comes one. By the way, I should have said at the beginning that the talk I'm going to give also involves many people that, uh, you may know and some of them even in the room. And in fact, the next one I have to mention is Autoworker Walker, who in 1991 rediscovered the idea of quantum solutions, which is the same. Sharing a secret key between two partners but using now entanglement. So using these outcomes not pre-existing that they associated to inequalities.
I think I can do better than just reading what Atwater wrote in the paper. You see, the eavesdropper cannot elicit any information from the particles while in transit from the source to the legitimate users, simply because there is no information encoded there. The results do not pre-existed the measurement. Then the information comes into being only after whatever noticed that at the time that you've stopped it will still a man later go to work on Eve.
But nowadays, probably they all should say they write something that is safer and inclusive. So you may say, Oh, fine, let's finish the talk here. We have an application for this strange idea that outcomes do not exist. It's helpful for key distribution, and he always gets complicated.
Because. The following year, the same two guys as before Bennett and Bass, plus David Mermin, probably well known to some of you, were known in physics, teamed up to stress that the protocol of EQ is essentially the same as the B. The 84 protocol is a sort of the entanglement based version, and the idea is somehow the following if you consider that all these things belongs to Alice. Then. Well, trust me if you don't.
But this is not it's not that difficult to prove that the produced to entangle pairs and least measures one in the ex based on the Z basis. What happens? Well, she gets the result and automatically the photon that goes to Bob gets split up in the same state. Okay. So somehow you can consider that this thing is the source of Alice. The only difference being that in the novel Dignity for All, Alice actively has to choose the basis or in fact, the state he it's chosen.
She can choose the basis, and then the state is chosen at random by the quantum system. But it's exactly the same. And indeed, if you look at the correlations, the conditions are the same. So if both Measure Z, they get either both plus or both minus. If both Measure X, they get either both plus or both minus. If they measure in different bases, they get random things. So it seems that. This idea. Somehow the idea of outcome do not exist.
Yeah, it's useful. But you get diluted into the into the same scheme as measure and modifies the state and then well whatever. Let me go be for wanting theoretical quantum redistribution because these ideas have been very, very influential. In particular, the dis translation between entanglement based and preparing measure schemes have been the key to what is called the proof of unconditional security. This is a technical world. Don't take me to face value is not is unconditional.
Love is the same word, but it doesn't carry completely the same meaning. Yeah. Unconditional means that you don't make any assumption on the eavesdropper. Unconditional love normal. You don't make any assumption on your partner. So somehow. So Heisenberg may be doing something wrong and he's not covered by these proofs. What is proof cover is that whatever the eavesdropper does in the middle. Assuming I listen to everything very well, we can guarantee security of the polygraph.
The first stretch proves just what I like. History. So the first such proof 1998, my youth. Nobody understood it. In 2001 shot in basket this one people understood it was much more easy to follow. Nowadays the framework that we use is has been developed if not well knowing this P.G. theses in those years. And again that's a statement about unconditional security.
Let me say it again that a quantum solution is probably secured against the most general attack of these Dr. is expected because the security is based on the laws of physics and whatever, however powerful is not very he or she cannot break the laws of physics. So. When I entered this field in the year 2000, quantum information was many told win thanks to the following thing. Essentially, you know why these have been very useful? Essentially because they shut up the unbelievers, right?
So those who still hope that quantum mechanics could be formulated in terms of local variables and keep to determinism and whatever, you know, that's it. We've proved them wrong. Now. Shut up and went entanglement in theory. And so for applications, we don't need them, we need entanglements in other things. So essentially that was a situation of inequalities at and your 2000 2005. But something happened. And what happened is these.
These guys, Antonio Seem, who's in Barcelona. Nicolas is on Geneva. At least my son is was in UCL and this moment. They come up with the following observation that is not that deep. Once you notice it, is that the ideal coalitions of these b b 92 Protocol VBA 92 can be obtained classically without any quantum. And here how you do it. So to this source, whatever it is, is sending two classical beats to Alice and two classical beats to both.
Classical zeros and ones. Now suppose that these beets are built in the following way. One possibility is that both of these beads are plus and also both bob beads a plus. Another possibility is that the first bit of is plus and the second is minus. And the same for Bob. The other possibility is the opposite of these, and the last person needs all of us. So these are really classical. There is no quantum physics here, right there. Just I've just written down the list.
The results are pre-existing. But look at the correlations. But if both measures, then they have the same result. If both Measure X, they have the same result and if they measure X instead they are completely randomised. So this should ring alarm bell, right? We have unconditional security proof that according to quantum theory, the laws of physics prevent an eavesdropper from knowing these. And these can be done with classical physics.
No, quantum, nothing. So what's happening here? Are all the security proofs wrong? And the answer is no, they are not. They were correct, but they gave a hint before what went on, that there were some assumptions. So the assumptions the assumptions on the devices are very simple. So remember I told you we moved on conditional security against these doorbell. But we made assumptions on what that is. And probably and the assumption was that the source of producing quantum bits.
My source now the classical one is producing two bits, two and two. So they mentioned four. If I stick to one beat even once, only quantum beat and there are something about the measurements, then they are conditional security proofs go through. Then those correlations you can only achieve with a maximally entangled state of two humans. So you see how an undesired assumption or unnoticed, obvious to everybody, everything is within quantum information anyway, right?
But those of us, for me, the quantum information, we love humans. So let's work with cubits. Well, that was an assumption. And now maybe you see where I'm going. So the bell violation is device independent. The Bell violation does not need me to assume that these are fraudulent, that these are them. So these are cubits. So can one look back and now have a security proof of cryptography based on the violation of a inequality? And the answer is yes, we can. We are in American politics these days.
So that's the paper where the word device in the pen was first used. The collaboration between people in Barcelona, Geneva and somewhere in Belgium. In Brussels. Uh. Well. Well, not too much. I understand what we have done there in the paper. I just want to stress that by sheer coincidence, this colloquium comes as close as it gets with the 10th anniversary of the submission of this paper. I didn't plan it, but I noticed after when I printed this slide.
Okay. So let me summarise the story so far. Here it is. As you can imagine, there's much more coming up because a whole bunch of empty stuff here. But so essentially cryptography. We had BP 84 E 91 cap 91 using bell inequality. So somehow we put it at the intersection between cryptography and bell. There is this sort of critical comment by Ben Emberson in Melbourne saying we don't need these bell inequalities. It's exactly the same as the BP 84 thing. These are these sort of security proofs.
However, I think his own my son has noticed that BBM 92 has these. The correlations can be obtained with purely classical means if you don't make this assumption of cubits. And this leads to the device independent paper. Now, let me complicate a little bit the story, because. It actually went like these. So I think she's on it must honest didn't get their idea. Waking up in the morning one day and having a stroke of genius. And the story is extremely convoluted. So this path.
Is Will. And I'm going to tell you what this puff piece. So. If you have to bring back something on the stove, please remember. How the [INAUDIBLE] did they find this very simple idea coming from such a complicated path? So what trigger that these paper. Is this paper. And what we get is what is this? What's let's start here. And even before starting here, I have to introduce a notion because oh, there was another thing that I would mention later.
There's something else and you'll come back later. Are you going to use the notion of no signalling that what is no signalling is these? It means that Anderson, Bob can be correlated, but. They cannot use the coalitions they share to send a message. Let me give you an example of not seeing any resource share the randomness. If by quantum key distribution or by just giving my piece of paper to Vlatko, we share a common list of things.
And then I go to I go back to Singapore. Vlatko stays here by by doing anything on my piece of paper. Nothing will happen on Vlatko side. I can change the digits. I can burn the paper. I can lose it in the play. Whatever happens, Black doesn't notice anything. This means it's a resource. We share some correlations, but is no signalling. He doesn't. What did I do on my side? It doesn't send a message to him. I need to send something to him in order to. To saying something.
Yeah. Now, as you certainly know, it is a pretty good example of why we do have this pretty good example because as you certainly know, quantum entanglement is also of this type. If I share entanglement with someone, if I make measurements on this particle, the statistics on the other side don't change. I cannot send a message by just measuring on my side a half of an entangled pair and hoping that the guy that keeps the other half sees anything changing on his side or her side.
Okay. So sharing an entangled pair is another signalling resource. And. By the way, I should mention that if I can send a signal the evolution of why these trivial. If you don't know what he's up to, maybe not obvious, but if I send a signal, any coalition can be established. I can tell you a little. If someone asked you this question, reply that way if someone asks you the other question is by the other way, so any kind of combinations can be established. If I can say nothing.
So what is intrinsically interesting with quantum mechanics is that the bilinguals are violated by these entangled states that are not signalling. So there is no message going on. And as you know, the experiments are arranged in such a way that if there were a message that would travel faster than light, that's the reason why to shut up the crackpots. They organise the thing like that. But if you believe it to be in quantum theory, you already know that there's no signal going on.
So why introduce this notion? Because the 1984 paper of Perpetual and Haulage is essentially these. He's asking the question. Can it be that the physical principle that defines quantum physics is the following We can violate the inequality without signalling. What do you mean by physical principle, defining quantum theory? Well, how do you define quantum theory? Hilbert spaces operators. Linear unitary reversing. This is doesn't sound like very physical, right?
It works. And the first one to say don't want to modify it easily. Okay, but it works. But we don't know. Knowing that we have to go beyond classical theory. We don't know why we ended up with a Hilbert space, with a vector space, and not with a more complicated stuff. So I could be completely different. One. The fact it works. And of course, the old kind of construction, see if I can justify these. But so these guys ask this question is it possible that quantum theory ultimately is the most.
Or the unique possible physical theory in which we can veritable inequality without signalling. And they came up with Answer No, it's not. So when there is an object, you forget about it. But essentially the answer was the no signalling physical principle identifies more than quantum physics is bigger. Or of course, as usual. Other people had the idea before, but they were duly cited, but forgotten later. Now this is just a. So here is what?
Remember we had two papers the HK, but I probably can't to Barrett only if he's in the home, but otherwise he's looking the across the road. Jonathan Barratt And Lucien Hardy is in Perimeter Abbey on Kent is in Cambridge. So they ask the following. Very interesting question they asked. Given that. We can. Let's look at this. Suppose that one day quantum theory is superseded by some other more general theory.
But this idea of no signalling escape so you still cannot communicate faster than light in any possible frame is just that with these correlations. But maybe more than those that are allowed by quantum physics, can we still prove security of cryptography? You know, at the end of the day, all the security of quantum cryptography is based on the fact that quantum theory is correct.
The day when quantum theory is proved wrong or partially incorrect, we will have to revisit all the security proofs of quantum crypto this guy has asked. Suppose a quantum theory collapses or is replaced by something more general. But nothing. Nothing is kept. I always see safe in cryptography and they answer somehow yes. And so here you may ask. Well, but if they ask this question with the story I told you now, why I didn't throw a natural from here to ballot.
How then can. It sounds like these are the guys who are going to rescue from oblivion. Not not oblivion. Strictly speaking, he was very much cited. But from being confused with the BBM 92 with the argument, the answer is this is for me, one of the mysteries of my field is a small mystery and I think nothing very big. But they got it completely wrong. And these are very smart people. But somehow the pressure of.
The custom read what they've written. It's interesting to note is that the quantum key distribution, whatever what it performs, untestable inequality, it may appear as if nonlocality the violation by the inequality is played a crucial role there. However, if not, why or because there we use the inquiry for a different purpose. And as proved by VBM in 92, it doesn't matter. So why they got it song? I don't know. I'm very happy because it allowed me and my friends to be the ones who got it right.
So here is a summary of device independent distribution. What happened after all these things? Very quickly. So we have Britannica in 2005. They have a general security proof, but zero key means you can get one bit of security after infinitely many rounds of the experiment. Normally you don't have too much time to wait. So essentially zero, right. These guys, they had a finite field, but only some attacks against these no signalling adversary.
These advancing is more than quantum and then whatever. And. Well. Not finished yet. Now what happened? If we bring back our good quantum theory, we say, okay, forget about all these this thing we like. Let's believe in quantum theory for the moment. And so they thought that can only distribute resources are compatible with quantum theory. That's the ideal back up. Of course at the beginning that's the idea of this device, independent data.
And now recently there have been developments that maybe these words are a bit too technical for most of you, but for some of those who you can read them, there are very interesting developments where essentially they proved that the and condition security proofs can be done and they are very similar to the result we got here. So it works and well about experiments and in in key distribution it's a bit difficult in for for device independent. Anyway, the show goes on. So we are here now.
Let me summarise. So here we have a cup of pasta and only here in the middle I put everything. I put Bartunek and I think it is on my Sundays and in our paper, all this bunch of things. So what happens late that what happens later is that this idea of device independent took off. So finally, BELLINI Quality is really useful for something. You can test the violation or the security of your protocol, cryptography, without knowing anything of what's going on. It's pretty remarkable.
You can buy two black boxes and test without knowing anything of quantum theory, without being experimenters, without opening them. You could test in principle that these guys give you unconditional security according to laws of physics. Black box security proof. That's pretty nice. And people started thinking. And one thing that happened is these device independent randomness generation you know handling this is how saw. In many cases, right? You can use it to run Monte Carlo simulations.
You can use it to uh, uh, for cryptography to, to, to generate keys. You can use for many other things you can use to test software that are too complicated to test anonymized benchmarking and whatever is called. So what happened there? Well, people noticed that the. You can assess randomness in a device independent way. Now, wait a second, Mr. Billy. How, counsellor, this comes after cryptography, right?
Because at the end of the day. Quantum key distribution means generate a random list between two people. Random for everyone. Yes, I. So, Alison. But nobody else knows anything about this list. Now, if you can generate randomness at a distance, you can simply generate randomness. Without the distance. Right. So it logically randomness should come before you could. It's the idea that the results from all three exist.
So when I make a measurement, I generate randomness and I can use this beats to generate randomness. So what happened there? Okay. Let me tell you what happened is that indeed, there were before some random number generator based on quantum devices. This one is commercial. Year on the year 2000. But what happened is that in order to certify these devices, you need to open the box. And now at this point, you see there is no advantage of being quantum.
Because I can generate randomness that we need in the lab just for fun to show to the students by taking a resistor. Most of you probably know that. Connect the resistor to a voltmeter without current. You say what is the voltage? Zero 12 zero. Yes, but there is thermal fluctuations and if your voltmeter is good enough, it sees fluctuations. You can generate random numbers by thermal fluctuation of the resistor. What's the point of that is normally you catch the radio.
Well, it's an antenna. And so you normally so you have to get rid of the radio. So bottom line, if you have a device that you need to characterise. Any way to go for Quantum or to go for classical? It doesn't really change much. What may change is whether this guy is much faster than his sister is much more stable. You know, a lot of this stuff depends on the temperature by definition. And so maybe this guy doesn't depend too much on the temperature. So there can be some advantages.
But I'm not very I mean, this one is cheaper, is much better, is not done, by the way, now, because this store is much cheaper. This guy is by €200. So. So before device independent, there was no real advantage in generating random numbers by quantum physics. There was only an advantage in generic. These are the numbers at a distance. That was. Yes. That that cannot be done classically. But in generating them here.
Who cares? Now, of course, now device independent does provide the quantum advantage because now with device independent generation of randomness, I can generate randomness without opening the box. I can certify that my device is producing randomness. Without knowing what the device is actually doing. And this is pretty cute if you think about it, because it contradicts these. Which is the most accurate, scientifically accurate cartoon of Dilbert.
All the others are managerial actuarial, of course, but this one. So this is the random number generator of the accounts department. And then producing this list? Not so well. I'm sure that's on them. That's what you can now be sure. It's perfectly correct. It's perfectly correct for what the philosopher called output randomness or. Part of the randomness, which means by looking at the list of what comes out, you cannot justify randomness.
But what about process randomness to certify that the process. Is generating these numbers without being pre-established. And the process is that's what quantum mechanics can certify for you. Device independence. So we devised a certification. We could certify that this guy, even if it happens to be generating a list of zeros for one hour, it's actually really generating a random pulse, of course. I mean, there is a bit of Bayesian reasoning here, right?
If I. And what if a generic New Zealand company that can operate in quality? So at some point the guy must polish some ones, but. You get the idea that in principle, given only the least, you can simplify the arguments of the least. But with this violation of any record, you can testify that the process is really generating randomness. Why? Because the process is not reading from pre-existing values. Okay. Great. So of course, yeah. I mean, I cited only one or two papers, but, you know.
Papers and money. I got arrangements for working on the got a grant to work in tandem. Other people need. Fine. So here we are from consumption. We've got randomness. The last thing I want to mention is this. The other side, what is called self-testing. So you may remember I left out this work of my SNL because in fact, I will tell you why. If in fact, they didn't really go to key distribution, that they wanted to go here and they inspired us to go here.
But what they proved is these and another people at one of only some of the best was in Bristol, by the way. So many British people, people working in here in this country. Daniel only he is in iceland and they came up with the idea of testing. And let me tell you what that is. So the best way to introduce Self-testing is by not by citing these people by slightly later, even by a mathematician genius called salesman who has proved essentially everything that we are.
We have to prove to the point that nobody has noticed them for some time. And then and now we are discovering all these results is the greatest danger of people working in this field to believe they have a theory. And at some point you check people to see one of 1994, you find, oops, he had it. Okay. So that's the result of that. That's what citizens summarise is. Let me read for you. Let's don't don't read everything. By the way, the abstract is very cute right after these.
This is the whole abstract for all the six new theorems without proof and 11 problems are presented easily. You build inequalities. Okay. And look at. Look what's written here in the square. Each State maximally violating the Belzer legislation of quality. So these inequalities I told you something like A is smaller than something and then you can violate by showing that is it becomes bigger than something and at some point it cannot become too big.
So there is a boundary there. So each they devoted maximal decision is essentially the same as the singular state of a philosophy one particle. Finally, a language that refuses to get on the stand. Remember, spin half particle state of x equals zero of the sum of two angular momentum one half. For quantum information. Up. Down. Minus, down, up. The same state as two electrons in a molecule in the same orbital. When you have anti-Semitism, that's as simple as it gets.
But the signal is strong, you see, because it's not the opposite. It's not. If I have the singlet state of the particle, I can maximally violate the clinic. That's a trivial exercise for quantum mechanics. One. It's the opposite. If I see that violation, it can only come. From that state's. I see a single number. Or some statistics. And I know the quantum state. And the measurements. He doesn't mention it, but I also know that the measurements are neutral, unbiased places.
It's pretty nice, right? Classic. A signature of a quantum state. You see statistics. You know the state you're going to think of. What? Of two photons of two atoms or two Bose-Einstein condensate. There is somewhere here and somewhere there a cubit. And they are maximally entangled. And. Now you see this guy gives a serious person gives the references. There is a pompous quality in 92, but he finds S.W. 87. Page 2442.
Now this looks suspicious. When ever have you seen someone signing a paper giving the page? Why does he do that? Oh, well, there was a reason. And the reason is that that paper is this one. If you don't have them. That is one of the best mathematical physicists around. At the time, he was a young fellow. And so these guys like Benny Liquides and Quantum 51 General setting a fight with the Alpha Sea Star Algebras, whatever it looks like a bit mathematical.
And indeed, you have to go to the page that the citizens kindly point out to you to find the following theorem. That's the same theorem as I told you before. Okay. If is equal to square out of two, which means the maximum element 18 the corresponding elements whatever from realisation of power has been matrices never implies that something these are precisely realising that what neither the description of this experiment. With some systems algebra notation. So that's the theorem.
That honestly, very few people in fact, how did I get to know about this? Because once I was giving a talk about this topic and Vanna was in the room and he told me, you know what, I think I proved that thing before. And then I took some. He must be one of my papers. I stopped the browsing. I saw how to find. Okay. This is citation. Finally is here for you. So. Okay, that's the in fact, that's the first proof of that statement. Then when all is calm and clarified the thing, read their upset.
We identify all states of two particles which are massive inequality, the straightforward extension of a single state or to spinoff. Okay, very clear. These I can also understand, and it's essentially the theorem that I was mentioning before. Now, this series of papers has no connection whatsoever with applications. This is a paper about inequalities, foundations. We have this, but inequality, if it's maximally violated. It's only state. Now the other side, there was the scalp, April 1998.
Here it is, published in Foundations of Computer Science Proceedings of the Conference. And here is what they do. Title Quantum Cryptography, Imperfect Apparatus provided a part of whatever he's missing, but he's perfect. The proof of these people is to remove the last assumption. And they have these search checking stories by now. You probably guessed what they're trying to do. Um. Now what happens in this paper is that. They they come from cryptography.
They don't mention Bell at all. They just take the 84 protocol or amend it, and they add something to add to save the day, to remove this fact that the coalitions can be obtained in a local way. But again, they they they didn't mention it like that. It's extremely complicated. People are going to read. They wanted to do device independent UK. What they ended up proving is self-testing is exactly the same as my perpetual information.
The other is namely that their criterion certifies uniquely the singlet state. So incidentally, there is something also very interesting here. For those who are computer scientist, you may ask if they are dealing only with the ideal case. Why do we speak of imperfect apparatus? An experiment I would have said imperfect. The bathroom is not ideal either. They have noise and so on. If a computer scientist means something else, it means these.
It means that one that works perfectly. But you don't know how. So it works perfectly for the task is just you don't control it. That's what they meant by imperfect operators. It's definitely not what an experiment would call an imperfect apparatus, because I think any parent would be more than happy to have something that doesn't know exactly how it works, but it works perfectly right. So anyway, here we are. These are device independent self testing at a glance.
We have myosin yo one criterion, one signature for the quantum states CHC another signature of the two classical fingerprint things of the single state and complementary measurements. And just to tell you what the state of the art is of the many kind of works in this field. So, first of all, now we have these classical fingerprints of every pure bipartite song written in Tango State. So meaning. You give me any state of properties?
And I can write down for you a classic statistics that if you go to the lab and you see that classical statistics, you know that state inside any pure state, your choice and many multiple state. So if you have three party, four partisans on the multiple, they think that democracy with robustness bounds. So, of course, you know, an experiment. They may say a sorry, the maximal violation of values quite really cannot make.
I can make the maximal violation minus epsilon. Does your criterion collapse? And the answer is no, it doesn't. If it's minus absolute, I can tell you how far your state is from the ideal case. So everything is fine there. We have the robustness and the computer scientists love this thing a lot because you can enchaine these proofs in a very complicated way to prove ultimately blind quantum computing. So you could think of my box and say, this is a quantum computer.
$200 billion. Are you willing to pay for it? Well, let's check it first. How do you check it? Well, you can check it. Look, it's not very practical, isn't it? But ideally, you check in a blind way without trusting what the manufacturer has put into the computer. Just sending things to check. Violation of inequalities. And if it does, at the point you say, okay, you can buy my quantum computer. And these these proofs must rely on this idea of some testing.
Essentially, you say, yeah, I test that the claim is correct by checking the states. So I'm finishing. Let me give you just the last glance to this nice field of device independent. There's much more. In fact, there's something that two dimensional witness entanglement certification. There is what is called semi device independent, namely, you know, once you start with all the assumptions and remove all the assumptions,
then someone comes in, say, can you put back a few of the assumptions? Like, you know. Yeah, I maybe I trust a little bit. Maybe I passed my measurements, but not my state. Maybe I passed that. This is a photon polarisation, so it's a cube, but I don't know my measurements.
So we can put back some of the building blocks and create certifications that intermediate right between the the standard quantum one way I believe I know everything and the device in the final I believe I don't I think I can put something in between anyway. So that goes the story. You needed the field of bell nonlocality or violation of local realism. You needed the idea of cryptography. You need to combine them and then you need to do it because if you did it when it went out of here.
So this intersection was very small for many years and it was actually an easy, crazy way by passing through these no signalling adversaries. And where they're one day quantum mechanics will collapse the. We've been following the randomness and this beautiful notion of self testing. So just keep in mind that this idea to be a quantum state, complicated as they are, they have a classical signature. At least they entangle pure ones. And okay, here's the summary that I essentially just did.
I just finished with a photo of my group. Very friendly and competent people. And I may have a couple of positions available if someone is interested. And with these, I thank you for your attention.