 OK, yeah, works. Right, good morning, everybody, at this ungodly hour. It's really quite early for me. And for every theoretician, I would say. And so I will give you a few lectures on the whiteboard, no slides or anything like that, where I will try and explain to you the concepts of information, correlations, and how they are resources. And not only in quantum physics, actually, but also in classical physics, they can be understood from a point of view of resource. And I will explain to you what I mean by that. And then I will move on to entanglement theory and how one quantifies entanglement, how this theory was developed. And then depending on how much time I have, I will also move on to more general aspects of coherence theory, resource theories in general in this field. But let's see how it goes. And I would like to also echo the comments of the two speakers before me that gave the introductory remarks. I mean, questions are, of course, welcome. I'm also around. So you can also grab me while I'm having my coffee during the coffee break. The point that I'm here is that you have the opportunity to interact with me. That's actually the real reason why I'm here. Not necessarily so much directly the lecture. You can also read some of these things simply in a book. But this is the opportunity to ask me and discuss with me. So no shyness or anything. Just wherever you see me, you can grab me and ask me. Right. So let's start here. What is information? So that's a question. And that's actually a question that had been asked already indirectly in the beginning of the 20th century, maybe. But it was really one particular scientist who really formalized this notion. And this was Claude Shannon. And so he in, I think it was around 1948, roughly, he made a very, very massive contribution by writing his theory of information. So in a way, this is like, I mean, he is kind of the founding father of information theory, just like, let's say, Heisenberg, Bohr, and Jordan Schrödinger were the founding fathers of quantum physics. So he really started this and turned this from some fuzzy notion into something that is rigorous and that one can really do maths and physics about. So which approach did he take? Actually, there is a range of approaches. But he liked to think in terms of communication, which was actually a good way because nowadays, communication, transmitting information across distances is a very, very important thing. I mean, you all use the internet. It's using all the time information transfer. So he was asking himself indirectly the question, if I have here, let's say, an information source, which can, for example, be a little man sitting in here reading a book and transforming this into a string of zeros and ones. And then he takes physical particles to send those zeros and ones out of the box to someone else, to you, for example. So he will have here 0, 1, 1, 0, 1, 0, 1, and so on. And then maybe somewhere here, there's a receiver. So here's a receiver. OK, so then what he assumed, which is not actually strictly correct, but in general, but to make a simple framework, he assumed that each of these zeros and ones are independent of each other. There are no correlations between those digits. But to the outside person, initially, they will look random. So each one of these physical objects here will, with a certain probability, be in state 0. And with certain probability, it will be in the state 1. So there will be a probability p0 and a probability p1, which is 1 minus p0. So that's kind of the basic setting. So this is an information source here, which prepares physical particles in a state. And actually, you may remember, this is how probably in quantum physics, it was explained to you how a preparation apparatus works. But actually, I mean, he was already thinking in these terms in the form of information theory. So now, what was the next question that he asked himself? So he wanted to kind of define how much information there really is in each of these particles, each of these digits. And then he asked himself, OK, how can I, how can I quantify this? So he started to think in terms of very simple examples, perhaps. So for example, he thought, OK, let's take the example where the probability that you get the digit 0 is actually 0. And the probability that you get the digit 1 is 1. And then that means that each of these particles that are coming out here will be in the state 1. And then intuitively, you would say, actually, that doesn't give you all that much information altogether, because after you get the first digit, you would say, aha, interesting, it's a 1. I learned something. The second digit comes, and it's again a 1. And you say, mm, this is interesting. OK, I learned something maybe. But after you have a few hundred digits of 1, you will actually say, well, this is kind of getting boring. I'm not even surprised anymore. I don't learn anything new. It seems that this information source will only produce 1s. So I can already predict what's going to happen afterwards. And so in a sense, you could say, I could characterize this information source in a very simple way. I can say, this is information source that only produces 1s. And then you have fully characterized an infinitely long string, even of digits that are coming out. And so that's an intuition, but it's not very quantitative. But let's take a brief second example, the case when p0 is equal to p1 equals to 1 half. That means each digit with a 50% probability is 0, 50% probability is 1. So now you really cannot say what the value of the digit is. It's completely random. So every time the digit comes out, you will say, aha, I learned something new. And I cannot predict what the next one will be. So every time, you will learn something new. So you will actually gain in the sort of intuitive sense information, new information with each digit that is revealed to you. So in a way, intuitively, you would say, there's very little information per digit. And there's a lot of information per digit. But this is only hand-waving. And then Shannon thought, OK, how can I make this a little bit more rigorous? And so I moved the receiver a little bit here, just to have more space. Then he thought, OK, so maybe in this case here, I could actually compress this message a lot by basically saying all the digits are equal to 1. That is one sentence. And that characterizes an arbitrarily long message. Here, you cannot really make this message shorter, because every digit you have to specify, basically. So he started to think whether maybe the compressibility of messages somehow can help him quantify in a quantitative way, really, how much information there is in a sequence. And that's exactly what it did. And that proved very fruitful. And this also proves very fruitful quantum information. So data compression. So let's try and see whether it is indeed possible to compress some messages more and some messages less. And so to have a look at how this works, I give a very quick example. I hope I get it right now. So let's take this one. Well, now I take an example that is somehow sitting halfway in between, so we don't really know what the answer is. So I have now an information source that spits out its particles in state 0 and state 1 with these probabilities. And now what I want to do is, I want to have a quick look at what happens typically. Well, typically what happens is that you have, in a message of length 4, where you have four particles together, typically roughly 3 will be in state 1. And one of them will be in state 0. This is true with a relatively large probability already. It becomes much more accurate when you go to very long sequences. So if you have 1,000 physical particles, then really very close to 750 will be in the state 1. And about 250 will be in the state 0. And the variation around this will become very small. And so Shannon noticed this. And then he decided to say, OK, why don't I just enumerate the typical, typically occurring events and the rest I kind of disregard? And how can this help? Well, let me take this example. So I have 0, 0, 0, 1. Oh, yeah. 1, 1, 1, 0, sorry. So these are the four kind of typical sequences. Each one of them has a certain probability. And so they have, I don't know exactly how much, but let's say, they make already 70 or 75% of all the possible sequences of length 4. I mean, in terms of probability. OK, so then he said, OK, let's enumerate them. That means let's call this one, this, actually, what's this? The next one, you call this. Next one, you call that. And the next one, you call that. So this is just in binary 0, 1, 2, 3, 4, 0, 1, 2, 3. So now we have something that was before represented in four digits. We have now represented in two digits. So that's shorter. Of course, we make a little bit of a sacrifice here. Because if I would really enumerate all the possible sequences, I would need four digits again. But these are the ones that happen most of the time. And so what he then asked now is, OK, if I take more and more sequences, can I make sure that can I basically enumerate all of them with a smaller number of digits? And then what I would do is I would not just send out these messages to the receiver, the original one. I would have a little encoding stage before that looks at longer blocks, enumerates them. There's a code book, basically. And then I send the enumeration to the receiver. And the receiver has the same code book, and it decodes it again to the long message. And so this has become more efficient. Now, this can only be a useful concept when, for very long blocks and very long messages, this becomes lossless so that you really do not make any, that you do not lose, let's say, sequences. So let's see how this might work out. So the first thing, the first intuition here is encapsulated in this. So this is the set of all sequences. And then here you have the typical sequences. That's how they are called. And here you have the atypical sequences. Typical sequences are the ones where the rate of zeros and ones is roughly the one that is prescribed by the information source. So roughly n times p0 zeros and n times p1 ones in the sequence. I say roughly because you have to admit that it's also maybe n times p0 plus 1 or plus 2. Roughly n times p0 plus or minus the square root of n. So those make the typical sequences. And the rest are the atypical sequences. And typical sequences mean that these will, for very long messages, become essentially with probability 1. And these ones will occur with probability that goes to 0 the longer the sequences are. So now let's enumerate the typical ones. So how many typical sequences there are? So now we make a long 1, 1, 0, 1, and so on. We have a long string of n information carriers. And roughly n, 3 quarters times n of those are in the state 1. And 1 quarter n are zeros. So how many permutations are there? How many sequences are there? Well, that's easy to determine. So that's the n over n p0. So n factorial divided by n p0 factorial n p1 factorial. That's the number of sequences. Well, OK, that doesn't tell you very much at all, actually. So let's work out how many these are in terms of bits. So I take the logarithm to base 2, and then we'll see what this gives. So the logarithm of n over n p0 is, well, let's take the logarithm a moment later. So let's take Stirling's formula. So n factorial is n to the power of n e to the minus n. And I have to say roughly, because it's an approximate formula. More strictly speaking, there's a 1 over square root of 2 pi n in front. But that doesn't really matter for us. This is actually a remarkably good formula. When you put it into Mathematica, and you take this for n equals 200, and you form the ratio of this and that, you will see that it's extremely good approximation, actually. So let's see. So this is n to the power of n e to the minus n p0 n to the power of p0 e to the minus n p0 n p1 n p1 e to the minus n. So this, this, and that cancel, because p0 plus p1 is 1. So we have n to the power of n n p0 to the power of n p0 n p1 to the power of n p1. So now we are getting also making a little progress if we split this up. So we have n to the power of n to the power of n p0 p0 to the power of n p0 n to the power of n p1 p1 n to the power of p1. This, this, and that cancels. So what we are left with is 1 divided by p0 to the power of p0 p1 to the power of p1 to the power of 1. And of course, to be precise, approximate. Right. Well, that's still not so terribly helpful. But so now let's take the logarithm. So logarithm of n, well, the logarithm of the right hand side is n times minus p0 logarithm p0 minus p1 logarithm p1. OK, so if I want to find out how big this is in terms of binary digits, I take the logarithm to base 2. And this object here is n times what is called the Shannon entropy. So this is the Shannon entropy. OK, this quantity, when p0 is equal to 0 and p1 is equal to 1, then you have 0 times log 0. That's 0. 1 times log 1 is also 0. So that's 0. OK, if p0 and p1 are equal and they are a half, then this is exactly 1. OK, so this is already showing you now that the typical sequences are very long when the probabilities of the two values 0 and 1 are very similar. And they're very short if one of the probabilities is very small compared to the other. That's already starting to match a little bit our intuition. But why is this useful? Well, it's useful now because now we can say, aha, if we have this information source, we have a very long string, we enumerate them, we can reduce the number of bits that we need to enumerate the typical sequences by a factor that is the Shannon entropy. Before we had n digits, but now we can enumerate the typical sequences by n times Shannon entropy digits. And that is less. OK, so therefore you would say, and that's what Shannon said, the more we can compress a message, a string of these 0s and 1s, the less information each individual digit contains. So basically information content, so the more the information content per digit of a information source characterized by p0 and p1 by these probabilities is given by the compression ratio h of p0, h of p. The more I can compress a message, the less information each individual digit must have contained. Now I should say a few words because this gives you the intuition, so firstly maybe I should give you a reference because, of course, I have just given you intuition, you have to put in epsilon's and deltas, what happens with the other typical sequences, how many are there, how much information do you have, how many digits do you have to transfer in total. So this is not a proof. This is an intuition. And my memory is not so good at the moment, so yeah. So a great book on all this is Cover and Thomas, Elements of Information Theory. And well, some publisher I don't know exactly, but you can find that very easily. That's a very well-known book, and it's a very good book because these people, Cover and Thomas, are information theorists, so they are, let's say, mathematicians. But they do make an effort also to give intuitions before they actually go about proving stuff. So I think this is a very nice book that turns this intuition into a rigorous mathematical proof. Not very long. It's only one or two pages or so, it's very easy. So basically, you just have to make sure that you show that the probability that a sequence looks is typical goes to one when n grows, and therefore that this goes to zero. And then you can make the following encoding procedure. When you have a typical sequence, which you can just see because you count the numbers, then you look in your code book and you transmit the short sequence. If you have another typical sequence, which you can also realize, then you say, OK, tough luck. I sent the entire sequence to the receiver. And then the average amount of physical information carriers that you have to send is probability of being here multiplied with a compression ratio, plus probability being here multiplied with a compression ratio one, basically. And then you will see one times probability going to zero vanishes, and this is left over. And this idea one has to then put into a few formulas, basically. And that's written over there. But I think it's pointless to go through the exact details. This is the idea. OK. So that's fine. That sounds really very simple, but it was really a major step forward for information theory. I mean, this started really the mathematical rigorous formulation of information theory. OK. So now let's continue now. And now that we have this rigorous expression of what is information, we can now actually also go about and say in a rigorous way, what are correlations. And this is really something that is necessary, because, well, if you go to people in the street, let's say, without a formal natural science education, you will get the strangest ideas of what correlations are. And shockingly enough, even, let's say, more educated people may actually get this rather wrong. And this is because this is not a concept that is formalized at school, for example. And then one forms their own intuitions, and they are not necessarily right. So next title, what are correlations? And once I have done this, then I will actually tell you how this is connected to resources. So I will actually make use of correlations to do some things, to do a job for us. OK. Right. What are correlations? So let's take, first, some example. Now we need to have two information sources. So one and two. OK. And they will be governed by some joint probability distribution. So now it's a probability distribution p of x and y. And as we talk about binary digits, it will be the probability of both expanding a particle in state zero, one in zero, one in one, and so on. OK. So now let's take a first example of an information source. So that's a source which has p of zero equals one, and all the other results are never occurring. So p of not comma one, p of one comma zero, p of one comma one are zero. OK. Are these two sequences correlated? Well, I mean, yeah, why not? I mean, when there's a zero here, there's a zero here. So clearly they are correlated somehow. That's what you would normally say. And I don't want to make this trial of raising hands and so on, because normally nobody raises hand for any of the possibilities. So question is, is this correlated or not? So many people would say this is actually correlated. But this is wrong. This is really wrong. Because we have learned before this sequence doesn't contain any information. This sequence doesn't contain any information. What does actually correlation really mean? Correlation means that if I have, let's say, two systems, A and B, and they are in some joint probability distribution. If I make a measurement here, I look at the particle here. What information do I gain about the particle here? The more information I gain, the stronger the correlation. That is a sensible concept of correlation. And then if we look at this here, then we would say, yes, the digits are the same. But there's no information content. I've shown you just before that there's such a sequence where every time the information carrying particles since they're zero contains no information. It means looking at this particle here, you don't learn anything new about this one here. So this is a setting that has no correlations. Correlations between information sources. Because information by about source B by measuring A. So that's source A. That's source B. OK. And that is really a fruitful way of looking at these things. So now let's take another example where actually the situation is really rather different. So let's say we have a situation like that. And the situation is such that the probability that both particles in state 0 is a half, and that this happens to be the same as the probability for both particles to be in state 1. And then the other possibilities never occur. So that's example B. Now this is really rather different now because if I just look at one of these strings, actually with 50% probability I will find a 0. With 50% probability I will find a 1. So very disordered, lots of information content. Now if, however, I have these two sources now and I look at this particle that comes out here, I learn a lot about this one because when I find 0 here, I know it's 0 here. All of a sudden when I'm allowed to look here at this digit, there's no entropy and there's no uncertainty anymore about the particle in the other sequence. So I have changed the information content a lot. So here we have very strong correlations because measuring one source means I change the entropy and the information content of the other source by a large amount. So that really now becomes our measure of correlations. So a quantifier, well, how to quantify. So the amount of correlations between the information sources A and B is given by the, so I'm a bit, yeah. So by, well, we compare the, I like it more with B, so sorry. We compare the entropy of the source B without measuring source A with the entropy of the source B conditional on having measured A. And now I have to define this. This is clear because I showed you already. I have to define what this actually means here. So H of B A is, so, sum. So we make a measurement of the sequence A, so that has possible outcomes x equals 0 and x equals 1. They occur with a probability P of x, which is the marginal probability distribution of the joint probabilities. And then we compute, for each of these outcomes, we compute the entropy of the associated sequence in this information source B. So that's sum y P of x, y conditional on x, minus sign before, logarithm to base 2 P of y, x. That is the entropy of the conditional probability distribution of y, that's the second sequence, after having measured the first one. And that makes sense because this is really what I explained to you in words. I take a measurement, I select a subsequence of the second source here. I compute its entropy, I average it over all possible measurement outcomes here, which I cannot predict. They are random. And then I compare what I would have had without measurements to the situation with measurements. And the change in entropy is our information gain. And well, that's now a quantitative definition. This actually turns out to be the same as, that's an exercise. So entropy of sequence 2 plus entropy of sequence 1 plus minus the entropy of the joint probability distribution. This and that is the same. And that's also nice because now you see it's symmetric. If by measuring A, I learned something about B, I can learn the same amount about A by measuring B. And that's you would feel that this is a natural thing to have. And so this comes out of this here as well. So that's a measure of correlations. And it has a name. So that's the mutual information. And the name really expresses this fact that it's a symmetric interplay between the two. And it also expresses that one source has information about the other. So that's how we quantify correlations in classical information theory. So now I want to see what this is actually good for. What do I want to say? The first example. Ah, yeah, yeah. So that's right. So if you have this probability distribution, so actually this is good. We make an example out of that. Because we have to see that it matches our intuition. So example A. Well, the joint probability distribution has entropy 0, because it's probability 1 and then 3 times 0. So H of A, B is 0. Now we compute the reduced probability distribution. Well, it's still such that with probability 1, we get the digit 0. With probability 0, we get the digit 1. So the entropy of that reduced probability distribution is also 0. And of course it's the same for both information sources. So therefore, I is equal to 0. B, this is kind of the extreme case where we would kind of agree that there should be maximal correlations. And indeed, the joint probability distribution has entropy 1, because we have probability half for this event and for this event. And so it's a half times log a half plus half times log a half that gives a log a half with a minus sign. So it's log 2. Sorry, and that is 1. And then the reduced density matrix, well, if we look at it, the probability here to get a 0 and a 1 is equally split. It's half, half. So therefore, the entropy is also 1. And it's the same in both sources. And so therefore, I is equal to 1. So it matches our intuition. No correlations, very strong correlations. And in fact, this may also be a nice little exercise. You can try and show that for binary sources, so that only have digits 0 and 1, the maximum value that you can have here is 1. So this is really the maximum. That's also exercise. It's not very difficult. You have to differentiate a little bit and so on. I think it's not difficult, but OK. Fine. Now, so far I have talked a bit about correlations. So that was one of the words in the title. But now I would like to speak a little bit more. I mean, I would like to drive. This, too, was this idea of resources. So correlations as resources. Right. Now, actually, a resource always comes from a constraint that we impose in the world. And so now I will talk about something very, very natural actually, because you're doing it all the time when you use the internet, in fact, you do cryptography. So at least when you put your credit card number into some web form or so, you really should hope that it's encoded and transmitted. So I would like to speak a little bit about cryptography because that is intimately related to correlations. So what's the goal of cryptography? Well, again, think about the internet. You are sitting in the privacy of your house where you have full control over your information. And the bank that you send your credit card detail to is also a private setting which also has full control of the information and nobody can break in. Let's assume that. But unfortunately, the bank is very far away from you. So therefore, you have to send your information over a channel that is public. Someone can just listen in, tap into your phone line or whatever. And so you have actually the challenge of transmitting secret information via using only publicly available channels. So make it more extreme. So I'm here. Fabrizio is there. And I would really like to talk to him about something secret now. But I cannot really go there. And even if I do, and if I whisper, you can still hear it. So we have a problem here. I want to exchange a secret with him. I have to use sound waves that all of you can hear. That's the fundamental problem of cryptography. So there is this constrained secret information public communication channel. How can I nevertheless achieve secret communication between the parties? So task, let's say task, transmit secret information sender to receiver. Challenge or let's say a constraint, transmission device is public. That means it can be read by anybody. Question, how to achieve task under the constraint. So that's a very, very natural setting. It's an example. And actually what resource theories are about to treat general constraints that you suffer and a certain task that you have to achieve under this constraint. And to then do this, it will cost you something. You need a resource for it that you can use up. So what is the resource here? So you need resource. And what is this? Well, it's secret correlations. OK, so now these are just words. Let me put this into action by actually giving you an example. So I'm not actually sure that this is also from Shannon, or so I don't know. Maybe this came later, I don't know. But anyway, it might have. I'm sure that Shannon was clear about this, even if he hasn't written it in his book. So what do we do? So here is A. I mean, I will start probably calling this person always Alice. And this person I will call Bob, so sender, receiver. And so let's say by some miracle, or not miracle, by some preparation work, they actually share some correlations that only they know. So how is this done? Well, actually, even nowadays, in certain cases, this is done in the following way. People take random number generator, and they generate a random sequence of 0s and 1s with the probability 1 half. And then they copy this sequence. And they put it in a suitcase. And they chain to your hand, they enter a private jet, and they fly from Washington to Moscow, for example. Because then you can be reasonably sure that these correlations that you have established are actually secret, only the sender and the receiver at the other end knows them. So then we have a situation like this. So here's 0, 1. I'll do this like this. This is too regular. This is what's also called the code book. It's a one-time pad it's called. And why it's called not just pad, but actually one-time pad will become clear in a moment. So now, actually, Alice may have a message. So 1, 0, 0, 1, 1, 0. Let's hope I got this nice. So what does Alice do? She takes this bit of information from a message. She makes an addition, module 2, and she gets the result 1. Does this with all the other sequences. So she gets 1, 0, 1, 1, 1. It's a little bit boring, but anyway. And what have we achieved by this? Well, now this string of bits and the original message string of bits have no correlations whatsoever because I added to them something which is completely random. So if this is a 1, then there's a 50% probability that this will turn out after this procedure to still be a 1. And a 50% probability that it's a 0. So completely random. And so no correlation anymore with this. Fine. So now this digits we now send publicly over our public communication channel to Bob. So Bob gets this 1, 1, 0, 1, 1, 1. And he does the decoding procedure. That means he adds module 2. The first digits of his code book result is 1. And he does this with all the other digits. And he finds 0, 0, 1, 1, 0. Exactly the original message. So he has been able to decode this. But none of you who have seen simply this part here that has been transmitted over public channel was able to learn anything about the message. And that's actually another little exercise that you can do. I have described this procedure. You should show that the correlation between this sequence and that sequence is 0 when you quantified in terms of this correlation measure. So really, correlations between this and this is strictly 0. So therefore, you can also not learn anything about the message. So you have achieved something useful. You have achieved secret communications over a public channel. That's fine. And now I mentioned that you use a resource to achieve that. This is the secret correlations. And I also mentioned that actually you're using them up in the process. So let me show this, too, actually, that actually really you are consuming these digits, these random correlated sequences in the code book. For example, why don't we just try and be clever? Because it's very expensive to fly someone from Washington to Moscow with a code book. Why don't we just reuse these digits? They are completely random. Why not reusing them simply? OK, so let's do this. So we have choose something clever here. Not really, no. So let's take one digit from the code book, x. And I take one message bit. And I add the modulo 2. OK, so that's one digit, the first digit of the code book, for example. OK, and now I take the second message bit. And I also add modulo 2, the first digit of the code book. So I reuse, well, my code book, so to speak. Now, can we now learn anything about the message? Well, the answer is actually yes. Namely, a simple calculation, m1 plus x plus m2 plus x. So all the additions modulo 2. Well, now this is something you either know or you can convince yourself about. Well, also addition modulo 2 is commutative. So therefore, we have m1 plus m2 plus x plus x. So x plus x, well, if x is 0, the outcome is 0 plus 0. If x is 1, the outcome is 1 plus 1, modulo 2 is 0. So actually, this is m1 plus m2. So now actually, we have learned something about the message, because from these two transmissions, we can actually find out whether m1 and m2 are equal or different. If they are equal, then this sum is actually 0. If they are different, the sum is 1. So actually, we have gained one bit of information about the message. It's not the full message, but it is one bit of information. So the first time we used the code book, it was completely secret. We could not gain any information about the message. I have reused the code book. And now I can learn one bit of information already. Namely, I mean the same as if I would have sent one half of this information to a completely public channel. So it's clear that these correlations that initially were secret, they lose this character of secrecy. And I used them for communication across a public channel. So secret correlations are being used up when employing communicating secret information across a public channel. And in fact now, so I've given you again the basic idea, if I try and transmit one bit of information using such secret correlations, I actually use up one bit of these secret correlations exactly. And then one can, I mean you can convince yourself from this example, it's just pretty obvious. Now it becomes a little bit less obvious when you actually have a message here that doesn't have equal probabilities. One way of showing this would be to take this sequence and first compress it using this compression scheme that I've shown you before. Then the compressed message will actually be completely random, it will have probability we're half each way. Then you can go back to this example, and afterwards you decompress, so to speak, and you show the theorem. So again, I don't, well, right. But it is useless for me now. I don't learn about the key, but whenever I try and use it now for information transfer, it will not be allowing me to do secret information transfer. So this is the point. Indeed, I mean from this calculation I learned something about the message. The problem is that the message is secret. If I was to know the message, then I can actually learn something about the key, but this is anyway clear. So the key point is that there are still correlations. This is actually true. So this is a good point, actually. The correlations are still there. But they have changed from secret correlations to publicly available correlations. So that's the difference. The correlations are still there. I have not learned the value of x, so therefore there are still the correlations there. But they can now be, for instance, we can be used by everybody. So this shows you that if I have a constraint, namely public communication channel, I have a task that I want to achieve. I need a resource to do so. In this case, this is correlations, and they will be used up in the process. So a little bit like the petrol you've put in a car, it will be used after a while. It's gone, basically. Well, it's not completely gone. It has been suffered some chemical transformations and so on. The atoms are still there, but it has changed its character in such a way that it cannot really easily be reused. So you have to make a lot of effort then to make new petrol, basically. So here it's the same. You have to make the new effort of sending a gentleman or whatever in a plane and fly it from Washington, Moscow. So right. So how am I doing, actually, for time? I think, OK. Good. Very good. That means that I might actually cover most of the material. I would prefer to, well, I don't know. What do you think? I mean, shall I give you a break? Or shall I just carry on? Now there's a natural break here. Now we'll start to talk about quantum physics. OK. A break, hands up. No break, hands up. Oh, that's the majority. Good. But I will not teach 120 minutes here. So I mean, don't worry too much. OK, well, whatever. You will stop me when I'm supposed to stop. OK, so now I made this effort of explaining all this classical stuff, because in a way, ah, shit. No, no, I wanted to tell you something else about classical stuff, I think. No, let's leave that out. So I made this effort of explaining to you these things so that it maybe comes a little bit more natural that the same structures actually appear in quantum physics as well. And in fact, a lot of, actually, it's interesting that there's a fair bit of entanglement theory of quantum correlations that actually mirrors quite closely, even mathematically, the theory of secret classical correlations and how we manipulate them. But I will not go into detail, because in the end, in the classical part, I will just talk now about the quantum mechanical part. But it is really a very, very close analogy as long as we talk about one sender and one receiver. So as long as we talk about correlations between two parties, it becomes very different when we have three parties or more. And I believe that I think Barbara Krauss will talk about multiparticle entanglement, right? So there's a lot of sort of, you know, that really deviates quite a lot. Anyway, but all this stuff started with bipartite correlations. So that was 1948 and close by. Now we make fast forward to 1995. And so what I will tell you about, you can also read in a very mathematical way, of course, in a paper, 54, by Benjamin Schumacher. So he actually translated really mathematically rigorous some of these ideas from the classical setting to the quantum setting. I'm not 100% sure. I mean, there were people before who already started to realize there is something, and they wrote down quantum versions of these expressions. But let's say Benjamin Schumacher really wrote down mathematical proofs of ideas of data compression, for example. So the classical analog of this compression of information sequences that I've shown you. And that was actually very early on in the formal development of quantum information theory. And actually Benjamin Schumacher was also the way the person who coined, together with Bill Wooters, coined the term qubit. So there's the bit in classical information, the qubit in quantum information. So he coined this term. But anyway. OK. So now I will develop this. But of course it will be just slightly more complicated. Mathematically it's a fair bit more work, but I mean the intuition is very similar. So again, I have a box. But now it's not a little box where someone reads a book and reads it out to you or something like that. It's really a preparation apparatus in quantum physics. That means it is a machine that prepares, with a probability Pi, a particle in the density matrix rho i. Often this is explained in terms of, let's say, a pure state, but it may also be a mixed state here. And so therefore, particles come out, and there might be rho 1, and so on. OK. So it looks very, very similar, except that now we don't have digits of 0 and 1. But we have quantum states, density matrix is rho 0 and rho 1. OK. So I wrote this down, but I think it will be simpler if I explain this initially with pure states. In the end, it makes no difference, but it just makes things a little bit more natural for you maybe. So we know that this source actually is described by a density matrix. So if you just look at each of these particles, the best thing you can do is to say this particle outside is in this density matrix. Probability with which the pure state is prepared, forming the projector, adding up with the probability weight. And that's really the best thing you can do. Every experiment that you can do outside of this box can be explained, can be calculated, the outcomes can be calculated by using the density matrix rho. So for example, you measure any observable on this particle, the expectation value will be the trace of observable times the density matrix. And that's all that we can know in quantum physics. We have observables that we measure and we measure expectation values and probabilities, let's say, of projection operators. So this is a full description of the system. Again, I assume that subsequent preparations are independent. So that's, again, important. So that's sometimes called IID. So independently, identically distributed. So each particle is governed by the same probability distribution and possible choices of pure states. And the preparations are independent. Now, you can generalize all these things to sequences that have correlations. And you have to do a lot more work and so on. But let's ignore this again. Well, I cover in Thomas, treat also those cases. So if you want to really read the whole book, you will also learn about cases where you have correlations and so on. And this is actually typically the case. So if you have language, there are correlations between the letters. So it is very likely that after a C, well, in German language at least, that after a C, there is an H or a K. But it's very unlikely, actually, I think the probability is probably zero that after a C is an X, except for maybe family names or something like that. So languages have correlations. But let's ignore this. We want to boil it down to the simplistic. And now one can ask the same question. What is the information content, but now the quantum information content of such a sequence? And that will allow us to define the notion of quantum information. And well, how do we do this? Well, I mean, you can say that Ben Schumacher was not super imaginative. He just opened Shannon's book or some other information book. And he said, hey, there's compression. Let's do the same. And that's what he did. But this shows you also something. I mean, sometimes looking at old books and just doing the same again, but in a completely new setting, can actually be pretty fruitful and can make you actually quite famous. And can really be of benefit. Because in the end, a lot of quantum information theory also did the same in some way. I looked at Kaua and Thomas and I thought, oops, this is interesting. Let me look at it. And then I realized, oh, and this can be translated to quantum physics. And then it became something useful in quantum information theory. Anyway, so no harm done by reading old books and using what they have done. OK, so quantum data compression. So basically, the job is the same now. We want to take a long string of quantum particles now. Let's say atoms or electron spin 1 half particles. We want to represent the quantum state that they are in by a shorter sequence of quantum particles without losing any of the quantum state that there is, so without destroying it. Oh, that's actually not such, one has to think a little bit. And I think the point of the achievement here is to realize that this is actually possible. Because if you think purely in classical physics, so what did Shannon do? He said, we have a string of 0s and 1s. I looked at the first one. I determined its value. I looked at the second one. I determined its value, and so on. Well, if you try and do this in quantum physics, you have a bit of a problem here. Because, well, psi 0 and psi 1 may not even be orthogonal. So then what do you measure? If you have an observable that has an eigenstate, let's say, psi 0, it cannot have as a separate eigenstate with a separate eigenvalue of the state psi 1 because they should be orthogonal. So you have a bit of a problem here, so you don't, I mean, you have to think a little bit more how to translate this from the classical to the quantum setting. And so what Ben-Jouma had did is, I was thinking about it a little bit, and he brought it sufficiently close to a classical setting. So the first thing that he noticed and remembered is that really this sequence here is fully characterized by the density matrix row. So if I take another source, and I give it probabilities qi and states phi i, and then what comes out is phi 0 and phi 0 and so on. I mean, well, OK, I should pay maybe phi 1 here. And if this happens to be such that actually row here is, well, qi phi i phi i, and this row and this row are the same, and physically there is no difference for you. So if you want to talk about quantum data compression, you may talk about this setting or you may talk about that setting, and you can try and run your proofs in either of them. So as long as you don't start to make measurements in bases and select one particular basis, then these settings become unequivalent. So one has to be careful now. We will use actually the fact that I can rewrite this in this way. But this is only OK if I don't do measurements on individual particles here. OK, so let's make use of that. I want to take this density, this source, this density matrix, and I rewrite it such that the states phi i are actually really orthonormal. So they form a basis, and in principle, if I would really want to, I couldn't measure on that basis. So now what you're going to do works assuming that this phi i are orthonormal. No, no, it works always. It works always. It's just easier to explain what I'm doing when I go there. So for that, it's important that I do not assume that I make measurements on individual particles, because then I really choose this basis. It is really only for explanatory purposes. It's easier to write down things when I talk about that particular basis. But the procedure works for any basis and any source, in fact. So that's important. So let's make this clear. For the sake of simplicity explanation. Here it might be different. OK, anyway. So let's say this is certainly true. And so basically what I've done is I've taken this row here, and I have determined its eigenstates and eigenvalues, and I wrote it in those terms. So I made it diagonal. OK? So in fact, I can make it even a little bit simpler, therefore, because these are eigenstates. I can make a unitary rotation of all the particles such that the eigenstates are 0 and 1. That makes no difference. It's a unitary rotation. I can undo it in the end. It's really just a different way of looking at things. So really, therefore, I can apply a unitary u on each particle such that u of phi 0 is 0 and u of phi 1 is 1. Now my goal in the end will anyway be that I will make operations that keep the state of these many particles essentially unaffected. So I can always undo at the end this unitary rotation, and I can go back. So this is also, it applies the same for the sake of simplicity of explanation. But it's not necessary at all. In fact, as far as I remember, Ben Schumacher didn't do it. And so then he had to write phi 0 and phi 1 all the time, and it became a little bit more cumbersome. But there's no magic here. Nothing of real importance has really happened yet. OK, but now, we are again very nicely close to the classical setting, because now our machine will spit out pure states which look like this. So 0's and 1's. OK, great. Now let's again look at what is typically happening here. Well, typically, we will have, when we have n physical particles here, n times q0 of those will be in the state 0, and n times q1 of those will be in the state 1. So n particles of which n times q0 are in state 0, and n times q1 are in state 1. But note, I have not yet said I measure them or anything like that. And I will not. Now, the important thing is now we know that, so typically this is what is happening. And that means that the Hilbert space in which these n particles are living will be spanned by all the possible basis states that satisfy this condition. States, so yeah, fall on the basis of the typical subspace. OK, so what do I mean by that? So again, I mean, actually it's the same thing here, so typical subspace, and here's our typical. OK, now I do not say I pick one particular basis state here. Now the only thing I'm saying is that essentially all possible states, all possible realizations of n particles can be represented with very good fidelity, so with very good approximation, as a superposition of the typical basis states. So a superposition. So all typical subspace, all states, yeah, well, I said this already here. So every element of this subspace here can be written as a superposition of these typical basis states. So that's the first observation. Remember, no measurement has been taken yet. So now the next step. Now I have to find some way to, well, reduce this space. I want to use less atoms, for example. So how do I do this? Right, so let's write this down. So I, again, take this simple example that I'd shown you before, OK, so slightly different, I think. So this is sort of, let's say, ah, sorry, Q, yeah. So let's say this is a situation that we have. So then we have these four typically occurring zero. So these are four basis states, yeah. And well, I mean, most of the time these are the ones that are occurring. That means also that the Hilbert space is essentially spanned by those four states. And that also means that, well, I can look at this sequence here. But if I have an identical situation here, I mean, identical in the physical point of view, then the sequence of, let's say, four physical particles will be spanned to a very good approximation by a superposition of those, yeah. So that's an observation. And that is a little bit more complicated now than the classical setting. So one has to do a little bit more work, that's what Ben Schumacher did. And again, this spreads over several pages, so I'm not gonna show you. I just gave you the key idea. Now, what does it do? Well, let's try again to use the enumeration trick of Shannon. But I don't want to throw away particles at the moment. Okay. Oh, I could call the first sequence like this. So for the moment, I've written down a new set of basis states. They are orthonormal. These ones are orthonormal. That means there is a unitary transformation between them. Okay. So now we do this, yeah. We apply this unitary transformation. We don't do any measurements for the moment. So we apply this unitary transformation. So now we have, again, four particles. And these ones clearly are non-trivial. These ones are over here. They are really, they are all in the same state. They actually represent a zero-dimensional Hilbert space. So they're completely irrelevant in a way. So all that we have done here is we have made a basis change such that this Hilbert space that is represented by these states is rotated into a Hilbert space that is actually spent by a smaller number of quantum particles. The weights and everything remain the same. And that's possible. So we're actually just looking at the Hilbert space in a slightly cleverer way. So now, again, comes the idea that, well, if we go to very long sequences, then the typical sequences are the ones that essentially spend the entire Hilbert space that appears with a certain finite probability. And the other ones, I mean, the unusual events where let's say all the particles are in state zero or so, they really happen in a very small probability. So what we do is we now take this setting here. We throw away those particles. We keep these, we transmit those. So they are sent. And then here, we take them, we apply u to the minus one. So the inverse unitary. And we regain the original states. Okay, so, sorry. Sorry, a bit more precise. It should be trying to be a bit more accurate. So this is what was sent. So now we add two particles. So add here. And then we apply u to the minus one. And we get the original states. So in this procedure, we do lose something, yeah? Because this tracing out, in some cases, will of course throw away a little bit of information because in some rare cases, one of these particles in this mapping will actually not be in the state zero. That will correspond to a small loss in fidelity. So we now don't represent exactly the original state, but only to a very good approximation. And what Ben Schumacher then did was to show that if we make the blocks larger and larger and larger, so n going to infinity, then the quality of approximation in this scheme, so when we do all these steps, will approach 100% and it will become perfect. And that's the close analogy to what Shannon has done. And the question now is, how efficient is this going to be? Well, I mean, it's not very difficult anymore. We know how many typical sequences there are. So therefore, if we have n particles, from information source described by row, then we need n times the entropy, Shannon entropy of q naught and q one, particles to transmit the entire quantum state with negligible error. So q naught are the diagonal elements of the density matrix when it's diagonalized. So the eigenvalues, yeah? So this is the Shannon entropy of the eigenvalues of the density matrix row. And so that is the same as n times trace of minus row log to base two row. Because this object you actually compute by first diagonalizing row, then you take the entropy of the diagonal elements and that's exactly the same. So, and this quantity here, n times s of row, that's the von Neumann, right? So because this was not invented, this, that you compute this entropy was not introduced by Schumacher. This is already, von Neumann spoke about this. He was already thinking about thermodynamic aspects of quantum mechanics and that kind of came up. But this is now nice. So this is the natural generalization of the Shannon entropy to the quantum setting. And so, and this is what von Neumann has not done. He has talked about it more in sort of a hand wavy way and so on. This is an information theoretical justification for this quantity and it tells you by how much you can compress quantum information by meaning that by how much you can take, go from n particles to a smaller number without seriously distorting the quantum state, okay? So that's, that's, yes, yes. Okay, and you see this worked in very close analogy to the quantum, to the classical setting. So, did I want to say anything else? No. Okay, right. So, Fabrizio, how much time do I, okay, good. Okay, so then? Sorry, 18. 20 minutes, okay, right. Okay, so now we want to explore the, now it becomes, this is really very closely related to classical information theory. Now we come to quantum correlations and now the mass starts because now it becomes more complex. But at the beginning, I can still try and start at least to develop it in an analogous fashion, but then there are many, many more facets to quantum correlations than there are to classical correlations, because in a way I've told you everything about classical correlations already. So, so quantum correlations. And now I want to turn this around, make it, build it up slightly, just ever so slightly differently. The classical correlations I afterwards showed you how they can be used to achieve, you know, secret communication of cross-public channels. And while I had these constraints and they were the resource. So now let's start directly from this idea of constraints and resources and then see what correlations might do. So, now, constraint. You see, correlations, quantum correlations will be a concept about a quantum particle here and a quantum particle there. And ideally quite far away from each other, okay? And certainly from today's technology, it is very easy to send classical bits, but it's very difficult to send a quantum particle in such a way that all along its path, actually the quantum state will be reserved. And so therefore, typically people say, we cannot do the sort of exchange of quantum particles. That's the, let's say that's a constraint that we have. So constraint is it's difficult to transmit quantum particles in a way that preserves their quantum state. I mean, it's not impossible, yeah? But I mean, it's really, really difficult. And I mean, so it's now, it's kind of 20, 25 years that people in serious are trying to achieve these kind of things and it's still difficult. So it's really, so I will come to that, yeah, yeah. So therefore, what we will assume is, let's say we cannot really distribute quantum particles, but what we can do is we can pick up the telephone and exchange classical information. And we don't care whether it's public or not, it can be secret, it can be public, I don't care. We can telephone very easily or just send emails or whatever or what's up nowadays. So what we can do is, so we can, let's say possibilities, we can exchange classical information perfectly and without limit, okay? And secondly, so that's part one that we can do, part two that we can do is in our local laboratories, we can store quantum information for as long as we like. We can manipulate it in any way we like. Well, that's also not really true, but I mean, it's easier than sending it over a thousand kilometers or so. Local, so arbitrary, local quantum operations are possible. Okay, so this is sometimes pictorially expressed in this way, so here you have your lab A, here you have your laboratory B. Well, okay, so this is the quantum particle and this is a nice house containing the lab. And then here, there's a big fat wall that doesn't let through any quantum particles, but only classical information. So this is called LO, local operations. And this is called classical communication. So what you can do is LOCC, that's what we always say. So local operations and classical communication. What we cannot do is exchanging quantum particles in a coherent fashion between these two laboratories. Right, so that's a constraint. And now the task is create arbitrary joint quantum operations between A and B. So that's now the thing that is obviously not quite so easy because one particular quantum operation would be to swap particles, I mean, to swap the quantum information from lab A to lab B and I've just ruled that out. So we need a resource to achieve this, okay? So the resource will be, well, quantum correlations, well, which we tend to call entanglement, okay? So that's what we will need. So now I have to tell you how this works, what is quantum correlations and all that stuff and how it's actually done, not only to transfer states, but actually also to do arbitrary quantum operations between the laboratories. So, let's leave that. Okay, so let's start with a simple task of trying to transmit an unknown quantum state from laboratory A to laboratory B. So transmit quantum state, state of a two level system of a two level system from A to B. And so really, I mean, completely random. So you pick with an equal probability any quantum state that you can possibly think of on two particles. That means if you would describe this as an information source with particles flowing out, they would be described by the identity matrix, okay? So how could one do that? So we would have here our information source and outcome, you know, any possible particles. Well, you could say, well, okay, maybe I can just measure them. But we know that this is not possible because I allowed you to pick any quantum state. That means you don't know in which basis to measure. So typically you will destroy one of these quantum states. And we really want perfection. I want really with 100% perfection that each of these particles arrives in laboratory B. So the solution is, well, to use entanglement that is provided to us before by someone. And the method is, of course, quantum state teleportation. And that is also, I mean, well, you will see that this is, well, I didn't write down the reference. URL, 1993, Bennett and Friends. And this is in a way our analogon to the transmission of secret information via a public channel. So now we transmit a quantum state in a situation where we cannot coherently transmit quantum particles. So it's sort of that's the relation. Right, so I will very briefly show how this goes because it's very easy. It also means that if you would have done exactly the same calculation that fits probably from here to here easily, you would have written one of the most famous papers in quantum information. So if you have the right thought, there are make the right observation, you do not need very much necessarily, very much complicated mathematics. Not every important result requires very complicated mathematical proofs. Actually, often the idea behind things is simple and can often be expressed in a relatively simple way, right? So yeah, so technical brilliance is not everything. So now how do we do this? So what needs to be provided to us is an entangled state of this form. So that means this is in laboratory A, this is in lab B, lab A, lab B. And you will have to forgive me that I will refuse to write normalizations and stuff like that. You fill that in yourself because it makes it so much more cumbersome. So now, let's say I have a state on Alice's side and the state is unknown means that the probability amplitudes alpha and beta are unknown to you. They can take any value, I mean subject to normalization of course can take any real or complex value that you like. So now the question is, how can I make sure that this state is available to Bob? I cannot just take this particle and send it over so I have to do something more. Okay, so for that, yeah, first thing I write this out, I multiply this out, so I have alpha zero, zero A. And, oh, there's a bracket missing here. Plus alpha, zero, one, A, one, B, plus beta, one, zero, A, zero, B, plus beta, one, one, A, one, B. Okay, so nothing has happened. Actually in the next step, also nothing will happen. I will just rewrite things. So, you know, I would like to rewrite this on the Alice's side in a particular basis. And that is the basis phi plus minus, which is zero, zero, plus one, one. And psi plus minus, which is zero, one, plus minus, plus minus, one, zero. Okay, okay, let's do this. And, okay, so this is equal to, I think it becomes a bit longer. And again, subject to, you know, normalization dropped in all possible places. So what we get is here, phi plus plus phi minus zero, B, plus alpha, psi plus plus psi minus zero, B, no, one, B. Plus beta, psi plus minus, psi minus, plus beta, phi plus minus, phi minus, one, B. Okay, just rewritten this on a new basis. Now I collect terms in a convenient way. So again, nothing happens really. Phi plus, on Alice's side, is correlated to, so, A times alpha times zero, B. And over here, we have beta times one. Plus, phi minus, which is alpha, zero minus beta, plus, psi plus A, alpha, one, one, plus beta, zero, plus, psi minus, alpha, one, minus beta, zero. So just calculation, no zero physics has happened so far. But now, thanks to rewriting it, we know what to do. Alice can do on her laboratory whatever she wants. So, she will measure an observable which has the eigenstates phi plus, phi minus, psi plus, psi minus with different eigenvalues. And so, there she carries out projective measurements on this side. If she finds that on her side, the state is phi plus, she knows that the state on Bob's side is actually alpha zero plus beta one, exactly what she had. At the moment, only she knows this. Bob does not know that yet. If she finds phi minus, she sees, aha, it's almost the right state, but there's a mistake in it, there's a minus sign instead of a plus sign. Okay, so let's write this slightly differently. Make a bit of space here. So, it's the same as actually the correct state and apply to it a sigma z poly operator. And sigma z is, in my case, this. So, let's do the same here. If Alice finds the state psi plus, she will see that, aha, the amplitudes are kind of okay, except they're in the wrong place. So, actually, this is sigma x, zero, oh, also almost the correct thing. And then the last one, psi minus a. And well, what is this? This is sigma x, sigma z. So, actually, in a way, even here, I have not really done anything because this is just rewriting. But now, if Alice finds the state psi plus, she tells Bob that she found the state phi plus and Bob therefore knows that he doesn't have to do anything, his particle will be in the correct state. If Alice finds the state phi minus, she will tell Bob and Bob knows, aha, she found phi minus. Therefore, I have to apply a sigma z operator on my particle and then I have the correct state. Pardon? But sigma x, sigma z. You're right, actually. So, here, so if she finds phi plus, psi plus, she will tell Bob, Bob knows, in this particular case, I have to apply sigma x and I will have to correct state. And well, here's also the same, he has to apply sigma x and then sigma z. So, the important point here is that Bob needs to know Alice's measurement outcome. So, this is one of four possibilities. He does not need to know the value of alpha and beta. The operations that he has to do, the sigma z and the sigma x, is independent of alpha and beta. Okay, so the whole procedure, therefore, after the measurement ensures that Bob has the correct state, what has it cost? Well, it has cost us two classical bits of information that we had sent from Alice to Bob and it has destroyed the entanglement between Alice and Bob. This particle here, this was kind of correlated across the laboratories and this is gone and now we have such an entangled state but purely in Alice's lab and after this whole procedure, these states are in a product state with the correct particle on Bob's side. So, Bob always has the same state and Alice has one of these four possible outcomes. And because the state on Bob's side is always the same, they're not even correlations anymore. They're just completely gone. There are no classical correlations, there are no quantum correlations, they're completely gone. So, we have used this resource of entanglement or this quantum correlated state here and we have used classical communication. So now, you could have asked yourself or can ask yourself, are these resources really necessary? Can I maybe do this trick a bit cheaper? Can I, for example, get away with less classical communication? Can I get away with less entanglement? Well, and the answer to that, boy, now I'm out of time, no? Okay, we can quit that. It's a good moment because this takes a little bit, it takes a bit more effort to make it clear. So, that stuff I will explain to you tomorrow. Tomorrow I will also then explain to you how to make general quantum gates very briefly and then I will go to the question of how to quantify entanglement and different ways of doing so and maybe I spend a few words also on general resource theories, what are conditions that we require for those. And yeah, okay, we will see how I progress. Fabrizio told me that if I really don't manage to go through, I'm apparently obliged to give another lecture. So, okay, thanks.