 I'll give it another minute or so, because people, yeah, I think so, I'll say I'll keep coming. Yeah, yeah, sure, sure, I'll start at the end, people still, yeah, it's only 16 now. It's terrible, I just spoke to the secretary, he has no idea, no clue what's going on, and there are about 115 registered, registered means that they're physically present somehow. So they should have a list going around to see if it's true. Yeah, yeah, yeah, yeah, I think this is really something. I think also most of these guys come from these countries. Yeah, it's annoying. Probably it's better if we start five minutes late now, because we have the entire day. You have an announcement still, don't you? You have an announcement? Good morning everybody. We have a couple of announcements before we start with the daily lectures. First of all, there is going to be the, we're going to shoot a couple of group pictures in front of the main entrance. So it's just on the left here where all the flags are, okay? Not these entrance on the left, the huge one, the bigger one. Okay, so that's about half past two in the afternoon. So it would be good if all of us are there. Second, we are going to circulate a list of the presences, because there is a certain mismatch between the registered participants who are physically present and those who are actually attending the lectures. So we just want to make sure that people who are actually following the lectures get their reward. We will now let, there is a simple sheet you just write in block letters your name and your signature, okay? And while we circulate it, we can start with lecture three by Professor Planny. Thank you very much. Okay, three and four actually. Three and four, right. So welcome everybody. It's even earlier than yesterday, but well, okay? So yesterday I told you about classical information and classical correlation, and I tried to explain to you the concept that these are resources to achieve certain tasks when you are suffering from some constraints by using this idea of trying to make transmitting secret information using public communication only. And I showed you that secret correlations were such a resource that allows you to achieve that. And that really formed kind of the structure that we also will be following now in quantum information, which I have started to follow because I started to speak about quantum information, which was defined as how much can you reduce the required size of a Hilbert space to represent the quantum state of an information source. And the compression ratio gave you the amount of information per physical particle. And then I started to speak about quantum correlations, but I started a little bit differently. I basically set a task this time, namely I had to transmit an unknown quantum state. And I did so under a constraint, namely that I can only ever conduct experiments locally in my laboratory. So let's say my laboratory here and the laboratory of Fabrizio over there. And I was only able to communicate classically between us. And so I could not just send the quantum particle with its information from one place to the other. And that posed a certain challenge and the constraint. And the resource that allowed me to overcome this was a particular quantum state that I wrote down. It was of the form where I can already write it. Again, it was the quantum state of this form that was shared. Okay, that's interesting. So I shouldn't come too close to the board. So that was this kind of quantum state here. And this is actually an entangled quantum state. That means it has quantum correlations. And that might not be very surprising to you because you see it clearly has correlations when I make a measurement on the side of Alice, so party A. I get in the basis zero or one. Then I get either the state zero or the state one. And whatever state Alice finds, Bob will find the same. So they're clearly in the measurement record, they're clearly correlations. And because they're local measurements, the only point, the only place where these correlations can be coming from is the underlying quantum state. So there are some correlations. And now these correlations allow us to do something that we cannot do by classical communication alone. Namely, we can transfer a quantum state from one place to the other. So these are really stronger correlations than merely classical correlations. So they are quantum correlations. Right. I think that's pretty much what we had. Then I showed you the protocol. I explained to you this quantum state teleportation protocol where I used this state as a resource. I put in here an unknown quantum state on Alice's side. And then I did a bit of rewriting and then I performed a measurement, a projective measurement on Alice's side. And then there was some classical communication being exchanged and as a result, end result of that. Bob, the other party, held exactly the quantum state that was provided initially to Alice here. So we achieved the task. And now what I want to do is, so I used some ingredients, namely classical communication and in fact I also showed you that afterwards this entangled state that was shared between Alice and Bob was gone and actually we only had some two particles that were entangled on Alice's side locally but there were no correlations left between Alice and Bob. So these were the outcomes so this was an executive summary of yesterday's lecture. But now I want to ask some simple question and you could also be wondering about actually two questions here. One is, could I have achieved the same with less correlations, whatever that precisely means, and could I have achieved the same with less classical communication? Was it really necessary that I communicate here the outcome of Alice's measurement from Alice to Bob which cost me two bits of information, two classical bits? Do I really need this? Can it maybe be that I need only one bit of information or maybe nothing? Does Alice really have to communicate with Bob? So that's not immediately clear. It could be that if I would have been cleverer I could have written down a different protocol that maybe requires simpler measurements that have fewer outcomes and achieve in the end the same output. And so this is something that we want to explore briefly because it is actually a principle that is very... the principles that will come out of that are rather useful. So it's often in quantum information protocols when you have an element of communication in them and you have local operations and classical communication you can often find out what are the minimal resources that you need by invoking principles such as by local operations and classical communication you cannot create an entangled state you can only create classical correlations but no quantum correlations. And the other thing is that there is no superluminal communication in the world. Okay? And these two principles actually tell us something about the resources that we need to achieve this transfer of an unknown quantum state from Alice to Bob using other quantum resources. So let's briefly illuminate this and as usual I will not go through very formal proofs I just give you a simple example that gives you the idea and then I give you a reference where you can read a little bit more about this where then the more the fully fledged proofs are there so that you can understand this more deeply. Okay. Right. So first question. What is the classical communication that is required successful quantum state rotation? And by that I mean not only the specific protocol that I have explained yesterday but that I mean any protocol that involves local operations and classical communication that achieves the transfer of an unknown quantum state completely randomly chosen from Alice to Bob. So I want to make a statement about any possible protocol of which there are of course infinitely many. So let's see what we can do here and so... Okay, so how does this go? For that I have to tell you something briefly first that was discovered a little bit earlier than the quantum teleportation protocol and namely a concept that is called super dense coding. So that was discovered in 1992 I believe and as I'm making this up now I didn't write down the reference but it was Wiesner I think so Bennett and Wiesner and this was in physical review letters I think it was in 1992 but there's really almost I think there's only one or two papers from these two guys and so you will find that quite easily. And they asked themselves a slightly related question about communication they asked if I send one quantum particle that is a two level system it's just like a spin one half particle how much classical information can I possibly transmit using those particles or this particle? And actually normally you would say well okay what can I do I take this particle I prepare it in the ground state or the excited state so 50-50 probability I send this particle obviously I transmit one bit of information and that should surely be the limit. Well the answer is that's wrong actually you can do more and so how does this go? It's actually very easy it uses entanglement again so we have here again so Alice and we have Bob and now let's assume that by some miracle or someone provided them with this they share initially two particles one on Alice's side and one on Bob's side and they are again correlated in fact I allow myself for them to share again two particles in this form so this in itself carries no information because I say I always provide the same state it's identically the same state so if I would have a machine that does this repeatedly then the ensemble would be described by a pure state and so overall there would be no entropy so no information so no information has been provided here so now in the next step Alice does one of four operations either so Alice would like to send two bits so either she wants to send the message 00 or she wants to send the message 01 or 10 or 11 so whatever something that Bob wants to know something with four possible outcomes and so what does she do so they have this state and now when Alice wants to send 00 she does nothing so that means she applies the identity operator to her particle that means nothing, doing nothing if she wants to send this she actually applies the sigma x operator the polysigma x operator that flips 01 that means actually if she applies this to her particle she ends up or the two of them end up in this state and the nice thing is this one is orthogonal on that one okay next step so Alice wants to send let's say this message she applies sigma z on her particle and then she gets the state 0 0 minus 0a1a yeah that's also a nice local operation Bob didn't have to do anything but this state is orthogonal on this one and this state is orthogonal on that one and then the last thing if she wants to send this message well she first applies sigma z and then she applies sigma x and that means that she will end up with the or the two of them will end up with this state and again this one is orthogonal on that one this one is orthogonal on that one because of the minus sign this one is orthogonal on that one now Alice has basically encoded her message in four orthogonal states that does not help Bob at all because actually to find out to really discriminate all these four states he needs to have access to both particles so therefore what Alice does and this protocol is now she sends the particle so this particle here will be sent to Bob so she sends one physical particle only the one that she's holding now Bob holds both of them and locally he's allowed to do whatever he likes so he makes a projective measurement where the observable that he's measuring has these four states as distinct eigenstates so he makes a measurement he can discriminate perfectly these four states so he can read the information therefore he can discriminate four states he can identify which one it is and so he can actually read out which of the four messages Alice has encoded so he has received four possible outcomes that means two bits of information only one particle was sent so this is called super dense coding because well I mean this is a bit so by using these quantum correlations here we actually this allows us for a single particle to achieve a twice as high communication capacity than a classical system would have a classical system that is not correlated with anything else so that's rather neat I mean it's not a very practical use of entanglement but I mean it's unlikely that this will really be made use of very much in technology but it is a very interesting example that you can actually here use correlations to enhance your information capacity so so that's super dense coding and now let's see what this how I can combine now this super dense coding with teleportation basically I want to show you that if I can I have the ability to implement the teleportation protocol I can use it to transmit two classical bits of information from Alice to Bob and you already see there's almost nothing that I have to say because the essential step here was the transfer of a quantum state from Alice to Bob that's exactly what teleportation does for you so how does it look all together so teleportation and super dense coding so we have now two entangled pairs one is I draw it really like so this is often when you draw diagrams about entanglement you make a wavy line between two level systems that represents particle zero plus one so that's our teleportation resource in the end so this is used for teleportation and then we have another pair of particles that's here our super dense coding resource also prepared in the state zero zero plus one so used for super dense coding SDC right so really and now the first step of our communication protocol is that Alice applies on this particle one of the four operations that she uses for encoding so identity sigma x sigma z or the product of the two and then we use teleportation protocol on these two particles so this is the particle to be teleported using these disentangled resource so what is the outcome of the thing afterwards well so now also exercise yeah go through what I explain you in words now with formulas because that makes you much more comfortable with it that it's actually really true so in the teleportation protocol we make a projective measurement on to one of these four possible states so in the end what we will end up with is some entanglement well some entangled particles that are sitting on Alice's side and another set that is sitting on Bob's side so this is on Alice's side again this is on Bob's side so it's not only the state that has been transferred in the teleportation protocol if you would look at the state of this if you would look at the reduced density matrix of this pair and you would look at it you would see that it's the identity matrix so it's not only that the particle here in the end is in the identity matrix all the correlations that this particle had with the rest of the world are also transferred so this was initially entangled with that one after the protocol this state has been transferred to here and all its correlations as well so in the end you have this correlated particle here and this is something you may just believe because I tell you but you have to compute that yourself yeah so that because it's not completely evident I mean because we are carrying out some measurements and you know it's not such a I mean calculation wise it's relatively trivial but conceptually this is not obvious at all that this should happen but that means now of course that Alice has prepared these two particles in one of the four states here in one of these four states now they are available here and now Bob can locally make a measurement on those identify which one it is and therefore he achieves two bits of information okay fine so teleportation protocol can be used to transmit two bits of information from Alice to Bob why does this imply that we also have to use two bits of information from why do we have to send two bits of information from Alice to Bob during the teleportation protocol so there's still something missing here in the argument so this is just a property that I showed you and now I want to invoke the principle that there's no superluminal communication now in general this is not the argument is a little bit more involved I want to consider a special case let's assume there was a teleportation protocol cleverer than mine that actually does not require Alice to send any information any classical information to Bob assume there is a teleportation quantum state teleportation protocol does not require Alice to send any information classical information right what does it actually imply if Bob does not need to receive any information he knows right away after Alice has carried out her measurement what sort of operations he may have to do to correct the quantum state on his side and he can do this without actually waiting for Alice to tell him her measurement outcome so he can do this instantaneously there's no uncertainty for him he doesn't need any piece of information to decide what to do right so this this means Bob does not need to wait for Alice to tell him about her operations so measurements unitaries whatever she did right fine I mean who knows maybe that's possible but now we combine this with this here because we also know that we can use the teleportation protocol to transmit two bits of information from Alice to Bob so now let's assume this exists and so Alice will set up initially this we have two entangled particles Alice will encode her information here then she does something else and she does this all in one moment very quickly and let's say Bob is a light year away so really far away immediately after Alice did all her operations here because he doesn't need to hear any classical communication from her and the protocol is assumed to work he will have this situation and he can immediately make a projective measurement here and identify which state he has and he obtains arbitrarily quickly the two bits of information that Alice has encoded in this state and that is faster than light communication and that cannot be possible I mean at least if we believe that the relativity is correct so therefore it's surely not possible to make a quantum state teleportation protocol without classical communication from Alice to Bob so that's the simplest case that I can think of to show you that there is clearly a problem now the natural question from you is of course well okay but how about Alice has to send one bit of information then the simple argument here doesn't work so well anymore now now one has to use information theory and error correction methods and so on so one can actually show that any protocol where I need to send two bits of information can be transformed into one that is of this type where you need to send arbitrarily little or actually vanishing information from Alice to Bob now that's more complicated because you need to basically use redundant so you need to use error correction and the basic idea behind the protocols behind the proof then is that well if Bob for example only needs to receive the proof of information well then he can start to actually make a guess so let's say it's really one bit that is required he can make a 50-50 choice and in 50% of the cases he will get it right actually and now if you have a communication channel that actually sort of works in 50% of the cases and gives you the correct result you can actually use what is called redundant coding where you send a message several times and then you know you make a guess every time and then you take the majority vote and then you actually with larger and larger probability you get the correct answer and so this is a method from classical information theory in fact that can be used to really show that any protocol can be transformed onto that and so this is nice because I mean initially you would say well quantum mechanics is not relativistically invariant it's not formulated like that so it's not immediately evident that such arguments should actually work directly but actually they do so here this is a nice example and it can so now instead of having to go through endless mathematical formulas trying to or infinitely often trying to find new protocols that might work better you have one argument and it rules out completely teleportation protocols that need to send less than two bits of information so that's nice and actually so as I said this can be used often such type of arguments when you want to remotely so between different parties implement some information protocol and you wonder how much information needs to be sent it's a good idea to analyze first how much information can be sent with the tasks that you want to achieve and that is actually typically one has to of course look a bit more carefully but typically this is also the minimal amount of information it has to be exchanged to realize this protocol that was a question yeah so correct question I mean this is not immediately obvious and when I say explain this in words that is not approved that's why I suggest I mean so the answer is yes it's exactly the same amount of correlations it's actually one of those four states that are written down there and they are locally, uniterally related to the original states so they have the same amount of correlations but actually it would be good indeed to write this down because actually I think the real question the really critical one is are there any correlations yeah when you do the calculation you will see that they arise there yeah right so now that's one question so that tells you about the classic communication cost oh there's another one wait a second I have to come because I will probably not hear you very well okay wait wait wait we can manage it with one bit of information but in the case of so we have okay so so right correct question so so the question was how can I use let's say redundant coding and repeating this protocol so I mean because once I put the state here taking one state and after the protocol it's gone yeah and so I cannot duplicate it so you always have to remember with these protocols that when I say I repeat it so that means I have a quantum source that can actually repeatedly produce the same state so there's someone that knows how to prepare the state yeah and so in that sense you can actually repeat this if someone will only give you ever one particle then you are completely right there's nothing that we can do but you want to have a situation where I share many of these entangled pairs I have a machine that prepares repeatedly for us several copies of that state and then we run the protocol but you don't need to identify what state they are so this is an important distinction right any more questions okay so now the next thing next question is how much entanglement do we need for quantum state teleportation maybe I mean so maybe do I really need a state of this form where I have sort of 0 0 plus 1 1 with an equal weight in between can I do with something less so for that we first have to kind of accept what is almost it's almost like the second law of thermodynamics that's kind of the second law well that's actually the first law of entanglement theory namely if you have local operations and classical communication you can create a lot of classical correlations but you cannot create quantum correlations so now what do we mean by that actually so by L O C C and starting with a product state product state between Alice and Bob so for example 0 0 0 0 so that's A B A B the most general state is of the form and that's called the separable state separable and this is classically classically correlated possibly so it's called separable I mean you know there's also the notion of separable Hilbert space has nothing to do with that so there's also I mean I just want to point out there is the concept of separable Hilbert space has nothing to do with that has nothing to do with that so I don't remember actually why it precisely was called like that but I mean well a kind of it's separable in the sense that you can always distinguish I mean I don't know it's a name now it's clear that you can make these states because this is a statistical mixture of product states I just make by do something locally and not talk about it at all then I have a product state and if I start from this pure state here and I make my local operations and then I will end up in some other product state simply now it could also be that we talk about what we do so Alice decides that she makes a unitary operation that Sigma X for example and she tells Bob and says Bob why don't you do the same yeah so then they may say Alice may say okay initially the state is this with 50% probability she does nothing and she tells Bob Bob don't do anything either then they end up with this and then with the other 50% probability for example I will apply a Sigma X operation so her particle will be in state one and she tells Bob Bob do the same and so Bob will do the same so now they have two possibilities either this one or that one and now they have to do something very active actually something that is not so trivial is namely they have to forget what they have done you will not have either this or that you will actually have Rho is equal to 0.0 plus so actually more precisely what they have created here is initially a state that may be written like this it's one particle that let's say Alice might hold it's her notebook that remembers whether she decided to do nothing or whether she decided to do Sigma X operation and then correlated with that is this state A B A B so that represents this information because now we can have a look at this notebook and identify which of the states it is so that's really the same as here this looks different to that there's one particle that is gone and then forgetting because what I mean by forgetting is that we remove our information of which of the applied identity or Sigma X so we destroy the notebook so actually to get this so let's call this Sigma then Rho is trace of A1 let's call this A1 here so that's actually how you get then this state here so that's actually I mean and this is not a trivial act erasing information is not something that just happens by itself it actually generates heat in the universe and I think probably Mauro wherever he is he may actually he may actually I don't know whether you will discuss any of this with the Landauer principle and so on okay but I mean so it's basically the fact that erasing information is not for free it generates heat that's called Landauer's principle basically and that's what you will learn about in Mauro's lecture in more detail anyway so this is how you create these these kind of states and well these are the only ones and these are states because they can be created by local operation and classical communication they only contain classical correlations okay now actually question is this state psi minus the project on the psi minus is that actually of this form well so is is this probable a separable state and the answer is well okay and that you can do by for example direct calculation fine so these states are the only ones that you can make this state you can only ones that you can make by local operation classical communication this state is not of this form okay so now I have to think for a moment okay yeah right so now I want to show you that you really do need states of this type so entangled states to actually make quantum state teleportation and again the full proof is a little bit more complicated let's show a simple example right so that's the quantum state teleportation protocol there's a little barrier here this is Alice's world this is Bob's world and typically we come with a state psi and then we do our business and in the end on Bob's side there's a state of psi why don't we do the following on Alice's side we have A1 and A2 actually we prepare two particles we prepare them in one of those states up there 00 plus 11 now we do the teleportation protocol and again I mean it's the same as I used here not only the state is transferred all the correlations that it's carrying is also transferred so therefore when I apply the quantum state teleportation protocol in the end I have a particle A1 here and this one this particle is transferred to Bob's side so now I have one of these entangled pairs crossing the divide between Alice and Bob now imagine this quantum state teleportation protocol would have worked with a state of this form something that I can make classically by local operation and classical communication if that was the case then actually only local operation and classical communication initially starting this state I would have created a distributed entangled state a state that cannot be written of this form and that's a contradiction and again now if you ask yourself well ok if I prepare some state here that is of the form alpha 00 plus beta 11 where alpha and beta are not exactly equal then you have to make a little bit more work but in the end a similar argument holds so this is another principle the principle that local operations and classical communications can only create separable states this allows you in all these distributed protocols it allows you to infer how much entangled resources you need to consume to implement the protocol and so well the first was superluminal communication the second actually really does play the role a very similar role to the second law of thermodynamics so that in thermodynamics well entropy can only increase well here actually well in fact correlations quantum correlations can only decrease when you act locally it cannot increase that's the second law so and there is a certain so there are connections now to two other physical theories relativity and thermodynamics and it turns out and that well quite possibly in the afternoon I will actually make this a little bit more rigorous in the sense that I will show you that thermodynamics can be interpreted as a theory that is really very much like the structure of of entanglement here ok but that will have to wait until the afternoon right ok so this is that then you have to do entanglement purification first which I will explain now and then then you have to invoke the fact that this is reversible and then you can transfer basically the situation to one that is about a maximally entangled state and so on so that's a little bit more work in fact so actually what I should also give you is another reference here so all these things all this story and actually also more what I will be telling you well I didn't give you lecture notes but not all but many of the things that I'm explaining to you are I'm not sure are written in this article which is specifically meant to be an article for non-expert but clever people so like you to explain some of the basic aspects of quantum information theory and so this is all the authors as well so you can find this also on the e-print archive I don't think that everybody has will have access to this particular journal it's not one of the really big ones this is a journal for articles for educated non-experts basically and so a lot of these stories also these arguments about the classical communication requirements and so on this is all explained at length in this article perhaps I could make this also simply available somehow to I don't know if that's possible yeah so maybe I download this and yeah it's probably slightly legal but let's see ok we will figure that out I mean in the worst case I think it's on the archive it's also I forgot the archive number but it's somewhere on the archive quant pH I think I'm not sure I think it's I somehow have a feeling that it's this number but I'm not completely sure but you can find it very easily when you search this yeah ok so that explains many of these basic principles so actually let's see I mean in a way I've ok so now I want to look further into these correlations because somehow I mean I always write down this state 0 0 plus 1 1 and this kind of seems to be a rather useful state yeah seems to have correlations and can be used in all these protocols and it's I'm always using this because in a sense it's the most valuable state that we have so first I would like to convince you that this is the most entangled the most quantum correlated state that there is when you share two spin one half particles so one on the other side and one on Bob's side so this is the most entangled state so the unit of entanglement so to speak so this state is the most entangled state of two shared spin one half so that's a little bit funny how can I say this when I haven't even told you how to quantify correlations I have not told you ok you know this is the amount and there's a formula for that and then I computed on this and then I somehow mathematically show that the formula takes the largest value for this state 0 after the 4 ok yeah that makes sense because they're always encoded in three digits yeah so that's the number anyway I see ok right so why can I say that this is the most entangled state of them all well so I stated this principle with local operations and classical communication you cannot increase correlations or you cannot make them from nothing now what would be a valid argument to say that this is the most entangled state well surely if for example any other entangled state can be made from this one by local operations and classical communication if that's the case then well surely this is the most valuable because everything else can be obtained from it ok again teleportation will show the way so that we have to do almost no calculation actually it's very very simple so again let's take this situation here so we have here the state 0 0 plus 1 1 and here's the divide of the world between Alice and Bob and now again I do the same thing as I did already previously in the argument I prepare on Alice's side two further particles and those are simply written in any I J K L it's just a general density matrix the most general density matrix of two spin one-half particles that you can think of so any state there is now I apply the quantum state teleportation protocol on these two particles and I've told you before any kind of correlations everything is transferred and so the outcome of this will be that we share two particles in the state row and on Alice's side here we will have some quantum state so we've started with this one shared between Alice and Bob we achieved in the end the preparation of an arbitrary quantum state between Alice and Bob so this one is sufficient to generate any other shared quantum state between Alice and Bob and therefore this is the most valuable and it's the most powerful and therefore the most quantum correlated at all because it can also create all possible quantum correlations that there might be and that's the simplest argument to show that this is really the unit of entanglement the most entangled state okay so right yeah okay that's right so in fact you can also show that and I had it noted down but I'm not going through that you can also show that it's not only possible to use an entangled state to make quantum state teleportation actually if you want to make any quantum gate that is distributed between two parties or any quantum operation any quantum algorithm that you want to do you can always implement it by providing some entanglement between the different nodes and then do local operations and classical communications and you will actually be able to implement the quantum algorithm you may have to provide quite a lot of quantum correlations it might not be enough to just have one pair of particles you may have to have two or three or many but it's always possible so and it's always possible to do this from starting from here and again one trivial way of seeing that this must be possible although it's not the most efficient way is we have two nodes and we want to make a quantum operation between them we use one pair of this maximum entangled state to teleport one particle from one node to this node then we allow it locally to do whatever we want and then we use another pair of these maximum entangled particles and teleport the particle back and the end result is that now we have a new transformed state under this quantum operation that we have applied so it must be possible this is a valid path but it's not the most efficient and actually so I don't want to tell you the details but if you look at we are you see this is 62 now I have to go back 7,000 that actually this paper gives a discussion of how many resources you need to produce for example a control node gate or a swap gate and various typical quantum operations to get out your quantum algorithms and actually the protocols that are given there some of them at least are optimal and the arguments that we are using for that are exactly the ones about the classical information exchange and non-increase of entanglement and so on and so basically we give these kind of arguments and at the same time we give a protocol that achieves those minimal resources and therefore it's optimal and so that's described in this work but I don't really want to go into that now because I'm seriously running out of time okay fine but I still haven't shown you how we really quantify entanglement and now this will now start this will again be quite similar to how we treat correlations and so on in classical physics and in particular it will mirror also this development that we that we quantify information by how well we can compress it and the same we can do with entanglement that will give us some measure and some for a particular case, namely for pure states and then we have to start to think what is it that we want to have as properties of valid entanglement measures in situations where we cannot so easily make such a construction that I will show you and that will actually be conditions that then we will also use as reasonable requirements for general resource theories because they will be very similar or the same constraints so now how to quantify entanglement actually well okay that's one way of writing it I like this one quantifying entanglement that was the shortest title of a paper that I ever had two words but it encapsulated exactly what we were on about so this is what we are going to attack now and this started in 96 I think no 95 apparently okay um PRA 53 2046 um that was um I forgotten exactly who were the authors I think Ben Advanced I'm Popescu and Wuttos I think but I'm not entirely sure but I mean this is the reference so you can find it so what I explain to you now or outline to you now is explained in here um but it's also you can read it in this article it's also explained there um because these guys were saying okay so how do we quantify entanglement so how much entanglement is in this state when alpha and beta are not equal to one over square root of two well how much is there well it's kind of clear in some limiting cases when alpha is equal to one and beta is equal to zero there's no entanglement has to be zero but otherwise who knows and then these guys were started to think and then they thought well okay why don't we do the same as we did for information we compress try and compress the entanglement and so how does this work so this is entanglement distillation or concentration in this case and they made an observation that was interesting so that challenge this give me uh a lot of copies and copies of this kind of state so two end particles we allow local operations and we allow classical communication and we want to transform a try and transform this kind of state into something of this form yeah because this we kind of accept this is the best state we can possibly have that's our unit and so we want to see how effectively can we transform this sort of states into these sort of states and by that we mean if we have n copies of that one then we will end up with n times e copies of that state and other particles that are uncorrelated and e this compression ratio will quantify our entanglement that's the idea so how does one do that that's actually quite easy well yeah kind of easy and um so again I will give you as usual I will give you the basic idea and then the nasty little calculations that you have to do on top of that I will just very briefly outline and then you can either do them yourself to have the satisfaction that you're as clever as these people or you look it up in the end yeah so what can we do so for that we make two copies so that means so we have here Alice Bob Alice Bob so what do I mean by this now so let's draw this in in a figure so what I by that I mean we have here A1 A2 B1 B2 we have two identical copies of the state alpha 0 0 plus beta 1 1 and now I would like to write out this quantum state to give you an idea of what we're going to do um let's write this out so this is alpha squared 0 0 on Alice's side 0 0 on Bob's side alpha beta in 0 1 Alice's side 0 1 Bob's side plus 1 0 Alice's side 1 0 Bob's side plus beta squared 1 1 Alice 1 1 that's the same as this so if both of these particles are in state 0 then these ones are also in state 0 ok so that's the first step so I just rewrote this somehow and now Alice is going to do a local operation actually she will make a measurement which is already a bit surprising because typically when you do a local measurement you will destroy correlations between Alice and Bob so Alice measures the observable let's say x 0 0 0 0 plus 2 times 0 1 plus 1 0 1 0 plus 3 times 1 1 1 1 ok I mean it's so it's an interesting observable in a way eigenvalue 1 belongs to the eigenstate 0 0 then eigenstate 3 eigenvalue 3 belongs to eigenstate 1 1 and the eigenvalue 2 is degenerate ok so what this actually means is that I count the number of particles that are in state 1 a little bit bigger font ok again ok so basically so in words this counts the number of particles in state 1 so and well mathematically that's written down like that so ok so let's work out what's happening so let's remove this so there's the outcome 1 and so that means we have to apply the project on the state 0 so that means the state after the measurement is this and that's pretty bad because that is a product state completely uncorrelated so we kind of didn't do the right thing here it seems the same actually if I get the outcome 3 then I end up with 1 1 analysis side 1 1 on both sides so also product state if I get this outcome the outcome 2 then I end up with the state 0 1 0 1 plus 1 0 1 0 and well ok divided by square root of 2 for a change I put to normalization and that kind of looks already like an entangled state it's not I mean it is an entangled state but we would like to really massage it a bit more and bring it really into a form like that and for that Alice and Bob actually locally on their side make a specific unitary transformation and the unitary that they are doing so it's ua tensor u well u tensor u and the unitary is such that it has an action on the state 0 0 action on state 0 1 1 0 and 1 1 and let's see what I would like to have here is something like this something like that and now I've used up some stuff so let's say 0 1 so these are also normal there's a unitary transformation that does that yeah so if I apply this then actually oh fuck this was wrong don't write it let me just I think this was okay and this was like that I think don't write anything because I might mess this up again yeah so 0 1 and 1 1 so for sure this is a unitary transformation so let's apply this now to see what I got this right so I apply this so then I find 0 0 0 0 plus yeah 1 0 okay that's correct so you can note this down that's fine and now I write this again so well now I have to carry so you see how annoying it is to carry a normalization so this is 0 0 Alice 1 Bob 1 0 okay before I say it in words so now what you see here is this particle here and this particle here so the second particle that Alice has and the second particle that Bob has is always in state 0 so actually what we have here is 0 0 between first particle of Alice and first particle of Bob plus 1 1 of the first particle of Alice and Bob and then 0 0 of the second particle so now this is really one pair of particles spin one half particles for example that are in our wonderful unit of entanglement and the other pair is in a product state okay so now you have seen of course sometimes this works great so we actually get something that is now a perfectly entangled state and we start it from something that was kind of with unequal weights alpha and beta and sometimes actually unfortunately we lose we kill our entanglement now you have to sort of start to say well okay what's the probability for these different outcomes and then you can say well what's the average amount of perfect entanglement that I obtain and here it is actually the probability to get this is 2 times alpha squared times beta squared it's a probability to get this and so therefore you can say the original two particles contained at least this amount of entanglement because it's a probability of getting a perfectly entangled state and let's say we give this the unit one is this optimal well surely not because I acted only on two parties so now basically what we are doing here is very similar to quantum data compression now if we have alpha 0 0 plus beta 1 1 n copies we can write this out and then this will be alpha to the power of n where we have a state where all the particles are in state 0 multiplied with all particles in state 0 so this is on Alice's side both side again and then we have alpha to the power of n minus 1 times beta and then we have 1 0 0 1 0 0 plus well the next possibility and so on so all the possibilities of placing a single particle in state 1 distributing this amongst the particles so there are n choose 1 of those states well so then in general we will have alpha n to the minus k beta to the power of k and then well we have k of those and then n minus k of those thus all the permutations of which there are n choose k but the nice observation here is they all have the same weight and now you can imagine probably I can make a trick like here making a local unitary transformation reordering things so that I get a certain number of pairs that are exactly in this maximally entangled state and a lot of others that are actually simply in a product state 0 0 so that's the idea and try and write this out that's a little bit painful actually but it can be done and then the question is well okay how many of these pairs of maximally entangled states will I get? well that's actually quite easy how many how many terms are in this here so it's n choose k if I now make this unitary transformation that does not change the number of superposition terms if I would be able to transform this in simply one of those and the rest are product states that would be two superposition terms if I have two copies then it's four if I have three copies it's eight so the number of maximally entangled pairs that I get is the logarithm of n choose k right so now how much is it on average that we have to kind of average this well which of these outcomes has the highest probability well actually now you impose the law of large numbers and you realize that the typical term that you are getting is one where k is n times eta squared that's kind of the probability that you find a one and then well you have here binomial law and so on and then you can go through the details and you see that this is the best so and by the law of large numbers the larger n will become the more certain you will be that you pretty much get this and so therefore this is logarithm of n over n beta squared and that is n times well the Shannon entropy of alpha squared and beta squared or equivalently it's n times the von Neumann entropy of the reduced density matrix of one of these states okay so that was quick it's detailed over there in that paper but that's the principle idea and now this is in full analogy to this dense code not dense coding the quantum data compression this is the compression ratio therefore we say that each of these particles contains an amount of quantum correlations or entanglement that is s times entropy of the reduced density matrix of one of the particles that's one half so that's already a good suggestion that this is the right formula but actually I'm not quite finished because what we really would like is that also the reverse process gives the same efficiency because if it does not then you know we would have very valuable and valid quantifiers of entanglement so what is the reverse process the reverse process is we start with some perfectly entangled states namely this many and by some local operation class communication we want to create n partially entangled states of this form so we really want to reverse this process is this possible yes it is and the knight is writing to that is writing to our defense quantum state teleportation mixed with a bit of quantum data compression so what do we do so we have our so these are n times n times s of row a where row a is the reduced density matrix of one of those and that's what we have and of course here is usual Ls and Bob and now let's try and turn this into that and to this end we start out actually with state alpha 00 plus beta 11 so we have lots of them but obviously we have too many we cannot just take this one use that maximum entangled state and make teleportation because then we can only transfer n n times s of those here and the rest will stay where they are so we have not achieved our task so we have to be a little bit more efficient on this side so why don't we do a very simple trick that we just used take these and if it's a very large number of them we actually make kind of the first step of this procedure namely on Alice's side we look at the particles we make a measurement we look how many ones and zeros there are and in that way we can compress the information on this half of the particles into a smaller number so in the end it will actually look like this so all these particles here there will be somehow so all the particles here will now be entangled with just the first few and I know that I can bring this down to n times s of rhoa because of this procedure and these ones here will be disentangled so there will be a lot of particles down here that are just in the product state and now I take these few I teleport them with these maximum entangled states so I bring this to the other side and I don't really want to draw this now because it will be a mess but once I'm on this side here I undo exactly this process I just undo this unitary transformation now that actually is the reverse process now that it's really asymptotically you really have to go through calculation now this is because in every step from here to here for every finite n you lose a little bit and you have to show that this little bit per particle goes to zero and that's you lose the law of large numbers and an explicit construction to show this but the outcome of all of this is indeed that well you have the process of concentration that has this efficiency you have the process of dilution which has the same efficiency so you can go forward and backward as often as you like so it's a reversible process and because they have the same efficiency really this justifies why we really like to call the von Neumann entropy of the reduced density matrix of a quantum of a two particle system as the entanglement content now this is only true when the state is pure when the state quantum state is mixed the von Neumann entropy is not a good quantifier of entanglement because in fact neither this procedure nor that procedure will have this efficiency actually the efficiency is lower okay so that's that's why we like to say the von Neumann entropy is a good entanglement measure for pure states I've forgotten what I want to say now right now you can ask yourself maybe so now I try and make use of kind of thermodynamic arguments again you know maybe one of these procedures can be even more efficient maybe I was just stupid maybe I could have provided some much cleverer scheme here or more elaborate stuff on Alice and Bob's sides more complicated operations and I could have had a little bit higher efficiency why not and that's actually not so straightforward to show because you again have to study all possible protocols of which there are many and you have to show that none of them is performing better so that's not very efficient so let's invoke the law of entanglement theory namely that you cannot increase entanglement by local operation of classical communication so we have here on the left-hand side you can view this as a process okay let's draw it like this let's assume that this dilution process really has this efficiency as of row A and let's assume that the concentration process is a little bit better so we can go actually from N to N times S of row A plus epsilon of maximally entangled pairs let's assume that so now I can make the following I start out with this and I use it to produce this many maximally entangled pairs so then I say okay I have S of row A plus N times epsilon these ones I use with a reverse protocol to go back to the original situation to my original N particles and this one I put in a safe deposit box in the bank or in an experimental physics laboratory and then I repeat this process and every time I repeat this I have left over N times epsilon maximally entangled pairs in addition that means I can run this cyclically as many times as I like and I will create more and more entanglement but all the operations that I've used are local operations and classical communications so but these cannot create quantum entanglement so therefore this cannot be more efficient this is a contradiction to our fundamental law and in fact you can really look at this in terms of engines where you say well I have a machine that does some useful work it takes these partially entangled states massages them works on them brings out these concentrated form like a distillery or alcohol or something like that what comes out on the side is actually classical information because we are making a measurement here it's a classical information record that is produced in addition on the side so it looks a little bit like this so here we start with N particles here we have N of SROA particles and here what comes out is information so information and the other process over there so this flows from in this direction looks actually pretty much the same but it flows in the opposite direction so it's S of N SROA then we start with maximum entangled pairs we push them up here we have N partially entangled pairs and again also in the teleportation we actually have to do some measurements and so on and there will be a measurement record and that's also again information that comes out and if you want to operate this cyclically actually then we have information coming out here information coming out here to make it truly cyclical we have to erase this information and it actually generates a little bit of heat so this is again this is kind of a slightly more thermodynamic picture of these processes and because one goes down in this way one goes up in this way we can combine them the net effect must be a decrease in entanglement and an increase in heat in the environment okay but they cannot increase the amount of entanglement here right seven minutes right that's correct right okay so now perfect okay that was for pure states and here we understand everything in a way what is the situation for mixed states and so now for mixed states entanglement this process is entanglement distillation and it has a certain efficiency which will quantify our entanglement and this efficiency we call the distillable entanglement this is a process of dilution which also has an having efficiency and that's called the entanglement cost so how much maximal entanglement pairs do we need to create an arbitrary distributed entangled state and now what one finds is the distillable entanglement ed of rho is typically well strictly smaller than the entanglement cost that means if we do this process and then afterwards we use what we get the concentrated entanglement and try and dilute it again we will not get back the original amount of entanglement that we put in but we get actually less and not only a little bit less that vanishes in the limit that might vanish in the limit of large n it is actually for each copy that we put in we lose a finite amount of entanglement so it's really we start with n and afterwards let's say we end up with half n of particles minus square root of n also which would be kind of tolerable so it's really a finite big difference so now in mixed state entanglement we really have a problem this is a perfectly operationally defined nice entanglement quantifier this the entanglement cost is also a perfectly well defined entanglement quantifier so we have two and actually I will show you in the afternoon there are many and they are all different in fact you can say something like the following roughly speaking although it has to have some additional qualifiers if you find another entanglement measure so let's say let's call it for reasons that will become clear in the afternoon e sub r of rho let's say it's really different to these two but let's assume that on pure states all the three of them agree and there is at least one such example and now you can say and now I tell you well actually on pure states they agree but in general they are different and so you may say yeah well who cares perhaps they impose the same order on the set of entangled states so by that I mean maybe they have something like if so if er rho1 is bigger than er rho2 so two possible entangled states and we compare them and see under this measure this one is bigger than that one maybe that's equivalent to distillable entanglement and also being bigger than rho2 or maybe the same statement here if that was the case then we would really not worry very much about these entanglement measures not being the same yeah because I mean they make the same order we can clearly say this state is more entangled than this one under all measures question is this true well the answer is no if this was true then this measure is identical to this one if however they are different then there's always a pair of states for which one entanglement measure says rho1 is more entangled than rho2 and another measure of entanglement says exactly the opposite and so now this is a little bit of a worrying situation if you think about I mean you would feel initially so now it really very much depends on which measure you take when you want to decide which state is more valuable than another okay so this is the proof of this is very easy I think it's here I think in 2000 so that's myself and and my student at that time now but maybe this is actually not so surprising because this procedure and that procedure are really quite different they are operationally describing a completely different setting one is concentration one is dilution maybe these other measures also maybe they correspond to some completely different operational procedure why would we actually expect that each entangled state is equally useful for any possible procedure that's not at all obvious and that's exactly what this expresses the value of entanglement now depends very much on what you want to do with it at least for mixed states for some tasks it's better to have this sort of entanglement for other tasks it's better to have that sort and by that sort I mean represented in this quantum state or that quantum state but that now poses a problem nevertheless because now you can say well there are what about two measures maybe there's a third one actually there are many more which mathematical quantities are decent measures of entanglement and which ones are not and to decide that next in the afternoon I'll write down some conditions that are sensible and then I will say anything that satisfies those conditions is a valuable entanglement measure and when I've done that then I'm kind of not really finished with entanglement theory but I've said enough about entanglement theory and then I want to use exactly those conditions and the principles that you have learned in the last few lectures here to actually speak more generally about resources and how we quantify those because it will follow almost exactly what I've shown you here except that I stopped talking about entanglement I just say a resource and I will stop talking about local operations and classical communication I will say the operations that are available to you and I will stop speaking about separate laboratories and distances and the inability to transfer particles I will just say a certain set of operations that is not allowed for us and this can be formulated abstractly like that and we can write down these conditions and then I can show you as the final thing that under this framework a lot of things fit so for example quantum coherence is actually a resource for a certain task and it can be quantified following these approaches non-classicality can actually be considered a resource and can be quantified in these manners thermodynamics can actually be formulated as a resource theory generally any setting in the world where you have a constraint where certain things are not possible for you can be translated into a resource theory setting and the nice thing about this is that of course these resource theories share some similar the same mathematical structure so if your proofs are sufficiently general and they really only make use of the fundamental aspects of resource theory they apply in all the theories and that's kind of neat so it's an overarching structure that allows you to make statements about all physical theories that arise from constraints and resources and that's kind of nice because it really encompasses pretty much everything that we use and it's a different way of looking at quantum physics because before we were looking at correlations and so on as a strange phenomena but now we really talk about all physical properties as are they useful for something how valuable are they and how can we quantify this how can we transform these resources and what can be achieved there and actually as I said this encompasses many physical theories okay so with that I close and so the announced tentatively announced lecture will have to take place because obviously I have not managed to do this yet thanks