 And we repeat anyway, what we did yesterday at the beginning. Let's start with a question with a five minutes after the talk. Okay, so welcome everybody. Good morning. It's the last day of the school, so I hope that you still fit. So I will start slowly because it's Saturday morning. So what I want to do first is to discuss briefly what we started talking about. So what did you do yesterday? So today we were discussing when can we transform a pure state psi into some other pure state phi via LOCC. And we found in the bipartite case that there are necessary and sufficient conditions that are actually pretty easy to be checked. Do you remember what the conditions were? Anyone? Say again. So LOCC, so we're talking about local operation and classical communication and then talking about the bipartite pure state. So then we learned, and probably you all knew already, that this is possible if and only if the maturation condition holds, which is that the Schmitt vector of psi is maturized by the Schmitt vector of phi. Now this is something that one can very easily check, because the Schmitt vectors are just the eigenvalues of the reduced density matrix. Okay, and this maturation condition is a set of inequalities that we have to check. Okay, so that's something very easy to do. I was there so easy. So at the beginning of yesterday's lecture, I was telling you that LOCC is something that is very difficult. Remember because we had this picture where we have Alice and Bob, which are spatially separated from each other. Alice does something, communicates the outcome to Bob. He does something depending on this outcome, communicates back and so on. And I was telling you that there are certain protocols where you can show that there are infinitely many rounds required. Okay, something very difficult in general. But what we saw yesterday is that in the case of bipartite pure state transformation, this difficult protocol boils down to something very simple, which is that just Alice does some operation. So Alice does some measurement, communicates the outcome to Bob, and Bob just applies a unitary depending on this measurement outcome. Okay, now because of the fact that this is this difficult protocol that we have in general boils down to something that is so simple, we can derive necessary and sufficient conditions. And in this case, these conditions are very easy to check. Okay, but this is only possible because this difficult set of operations that we have to deal with, obviously, can be characterized very easily. Okay, the bipartite case. The multipartite case is not the case. And what we started discussing then yesterday was, okay, so now we know which kind of transformations we can do. So this means that we know also what is the better resource. Right, so which state is a better resource? Because we discussed at the very beginning that, of course, if I can go from psi to phi via LOCC and LOCC are free operations, this means that entanglement that is contained in psi has to be larger equal to the entanglement contained in phi by the very definition of entanglement, if you wish. Okay, so we know now because of these criterion, which are the better states, which are the more resourceful states. Right, so psi would be better than phi. Because if I want to do something with phi, I can also do it with psi because I simply first transform psi to phi. Right, so whatever I can do with phi, I can also do with psi. But with psi, I might be able to do even more. Okay, so this is why psi is a better resource than phi. And what we have seen because of this maturization condition is that the phi plus state, so the state where we have all the equal weight, proportional to this, can be transformed to any other state. So this is true for all phi in the same Hilbert space, of course. Okay, so having the same dimensions. Okay, so this means that if I want to do an experiment and I have a bipodite system, then what I should actually do is to prepare the phi plus state because once I have that, I can do whatever I want. Okay, because if any of these states phi has some application, I can also achieve the same task, so I can do the same thing with phi plus because I first simply transform to phi. Okay, so this is why the phi plus state is really their optimal resource or if you wish, it's the maximal entangled state. And again, I'm saying their maximal entangled states because what I actually do is that I consider LU equivalence classes. Okay, so by phi plus, I mean any state that is LU equivalent to phi plus because LU equivalence doesn't change any entanglement, as we have seen. Okay, so this is the optimal state and now how can we measure it? So this was our chapter three, entanglement measures and monotones. And we started with the definition of entanglement measures and monotones for mixed states. Okay, so who of you recalls the definition? What is an entanglement measure? Exactly, and now we have it for mixed states, so it has to be that E of rho, everything that I can get via LOCC, right? So any completely positive map that is in LOCC applied to rho. Okay, so by this, I mean that this is a completely positive map which is realizable with an LOCC protocol. Okay, so this means that rho goes to lambda LOCC of rho with LOCC. Okay, so whenever this is possible, then our measure cannot increase. Okay, so this means any entanglement measure is not increasing under LOCC. We were also talking about the strong monotonicity condition which was that E of rho has to be larger or equal than the average amount of entanglement. So here we have that rho can be transformed via LOCC into this ensemble, P i sigma i. So this means that I do some protocol, I make some measurements and they are obtained with probability P i, the state sigma i. As we had yesterday in the example of pure states. Okay, so I start with my state rho, I do some measurement, I get with probability P 1, the state sigma 1. For the other measurement outcome, I would get with probability P 2, sigma 2, and so on. Okay, and of course this protocol could be many, many rounds. Let's just look at the initial state and the final state. And the final state is with certain probability in one of these states sigma i. Then for the strong monotonicity condition, we had that it's not increasing on average. So this has to hold for any transformation that is possible like that. Okay, now yesterday I was calling this entanglement monotone and I was asked by one of you that the original definition of entanglement monotone actually also states that the measure has to be convex. Okay, this is a crucial step, this is a crucial point that I want to make you aware of that some, I mean sometimes you might read entanglement monotone and maybe it's not convex. Okay, so you have to be careful there whether the measure that is considered is really convex or not. And the reason why I was introducing it yesterday just with this condition was because I want you to be aware of that fact and I want you to make you aware because when we discuss now about pure states or convex states and so on, convex functions, we will see when this condition is actually really stronger than this one. Okay, and so for this reason I introduced to you this entanglement monotone as something that obeys just that without it being convex necessarily. Okay, so maybe, I mean it's actually wrong to call this entanglement monotone because as I said the original definition contains convexity but to make the point stronger I will continue calling this now entanglement monotone just because it's shorter than saying a function that obeys the strong monotonicity condition. Okay, that's the same thing now for this lecture. Okay, so now let's talk about pure states because we are discussing pure states mainly. So what are entanglement monotones and entanglement measures for pure states? So first of all, what is the definition? Well, the definition we have it here, we have it for mixed states so in particular we also have it for pure states. So an entanglement measure, so E is an entanglement measure for pure states and this is crucial, okay, so you have to add that because otherwise it's not true. So it's for pure states if E of psi is larger equal to E of phi for any psi and phi which are in this way related to each other so that I can go from my state psi to my state phi, okay? So exactly the situation that we have studied before. Okay, so for pure states we only have to look at pure states obviously. This is why it's an entanglement measure for pure states, okay? And what we have to know in order to be able to see whether a function is an entanglement measure or not we have to know all possible transformations, right? Because we have to check that this inequality is true for all possible transformations. Okay, now in the bipartite case we are lucky because we derived that already so we know when two states are related to each other via LOCC in this way we know that the necessary and sufficient condition is this maturization. So then what is an entanglement measure? This gives us a very simple characterization of entanglement measures because E is an entanglement measure for pure states if E is what is called a sure concave function of the eigenvalues of the reduced state so of the eigenvalues of rho A which are of course the same as the eigenvalues of rho B because these are the Schmidt coefficients and what does it mean that it's sure concave? Well sure concave I mean what should it mean? If you look at the definition and if you use that we know when this is possible wake up or when this is possible we have to check that a function obeys this inequality for all states for which it is possible so what do we have to check then? Well I mean what has to be true is that E of psi is larger equal to E of phi if and only if lambda psi is maturized by lambda phi, right? Okay because if and only if this is true I can do this transformation so the only thing that has to hold is that and this is exactly the definition of sure concavity okay the eigenvalues of the reduced state again these are of course our lambdas right this is lambda this is our Schmidt vector okay so this is easy and this is solved okay so this is the complete characterization of entanglement measures for pure states we know exactly when a function is an entanglement measure for pure states okay so you have to think about it in the following way imagine that you work on entanglement and now you come up with a nice measure of course you have to check that it's an entanglement measure, right? so you have to check whether this condition here is obeyed now in general you have here LOCC which is something very difficult so you cannot check that so easily okay because I don't know what this is so how should I then check this inequality so in the bipartite case we are lucky because this is simple and so we can do that okay and we did it and we found this necessary and sufficient condition and for our entanglement measure this simply means that it has to obey the same order so whenever our state psi is a better resource then entanglement has to be larger equal than entanglement of phi obviously and so we are using what we have derived here in order to characterize all these functions okay now what about entanglement monotones E is an entanglement monotone for pure states E of psi is larger equal to the sum P i E of psi i again this is for pure states only and this has to hold for all psi and ensembles P i psi i such that psi can be transformed via LOCC into this ensemble it's clear it's just the same as what we had for mixed states now for pure states okay so what I do here again is that I start with my state psi I do some measurement I get with probability P 1 state psi 1 with probability P 2 the state psi 2 and so on okay and what I have to check is whether on average the entanglement is not increased okay so this is on average okay now first thing is an entanglement monotone for pure states an entanglement measure exactly so this of course includes also the case where P 1 is equal to 1 and all the others are 0 and then we have the situation where psi is transformed to psi 1 or psi if you wish and then this has to be non-increasing okay so an entanglement monotone or being precise a function that obeys this condition okay is for pure states is an entanglement measure for pure states okay now when do we have an entanglement monotone well so for this we don't know so much right because we have not studied this ensemble transformation we have not studied the transformation from psi to an ensemble we don't know the necessary and sufficient conditions there yet but what you can show and this is due to Gifre Vidal is that is an entanglement monotone for pure states if and only if E is non-increasing on average under unilocal operations unilocal operations so non-increasing on average is exactly this condition this means that it's non-increasing on average but you only have to check whether it's non-increasing on average under some very small set of operations because these are unilocal operations and unilocal means that Alice you just have to check whether if Alice does whatever on her system without Bob doing anything whether this inequality is true and then you have to check the same thing for Bob okay so you don't have to consider that Alice and Bob can communicate or anything you don't have to consider the case that Bob has to do something or can do something that depends on Alice's outcome you just consider the case where Alice does some general measurement whatever she wants and check whether this inequality is true and then you do the same thing for if Bob does an arbitrary operation on his system okay and the nice thing about that is that this is also true if we add Charlie and the system D and so on so this is also true in the multipartite case because you only have to check whether the condition that is non-increasing on average holds under this very restricted set of operations where you don't have to consider any classical communication okay it's a pretty strong result but if you think about it just think about it for a second then you can see it very easily because in matching I mean one direction is clear right if E has to be non-increasing under any LOCC then in particular it has to be non-increasing because you only want part of it as something because this is obviously LOCC so this direction is clear now the other direction that it's sufficient to check that this is non-increasing under unilocal operation you can see it as follows because these states here you see these states are the states that we get after Alice, Bob, Charlie and whatever did some measurement so I can always write this state as some A i1 then so B i2 and so on this is the state psi so something done by Alice, something done by Bob and so on now if the condition is true by four operators of this kind so if here is that entity this is a unilocal operation then I have this inequality right so I use this inequality and then use it for use the fact that it's also true for Bob in order to apply the next operation and so on and so on and in this way we will find for all these kind of states what we get so this is why this is a necessary and sufficient condition but you see already from here that the fact that in order to check what a function is an entanglement monotone boils down to something where I don't even have to care about classical communication shows a very clear difference between entanglement monotones and entanglement measures because for entanglement measures I mean look at this in order to check that I need to know what this is now in the bipartite case we were lucky because we know what it is okay so we have a characterization in the multi-partite case it's very difficult because we have no idea what this could possibly be okay but in order to prove that something is an entanglement measure I have to be able to check that inequality for a set of states that I don't know okay however if I check whether something is an entanglement monotone I don't care about that okay so I don't need to know what are the possible transformations that I can do because what I do consider there are just the possible ensemble transformations okay so I mean something more general but in order to prove that this holds I just have to prove that the function is not increasing under a very simple set of operations okay so that's much easier to do okay yes yeah so this is true for multi-partite system I mean both definitions are true for multi-partite this is of course only in the bi-partite case this is necessary in sufficient condition whereas this is true also in the multi-partite case and this is really a great simplification okay because now I'm dealing with operations that I can characterize before I was dealing with operations or here I have to deal with operations that are very difficult okay as we were stressing yesterday a lot so good and now of course if I have something as simple as that I can solve that and in the bi-partite case you can show that this is true if and only if E is and what is called extendable concave function the eigenvalues of rho E so again of the Schmidt coefficients if you wish extendable here just means that you can add zeros okay so if I have a function lambda 1 up to some lambda d then I can also include some zeros in addition okay so this is just because the dimension of the Hilbert spaces might not be the same okay so I might have that the dimension of system A is 2 and the dimension of system B is 5 okay or well in the case of bi-partite pieces doesn't matter but if I have an additional system C then the dimension could simply be different and this is no sorry what I'm saying now is something too but this is true in a bi-partite case of course okay and extendable means just that you might consider rho A in a smaller or a larger Hilbert space than rho B okay so this is why you have to extend it be able to extend it by zeros that's not so important what is important is that you just have to check what is the concave function of the eigenvalues of the reduced density matrix yes exactly so if I want to check the average quantity so it's not increasing on average as we have for monotones right then I have to check only the condition on unilocal operations and I don't care about communication at all okay yes yes yes and this is the yeah so this is also true in a multi-partite case exactly and this is what really simplifies this problem because here you have a different condition right so you're checking if something is non-increasing on average what we have as entanglement theory is that LOCC is the free resource and we cannot increase under LOCC this is this condition right okay so in the case of multi-partite systems again this is an important simplification it's a huge simplification and in the case of bi-partite systems we have a very simple characterization of entanglement monotones okay this is also due to Gifre Bidal now yeah now what is an example of an entanglement monotone I'm sure that you all know examples so one would be for instance the concurrence you all know the concurrence I guess no so maybe coming back to your question so maybe what is what was confusing here now is of course if I study entanglement monotones then I don't need this order to make sure that the measure is a monotone but of course what I want to have is an order okay so I want to know which state psi is more entangled than the other okay and for this I need to study the order which is induced by LOCC okay so I have to study this okay okay so an example of an entanglement monotone would be the concurrence which is defined as the absolute value of psi conjugate for two qubits sigma y tends to sigma y psi I guess that you all know that no the concurrence maybe you know it a little bit written in a different way but I guess that you know it and what I want to say here is that this is an example of something that is called an SLOC invariant okay let me now just mention that you will come to SLOC which has stochastic local operation assisted by classical communication a little bit later but this SLOC invariance allow you to construct entanglement monotones okay there's a general procedure of doing that and so this is one example another example which already would be this entanglement monotones EL which are defined as so D is the dimension of my Hilbert space of my local Hilbert space then this would just be the sum of the sorted Schmitt coefficients of my star entanglement monotones of my state psi and using the condition that we had this maturation condition we have written it down yesterday also in terms of these quantities if you look back at your notes from yesterday you will see that this condition here is equivalent to say that EL of psi has to be larger equal to EL of phi for all L okay so L going from 1 to D so this is of course equivalent to this okay so we have seen that we can do this transformation even only if the maturation condition is fulfilled which is equivalent to these entanglement monotones of psi being larger equal to the entanglement monotones of phi so that's very nice because now we have a set of entanglement monotones which in the case of pure states are also entanglement measures that allow us what I can transform psi to phi in the case of bipartite but it's obvious because I'm writing here lambda so there is no lambda in the multi-partite case not in general no so for pure states entanglement monotone implies entanglement measure and for convex function also but in general not this is also why I chose to really call this now monotone to stress that whenever it's convex it's really stronger but if it's not convex it's not necessarily strong okay okay actually talking about this there are examples of entanglement measures which is not this means that it is a measure so it's sure concave but it's not concave so it's not a monotone there are examples of that and I guess that you also all know them because these are just already entrapies I'll give you another example that is very physical in my opinion it's operational it is imagine that I have a state psi and I consider now all the states that I can reach via LOCC so this is a set of states and with LOCC I can go to any of these states in this set that's a very physical thing this tells me how well I can transform psi into other states how can I reach from a state psi now if you do that in that case then you can compute here the volume of this set and you can very easily show because of physical arguments and because it's operational you can easily show that this is the volume here is indeed an entanglement measure okay so it's not increasing under LOCC however this volume is not a monotone okay so it doesn't obey the strong ethnicity condition it's a very physical in my opinion very physically motivated measure which is not a monotone but it's for sure an entanglement measure okay so to give you an example for measures that are not monotones I was mentioning that okay now let me talk briefly about some applications of entanglement monotones because they have very nice applications because once we have an entanglement monotone for pure states we can construct from that an entanglement measure for mixed states this is point C applications of entanglement monotones for pure states okay so the first application is that I can construct construction of entanglement measures for mixed states okay and this is actually very simple and I guess that you also many of you know because the following holds you have an entanglement monotone so let E be an entanglement monotone for pure states then the convex roof construction of E is an entanglement measure for mixed states so E of LO which is the infimum over all P i phi i such that rho is P i projected onto phi i so all decompositions of rho the sum over i P i E of phi i this is what is called the convex roof construction this thing here is an entanglement measure of course for mixed states okay so if I have an entanglement monotone for pure states then I can use it to construct entanglement measures for mixed states okay so I can extend it and the way that I extend it is that I take this convex roof construction okay so I take the infimum over all possible decomposition of my state row and this can then be pretty easily shown to be an entanglement measure okay that's nice right because it's a good application we can use it to construct to extend this to the states of mixed states to get an entanglement measure what is of course not so nice is this infimum right because so in order to compute that this might be very difficult again you have to optimize over all possible ensembles and we have already seen in the first lecture that there are infinitely many ensembles okay so this is why this is difficult in general for some cases this can be calculated analytically like for instance for the entanglement of formation for example of this entanglement of formation which is defined as this infimum so the convex roof over all the compositions E of pi i where E is just the for Neumann entropy of the reduced density operator so I guess that you are also familiar with that that measures how much entanglement do I need on average to construct my state okay so what is the minimum amount what is the best way to construct my state as we have also discussed in the first lecture there are infinitely many ways and after that after you have prepared it you don't see how you prepared it so this information is lost and so you are asking what is the best way for me to prepare the state row and in the case of two qubits this has been there is an analytic expression for that something that you can easily compute giving some eigenvalues of some operators that you construct from your mixed state row okay this is due to Wooters this for two qubits analytical formula and this is due to Wooters okay so sometimes you can compute that even though in general it might be very hard to see infimum but nevertheless I mean this works and this is a way of computing things and of course having something like that could be very useful if you want to derive some bounds because bounds you can always derive with one particular decomposition okay now let me just briefly explain to you why this is entanglement measure and in order to do that I come to what I was saying already several times now, namely that if E so this is a remark if E is a convex function if E obeys a strong monotonicity condition and E is convex then E is an entanglement measure so convex functions that obey the strong monotonicity condition are entanglement measures why is that well let's see so we have an entanglement we have something that obeys a strong monotonicity condition so what I called an entanglement before a monotone before so we have that E of row is larger or equal than the sum p i E of sigma i for any row and p i sigma i for which this is possible via LOCC okay so this is the condition this is the strong monotonicity condition and E is convex okay but E convex means that this thing here is larger or equal than E of the sum p i sigma i but this is precisely E of lambda LOCC row sum over all the outcomes which would be lambda of row then of course I get this state and so our E of row would be larger or equal to this E of lambda of row which means that it's an entanglement measure okay so if it obeys a strong monotonicity condition and it's convex then it's a measure and as I told you before the richer definition of entanglement monotones was including convexity okay so then it was really something stronger okay okay now let me also tell you something else which is that if you have a convex function then you can prove that in order to show that this condition is true so that it obeys a strong monotonicity condition again you do not have to consider this complicated LOCC thing because you can again restrict yourself to unilocal operations okay so the theorem says that if E is a convex function so let me just remind you that E being convex means what? means that the sum of Ip of E sigma i is larger or equal to E of the sum Pi sigma i okay so if this is true then E obeys a strong monotonicity condition if and only if E is non-increasing on average under unilocal operations okay so again I just have to check that it is non-increasing if Alice does something it's non-increasing if Bob does something but I don't have to correlate the outcomes okay so this is again a huge simplification okay now without I mean I don't want to talk more about this because I finally also want to come to the multi-partite case let me just tell you one more application of entanglement monotones because this also holds also in the multi-partite case which is the success probability okay so this entanglement monotones can also be used to study the cases where we cannot do a deterministic transformation so we were studying before but I can only go probabilistically to some state this is what is called stochastic transformation we go there with a probability that is smaller than one it will be our second application which is maximum success probability so we are considering the situation that psi cannot be transformed with LOCC to phi and we are wondering what is the maximum success probability so what is this okay so what is the maximum probability in obtaining the state phi because I'm considering this situation that I have my state psi I do some protocol this is now whatever many rounds whatever I end up with p1 phi but here I end up with something else some phi2 and some phi3 and whatever and I'm wondering how can I make this probability here as large as possible or here I'm wondering what is the maximum success probability so how large can this value be and let me just say here that this is also related strongly related to SLOCC because SLOCC classes these are equivalence classes and it's defined so two states are called SLOCC equivalent if there exists invertible operators so this is in GLD invertible operators invertible matrices such that these matrices applied to our state psi gives me with a certain probability the state phi and they are called SLOCC equivalent if these operators exist and this p is of course different than zero okay so you can obtain the state at least with some probability with a non-vanishing probability okay now in the case of bipartite systems you can easily see that whenever they have the same Schmitt rank so whenever they have the same number of coefficients that are non-vanishing the Schmitt vector then you can always transform them probabilistically okay so they are all SLOCC equivalent so in the case of bipartite systems there is no there's only one class of SLOCC okay in the case of multi-partite systems however this changes again completely because there are several classes so we know that for qubit case for instance we have the chz class and the w class right they are not these are two separate SLOCC classes which means that you cannot go from the chz state to the w state via invertible operators okay so you cannot even reach the w state from the chz state even with the smallest probability it's impossible with local operations in a bipartite case this doesn't happen okay so now this was just to introduce SLOCC classes basically in this context because of course this means that these two states are in particular SLOCC equivalent right because I can go from psi to phi so one of these branches just means that I apply here some operator a then so b then so c and so on to my state psi and so they have to be in the same SLOCC class and what I care about now is what is the maximum success probability and what you can show is that it is given by entanglement monotones so you can show that the maximum success probability of going in general actually from a state rho to a state sigma is equal to the minimum of all entanglement monotones such that well let me write here mu mu of rho separable is equal to 0 so this is normalized to 0 of mu of rho divided by mu of mu of rho divided by mu of sigma okay sorry now I just noticed that I forgot to mention something because we discussed it already yesterday but we didn't write it down of course due to the conditions that we had the very beginning the definition of entanglement measures entanglement monotones for pure states for mixed states from the definition it follows out basically that the entanglement of a separable state is always minimal okay because you can always go from a state to a separable state whatever separable state you want because a separable state can be prepared via LOCC and so you can always without loss of generality choose this to be 0 for any separable state okay and so this is why I write it here so it just means that it's rescaled but it's not a restriction to the entanglement monotones so what this formula tells us that if I want to know what is the maximum success probability of going from some whatever this is also true micropathite so whatever state rho to some state sigma then it is given by this innocently looking expression so it's just the ratio of the entanglement monotones of these two states now I'm saying it's looks innocent but of course here a minimum of all possible entanglement monotones right? this is what makes it challenging because in general we don't know this whole set however in the micropathite case pure states you can show that the maximum success probability of going from a pure state psi to a state phi is equal to the minimum and now it just has to take the entanglement monotones that I was introducing before so you take this set of d entanglement monotones that you have and you just compute the ratio so these are d different numbers and you take the minimum of that which is something very easy to do now this closes the circle because we see that this is one if and only if this was true which is just the condition that the maturation is fulfilled which is the condition that psi can be performed to phi via LOCC as it should be so the success probability is one even only if I can reach the state with LOCC right? and this is what closes the circle so we see here that the maximum success probability which is now really doing an SORCC protocol not a deterministic protocol can be computed using entanglement monotones in general this is also true micropathite so if we would know all entanglement monotones we would have solved in particular also the problem of which states can be transformed to another state via LOCC because if this is one then it can be transformed deterministically even only if and the problem in the micropathite case is of course that we don't know all entanglement monotones in the pipadite case however you can prove that it's sufficient to look at just these entanglement monotones that we have introduced before okay so in the pipadite case that's very easy for the pipadite pure states again so this can be easily computed and you can know whether you can reach the state deterministically or what is the maximum success probability okay now one thing that I wanted to mention here briefly is if I do this transformation how does the protocol actually look like so if you look at the protocol in order to derive so if I have some state psi and some state phi how can I reach this maximum success probability so what is this protocol that I have to do here okay then you see in the in the pipadite case that what you do is that you go from the state psi to a state psi with LOCC so deterministically and then you go with SLOCC so probabilistically to the final state phi now what is interesting here is that this state is in the immediate state here occurs in a completely different protocol again namely the one where you want to study what is the maximum fidelity so I want to do a deterministic transformation from psi to some state and I want to optimize the fidelity of the final state with the desired state so just as a remark this state psi here that occurs when you do maximum derive the optimal success probability of obtaining the state phi it also occurs when you optimize a different quantity which is the fidelity okay so it's the same state which is interesting and not obvious okay so this was as much as I wanted to say regarding the applications of course now there are many more many more transformations that one can consider we discussed that yesterday briefly that of course I can also study now what is the transformation from some pure state to some mixed state or even more generally from some mixed to some mixed state and so on and so for pure to mixed for pure to an ensemble this is known there are some works by Martin Plainio and others and also Gifre Bidal and Michael Nielsen where they studied that and for mixed to mixed very little is known okay so this is a very difficult problem still sorry I have an hour okay okay well that's a pity I still wanted to talk a little bit about Michael about that may I have one more minute so let me just say that we studied now or we have seen that the phi plus state is the maximum entangled state now what happens if we go to more systems so imagine that I consider now three systems what is the maximum entangled state there well so according to everything that I was saying the maximum entanglement should be the optimal resource so this should be the state with which I can reach the whole Hilbert space like in the case of the bipartite we have the phi plus goes to any state in the Hilbert space so I want to have here a state which can be transformed into any other state in the Hilbert space so I want to get any other three qubit state but this cannot be because we have already seen before that the gx state and the w state they are not even SLOCC related so not even probabilistically you can go from the gx to the w and this implies immediately that there exists no maximum entangled state in the multi-partite case so you cannot have a single state which you can transform via LOCC into any other state in the Hilbert space that's impossible in the multi-partite case but what you can have is a set of states so instead of having this picture and saying okay from here I can go to everything else what I can do is say I have several states okay and now giving this set of states I can go to the whole Hilbert space so this can be transformed and I reach all the states that I want okay and this is what we call a maximally entangled set and you can characterize that so you can compute it what can they reach and so on and you can study there really also LOCC transformations in the multi-partite case okay so I will talk more about that in my talk next week so maybe you're still interested in that and maybe you can attend this talk okay so with this I'm sorry that I had to rush now thank you for your attention