 Okay, so first of all, I would also like to thank the organizers for inviting me here and giving me the opportunity to give this talk. I would like to talk about some new insights that we obtained within the last few years on multi-partite entanglement, in particular on SLOCC and LOCC transformations. And what I'm interested in is what is entanglement? What can we do with entanglement? And how can we quantify and qualify it? Now we started working on this topic several years ago, and the way we started was also, as I want to start here, I would like to go back to the very beginning of entanglement theory and talk about the origin of it, namely that entanglement is a resource theory, and it's a resource to become local operations assisted by classical communication. Then I would like to talk about briefly the bipartite case, so I will show you some known results, and this helps me also to make clear what is the difference between the bipartite and the multi-partite case, which is what I want to talk about next. In particular, I will show you that the generalization of the bipartite maximal entangled state in the multi-partite setting is a set, it's the maximal entangled set. And then I will talk about how we can quantify entanglement with some new operational entanglement measures. I will then talk about using a finitely many round LOCC protocol, and then I want to talk about the results that we have obtained for generic and bipartite states. Okay, so let us start with the very beginning. So we consider usually two parties or more, they are called Alice and Bob, and the important thing is that they are spatially separated from each other. Now they might share some resource states, some state sign, and what they can do because of the fact that they are spatially separated is that they can apply local operations and communicate classically with each other, so this is what we call LOCC, so local operations assisted with classical communication. Now what they want to do is some fancy things like for instance quantum teleportation or entanglement-based cryptography and things like that, and for this of course they need to use entanglement, okay, so you cannot do that by just applying LOCC, so by just local operations assisted with classical communication, you cannot do that, so you need entanglement for that. So entanglement is a resource to overcome the restriction that we have due to LOCC. Okay, so this means that entanglement theory is a resource theory, okay, so the separate states, the ones that you can prepare locally are the free states, this is what you can use freely, and the free operations are LOCC, that is something that does cost any entanglement. Okay, so the resource then is entanglement, so everything that is not of this form is entangled. And so when we now for instance consider the case that we have some pure state psi which we transform via LOCC to some state phi, then it has to be that entanglement, if we measure it with whatever quantity, has to be larger or equal of psi than of phi, okay, because I operate here with something that is free that doesn't cost entanglement, so this means that I can only decrease entanglement by doing LOCC. Okay, so this means that when we study LOCC, we get an ordering in the set of states in the sense of how entangled they are, okay, so if I can do this transformation then this state has to be more entangled than this one, and the important thing here is that it doesn't depend on the measure that you use in order to measure your entanglement, okay. So as you know in the multi-paradise setting you will have many entanglement measures and for any entanglement measure this has to hold, in fact that's the only requirement that you have on an entanglement measure, that is not increasing under LOCC. And let me note here that I'm talking here about pure states and my talk is about pure states and about single copies, okay, so I will restrict myself to that case. So we see that we get with studying LOCC, we get that partial order regarding entanglement theory and another important thing is that of course LOCC is also, I mean to study LOCC is very important for applications, just think about the one way quantum computer or things like that, distributed quantum computation or whatever, of course you're allowed to do LOCC and you have to see how you can manipulate your state, so what kind of other states do you get, okay, so this is why this LOCC is very central, obviously because it's the free operations that we use in our entanglement theory, okay, so let's talk about what is LOCC, so from a physical point of view LOCC is very easy to describe because it's just everything that you can do locally, okay, so this means that Alice applies some generalized measurements, she obtains some outcome, I transmits this information to Bob, who then depending on I apply some operation, communicates the outcome back, depending on that Alice applies again an operation and so on, okay, so this is the physical description of LOCC which is very clear I think because it's simply, I mean it's, it's the definition is physical, right, so this is what you can do if you have separated parties, now mathematically however it's very difficult because mathematically if you write down this protocol, you see that you will get a CPT map that looks like that, where these operators, the crowd operators have to have a decomposition into possibly infinitely many operators which are of this kind, so local, of course Alice does something then Bob does something then Alice does something Bob does something and so on, so these are local operations and a difficult thing is that you have to make sure that each of these factors here is an LOCC, so this is really a complete operation that Alice did at that point of the protocol, so this means that you have to make sure that each of these factors obeys the completeness relation, okay, so this is of course a very difficult problem, especially being aware of the fact that in general we can have here really infinitely many rounds, okay, so there are protocols where actually infinitely many rounds are required, okay, and then you should find that there's an infinite product of operators where all of them obey this condition, okay, so this is a mess that's really very difficult to do, the good thing is that in the bipartite case when we study pure state transformations, so I consider again a state psi and I transform it to some other state with LOCC, then what happens is that this difficult program boils down to something that is very simple because you can show that it's equivalent to do just an operation on Alice's side, she communicates her outcome to Bob who then just applies a unitary, okay, so this boils down to a very simple protocol, it's a very simple, also mathematical form, okay, that is very good because if we know how we can characterize LOCC then we can also study what kind of transformations we can do, we can introduce entanglement measures and so on, and this is exactly what happens because using this simple characterization of LOCC you can show that you can do a transformation from a pure state to another pure state in the bipartite case if and only if some simple condition holds the maturization condition, now let me note here that because of the fact that LOCC in the bipartite case for pure states is such a simple protocol you can show something very strong, namely that a set of operations is equivalent, you can show that whatever you can do with LOCC even if you would use infinitely many rounds you can do it with finitely many rounds, it's kind of obvious from here, but you can also show that for pure state transformation LOCC is equivalent to a much larger set or much larger, a larger set of operation which are the separable operations which can be written in this form as you see here, they are mathematically much much more tractable than LOCC, but in general they are larger than LOCC, okay, in this case pure bipartite state transformations they are the same, and on the other hand you can also show that all these protocols are of a very simple form which we called all deterministic, because I will come back to this a little bit later, but so do you see here that okay everything is equivalent in this case of pure bipartite state transformation, now also because of that reason you can identify also what is the maximum entangled state, because the maximum entangled state is the optimal resource, so it is the state with which I can do everything that I want to do, and in the bipartite case we have that the phi plus state, so this maximum entangled state here can be transformed with LOCC into any other state phi, so this means that victorially this is like that, so you can go from the phi plus state to any other state in your Hilbert space, okay, so this means that if you want to do whatever with a bipartite state the best thing is that you start with the phi plus state, because once you have the phi plus state then you can do with your free operation whatever you want, because you can first transform phi plus to your desired state phi, and then work with this phi, if this phi has some particular kind of application, okay, so this is very nice and closed, and let me mention here that of course local unidireis do not alter entanglement, so I'm saying here that this is the maximum entangled state, but I could have picked any other state that is a new equivalent to that, okay, and in what follows actually I will always consider a new equivalence classes, so I will pick one representative of an a new equivalence class, because local unidireis is just a basis change that doesn't change entanglement, okay, so this was the bipartite case, the simple case let's go to the multi-partite case, and let's ask ourselves first why do we know so little about multi-partite entanglement, what does it actually make it so difficult to address this problem, and what is different from the bipartite case, where everything at least regarding the aspect that I was discussing about before is very simple and is solved, so if you consider a multi-partite system then you immediately notice that there are several problems, one is obvious one, which is that if I consider for instance n qubits, then of course I have exponentially many parameters in describing my state, okay, I don't have that in the bipartite case of course, so the systems are scales exponentially with the number of qubits or systems that I consider subsystems, so this is why of course I have to deal with much more many more parameters, the second problem is that in a multi-partite setting we have different what is called SLOCC classes, so SLOCC is stochastic local operation and classical communication, and what this means is that if you want to go from this state here, the GZ state which is here, to the W state which is here, then you want to do it in a probabilistic way at least, so you might not be able to do it deterministically with LOCC, but you might be able to do it probabilistically, so that you can obtain at least with a certain non-vanishing probability, the other state, the W state, okay, now it has been proven many years ago that this is not possible, so there are different SLOCC classes which means that these two states are really separated from each other, you cannot go from one to the other even with an arbitrarily small probability, okay, this also means that it's very hard to compare these two different classes because you cannot translate one to the other, okay, so this is the second problem and it gets much worse when you go to more systems because then you will have to face infinitely many different SLOCC classes, okay, so something that doesn't happen in a bipartite case, the third problem is the question whether it is exhausted set in the three qubit, yes, so if you consider genuinely multi-partite states, otherwise you also have to bicep problems, okay, now the third problem which is a big problem is this problem with SLOCC, okay, so the SLOCC is, as I explained before, it's very difficult and you don't have a shortcut, so you cannot write this in a simple form, as I said before it has been proven that sometimes you might need infinitely many rounds, so SLOCC is not closed, it's mathematically a very difficult set of operations, the other thing is that in general it has been proven that SLOCC does not coincide with SEP, so the SEPTRAB operations is what I mentioned before, those that can be written in this form where you just have to check the completeness operation of the whole map, not of the factors, so it's much easier to deal with mathematically, but the problem is, and this is known since a long time, that SLOCC is strictly smaller than SEP in general, okay, so this means that not all SEPTRABOR maps can be implemented via SLOCC, so this also means that SEP, so the SEPTRABOR operations do not really have a clear physical meaning, whereas SLOCC of course has, okay, that's very bad because this makes it very difficult. Now, facing all these problems, let's go back to what we know and what we actually do want to know, okay, so what do we know? Well, first of all, we know that of course entanglement theory is still a resource theory, so we know what is entangled and we know what kind of operations we can do, so we know that this is not entangled if it factorizes and we know that if I can go from one state psi to another state via SLOCC, then the entanglement of psi has to be larger equal to the entanglement of psi, okay, so these are the things that we know. Now, what do we want to know? So if you look at the state, which has exponentially many coefficients, then it might not be actually desirable to find all entanglement measures, right? You don't want to have exponentially many quantities that characterize entanglement because what do you do with that? I mean, you would not evaluate it through a large system, right? So probably what we want to know is what are the most relevant states? So what are the states that we should care of? How powerful is the state? How can we measure certain kinds of entanglement of a state? And how can we find new applications of entanglement, okay? And this is exactly what I would like to explain in the following. And so first, let's discuss briefly how we can actually gain some insight. So given the fact that the partial order is induced by LOCC, it means that if we want to study some relation among states or want to address the questions that I just raised before, we will have to deal with LOCC even though it's very difficult, okay? So there is no way around it. I mean, you cannot do it otherwise, in my opinion. Another problem is, and what we see immediately is from the fact that we have two different SLOCC classes, we cannot have a maximally entangled state. So you can easily convince yourself that this fact implies that there cannot be a state in the same Hilbert space that can be transformed deterministically to all states in your Hilbert space, okay? Like what we had in the bipartite case. So this cannot exist. So this means that first of all, we have to generalize the notion of maximally entanglement to something completely different because it's not a single state. And so the way we address these problems was to look at this equality that you see up there which says that you have your set of LOCC, you have a larger set which is mathematically much more tractable, and you have smaller sets that are also mathematically much more tractable, okay? And so what I would like to do now is to focus on these two inequalities. On the one hand, I will study this here and we will see how one can generalize the notion of bipartite, maximally entanglement to the multipartite case. We will also see how we can then find some new entanglement measures. And on the other hand, I want to study this relation here and you will see that indeed in the multipartite case, I mean, everything that can be more complicated will become more complicated than in the bipartite case, okay? So you will not have any equality here nor here, okay? This is what we will do by looking at the finitely many round LOCC. Okay, let me just mention here what I mean with this all deterministic protocol. So this is a protocol where I start with a state psi, then I do some operations. So at least for instance, apply some operation and for each of the outcome, she will obtain a state that is a U equivalent to the first one, to this one, okay? So in each step, you obtain deterministically a new state. This is what I call all that, okay? So you obtain one state, then you process this state further, the next step you again obtain deterministically your state, okay? So let me stress the fact that here, I really care about LOCC, which is a deterministic operation, because it means that in each output, so Alice makes an operation, she obtains a one, get some state, and she obtains a two, get some other state. These two states in this protocol are a U equivalent, okay? And in general, if you want to do a transformation from psi to phi, then all the outcomes will be a U equivalent, that's the idea, okay? So now let me give a short outline of how I want to proceed in this talk. So first of all, we will address this problem of maximal entanglement, okay? So what should we call maximal entanglement in a multi-partite setting? And of course, there are several other attempts in the literature where certain aspects of the Bell states, of the maximal entangled by-partite state has been generalized to the multi-partite settings. What I would like to do here is to take the operational approach, okay? So what I would like to call maximal entangle is really the optimal resource under LOCC, under the restriction of LOCC, okay? And what you will see is that, actually we will have to introduce a set of states and we'll see that this set will be maximally entangled if I can go with this set to any other state in my Hilbert space. Okay, so then I want to briefly discuss what is actually the maximal entangled set? So how can we characterize it? How does it look like? And from this, we will obtain some new insight in multi-partite entanglement, especially for instance, that there exists isolation, which means that you have a state, a pure state psi, and you cannot transform it with LOCC to any other pure state psi, okay? Unless you do LU, but I exclude LU SSM. Moreover, we will be able to see that LOCC is not equal to set, even in the case of pure state transformations, okay? That's something that was not known before because this was, is a result from mixed states that I showed you before. Okay, using now this insight, we can then also introduce some new operational entanglement measures so they have a physical meaning. And then what I would like to do is to briefly talk about, finally many rounds, LOCC, where we will see that certain protocols that work in the bi-partite case will no longer work in the multi-partite case, okay? So there is a new aspect again entering the multi-partite business. And last, but not least, I want to talk about the generic and quitted states for which one can actually show some very strong results and one can also derive the maximal entangled set for this general Hilbert spaces. Okay, and then we will discuss the consequences about all these findings. So let me start with the maximally entangled set. So it's really the generalization of maximal bi-partite entanglement. So it should be the set of states with which I can get any other state in Hilbert space, okay? And actually it is the minimal set of states that does that because I don't want to use some states that I don't need. So I want to know the set of states for which it is possible to go to any other state via LOCC and there exists no state outside this set that can be transformed via LOCC to this state inside the set, okay? So it's the minimal set, okay? So it's the minimal set of states that is required to obtain any other state in the Hilbert space, okay? So this is the kind of natural generalization of bi-partite maximal entanglement. So pictorially, we have here a set of states. In the bi-partite case, we had just this. So we had one state and from here I can go to all the others. In the multi-partite case, we know that this cannot exist. There is not a single state, but there is a set of states from which I can transform to any other state in the Hilbert space. So this means... So these are now what I draw here is a single state here. It is one set, it's a set of states, okay? So this is side one, this is side two, side three, and so on. What do you mean by this joint? Yes, you can obtain a blue state from two red states. No, they are not this joint. Sorry, no, this is not. So there are transformations from here to there, for instance. Yeah. Okay, so you see that what I want to characterize is this upper line. Okay, so I want to know what is the set of states that I need in order to obtain the whole Hilbert space via LOCC. And I want to know the minimal set. Okay, and I want to know that because of course, if I would have this set of states at my disposal, it's as if I would have the maximal entangled bi-partite state. Okay, with this set, I can do everything that I want. Okay, now how can we address that? Well, how much do I have? Okay, so very briefly, what we have to do, what we want to consider are transformations from psi to phi, right? But I told you that LOCC is very difficult, okay? So how can we actually possibly characterize the maximal entangled set? Well, you can have a very simple observation is that if I want to go from psi to phi, then first of all, these two states have to be in the same LOCC class. So if I can do the transformation deterministically, in particular, I have to be able to do it probabilistically, obviously. So I can write it in this form here. And now imagine that I consider one of the outcomes in my protocol, okay? So I consider this m1 up to mn. These are local operations and they have to transform my state psi into phi, of course. Because for each outcome, I have to obtain the state phi. So let's write this equation. And if you do that, you see that at the end, you will have the disoperator, the m1s applied to these g's. If you invert now the h's on this other side, then you will see that this operator here has to be a symmetry of your state psi 0, okay? So the operator that you obtain here, which is some information about the initial and the final state, and the measurement that you have to do, has to be a symmetry of your what we call a seed state of psi 0. Okay, that's a very crucial point here. Because it means that only if you have symmetries, you will be able to do some LOCC transformation, okay? And now you see we can write down our measurement operators, which are the simple expressions here. And this s here is just a symmetry of your state psi 0. Okay, but now for instance, in order to study what are the separable transformations, okay? Now, even if they're not physical, mathematically, what are the separable transformations that I can do from, whether there exists one from psi 1 to psi 2, then I just have to use the completeness relation of my measurement operators that I just derived. Okay, and so the necessary and sufficient condition will be simply that the sum of m dagger m here will be equal to the identity, which is exactly this expression here. So this means that to solve whether there exists a separable transformation between psi 1 and psi 2, is easy because there is this, well easy, but there is this necessary and sufficient condition here. Okay, and this is a nice result that has been derived by Gillard Gore and Nolan Ballach. The problem, of course, is that LOCC is not equal to SEP. Okay, so we have to study LOCC, not SEP. So how can we then do that? Well, the idea is simple because you say, well, if you want to know which states can be reached, so from which states does there exist a state psi such that they can go there with LOCC, then you just start by saying, okay, let's first investigate those states, phi for which there does not even exist a state psi that can be transformed to phi via a separable operation. Okay, if there does not even exist a separable operation, of course, there also doesn't exist an LOCC operation. Okay, so this means that all these states here for which there doesn't exist one must be in the maximal entangled set in the MES. And for all the others, actually, we could derive the LOCC protocols which do the transformation. Okay, and this is the way how we identified the maximal entangled sets. And this is the result. So in the bipartite case, for two qubits we have a single state. Okay, moreover, in the bipartite case, there is no isolation. So every state can be transformed into some other pure state via LOCC. In the three qubit case, you have the maximal entangled set is this set here. Now, without entering any details, there are some parameters here. And it is a zero measure set. Okay, so it's just a three parameter family. And in the three qubit case, you have, so this set is enough to get the whole Hilbert space. Okay, so I only need to focus on this set if I'm talking about three qubits. The rest can be obtained via LOCC. But there are infinitely many states. Okay, so the set MES contains infinitely many states. And in the three qubit case, what you can do is you can always transform any state into some other state. So there is no isolation. Okay, but this was a huge difference now from the bipartite to the three qubit case. We go from a single state to infinitely many states, still zero measure, but infinitely many. Now there is an equally large, if not even larger step to go from three qubits to four qubits. Because for four qubits, what happens is that the MES, maximal entangled set, is a full measure. So this means almost the whole Hilbert space is in the MES. Okay, that's bad news. Because we wanted to study the MES in order to identify a very small set of states which we need to consider. So that we don't have to consider the whole Hilbert space. And now it turns out that the MES is basically the whole Hilbert space. Okay, well, but this is what it is. So almost all states are in MES. And the reason for that is that in starting from four qubits, you can show that almost no state can be transformed to some other state. So there is isolation. Okay, so you have a state, you want to process it locally in order to obtain some other pure state, but you cannot. Okay, you can just apply locally, there is. That's it. Okay, and this is why in this case, you get this full measure. Now, the good thing is that if you consider now the states that are in the MES, but that are convertible, so that are useful for state transformation, then again, you get a very handy set which is again zero measure. It has this very simple parametrization. Okay, so the states that are the most useful ones have again a very simple parametrization. Now, we also studied the three qubits case, which is very similar to the four qubit case, but what you obtain here is again some new result, namely that there are some transformations that you can do by a set, but you cannot do it via LOCC. Okay, so you can prove that even if you consider infinitely many rounds, it's impossible to do that via LOCC. Okay, let me now briefly discuss also the other insights that we obtained here. So first of all, we learned that there are very few LOCC transformations from pure to pure states. Okay, so it's very difficult to do such a transformation. So then the question is, is the set, the maximum entanglement set actually always a full measure when we go to more than four qubits? And now, as I don't have time to talk about more details, I will just give you the answer here. Indeed, one can show that the maximum entanglement set of more than four qubits, and so qubits and qubits, arbitrary dimensions, is always a full measure. So one can prove that generically, you cannot do any transformation. Okay, so LOCC is a very rare property that you can do in LOCC transformation. Moreover, we have seen that separable operations can be implemented. There are separate operations which cannot be implemented via LOCC, even in the case of pure states. Another thing is that those LOCC transformations that existed have always been of a very simple form. They were always these all deterministic protocols. Okay, so then you might ask, okay, is this also in general true? So is any LOCC transformation always of this simple form? Is it always all that? And the answer is no. So in the multi-paradise setting, you can show that there are certain cases where you need an intermediate step, and maybe I can just move forward there. So you can show that there exist states which you cannot transform with such an all deterministic protocol, which is a simple protocol, but you really need an intermediate step. So you need some step in between that goes with some probability to some steps I want, and then you can do the transformation to phi, and to some other state, you can do the transformation. Okay, it's something that doesn't happen in a bipartite case. Okay, but multi-partite, it does happen. Okay, now the other question is, of course, now that we have studied LOCC, is it possible to introduce some new entanglement measures? And the answer is indeed yes. So you can study some new entanglement measures, which have a very clear operational meaning because they measure how useful a state is for LOCC. Now let me skip that, and let me come to the generic case. So let me just stress that again, that you can show generically that the maximum entangled set is of full measure. And the way you prove that is by proving that generically, so almost all states in the Hilbert space, don't have a symmetry, a local symmetry. So it's trivial, the local symmetry of almost all states is trivial, it's just the identity, okay? Because of that, you cannot do any LOCC transformation because as we have seen before, the symmetry of the state dictates what kind of transformations you can do. Okay, and that's a highly non-trivial result, by the way. And using this result, you can also show then, for instance, what is the maximum success probability with which I can go from one state to the other, and you derive a really surprisingly simple expression, which you can easily evaluate. Okay, so let me now conclude because we have seen that there is only a zero measure set of states for which LOCC transformations are possible. Okay, so now you might wonder, okay, but then it doesn't make sense to study that at all, right, if there's only a zero measure state, and unfortunately I cannot do anything, what's the point in studying that further? Well, I see it differently, because if you look at all the set of states that have been identified that are physically relevant, so let this be maximum entangled states, stabilizer states, hypergraph states, matrix product states, projected entangled pair states, whatever, they are all of measure zero, okay? So what is interesting now is to study the convertibility properties of those states that are physically relevant, okay? Which are all of measure zero, so this fits very nicely to the fact that also the convertible states via LOCC are of measure zero. Okay, I'm going to skip this, and I come to the conclusion. So what we have seen is that after all, LOCC transformations among multipartite states is not that difficult, but the reason is because there's almost nothing possible. Okay, so there are very few transformations that are actually possible. Now in the three qubit and generic three qutrit and four qubit case, you can actually derive all the possible LOCC transformations, and with that you can also evaluate some new operational entanglement measures, and what I didn't discuss about is that if you consider the finitely many round LOCC, you can completely characterize all states that can be reached, and for higher dimensions you can prove in general that the maximum entangled set is a full measure, and that it's almost never possible to do some LOCC transformation. Okay, so this is, someone shows us that we can study LOCC also for the multipartite case. However, what could possibly go wrong compared to the bipartite case does go wrong. Okay, so all the simple things that we had in the bipartite case like equivalence here, equivalence here doesn't happen in a multipartite case. Okay, so as an outlook, what we want to study now is indeed this simple set, which has also a simple parameterization of states in the MES that are convertible. We want to focus on entanglement properties for physically relevant sets, and also introduce some other operational entanglement measures, and of course a big thing will also be to study more general transformations when you consider something more than just going from a pure state to a pure state, or generalize it for instance also to Gaussian states. With this I would like to thank the people in my group and Julio de Vicente, who has been involved in many of these projects, and you for your attention. Thank you. Thank you very much.