 So thank you very much for the invitation and giving me the opportunity to present my work. So the topic that I will talk about is entanglement and coherence and quantum state merging. This is a joint work of myself and co-authors from IGFOR and also with Andreas Winter from autonomous university in Barcelona. And so it will be very related to the first talk today to the resource theory of coherence that I will also briefly review. But I will not talk so much about this generalized resource theories of superposition at this moment. OK. So in the first part of the talk, I will tell you some results about the resource theory of quantum coherence. So we will talk about incoherent states and incoherent operations. We will talk about how to quantify coherence and we will also talk about quantum coherence and distributed scenarios. This is very much related to entanglement to local operations and classical communication but more with respect to quantum coherence. And in the second part of the talk, I will present the results, which is related to quantum state merging. We will talk about standard quantum state merging, which was already presented more than 10 years ago. And we will talk about the incoherent version of quantum state merging where these concepts of coherence come into the game. OK, let us start with the first part, the resource theory of coherence. And again, this is very much related to the first part of the talk that we had today. So we have a quantum state and we call incoherent, if it is diagonal in some preferred basis. So if it is a sum of our IPI and some states I, and if a state is not of this form, we call it coherent. And the set of all incoherent states is labeled here by a calligraphic I. And this goes back to the group of Martin Plano a few years ago, and there is an article by myself, Gerardo Adesso and Martin Plano, that is about to be published in reviews of modern physics. So a quantum operation is called incoherent, if it can be written as in such a form. So we have here a lambda applied to rho, and it acts here with a cross operator. So a cross operator multiplied to the rho, and then a cross operator, a dagger. And as we heard in the first talk today, these cross operators have to fulfill these incoherence conditions, or the crowd operator. Applied on each of the diagonal states must produce a diagonal state. This means that such quantum operations, so this overall procedure is a quantum operation, and such quantum operations cannot create coherence, even if you know the individual outcomes of your measurement. Because every quantum operation can be interpreted as a measurement, at least in principle. And I here labels the outcomes of the measurement. So there are other frameworks of coherence that are discussed in the literature that are slightly different from the framework introduced by Martin Plano and his group. And I will review on this slide the most important frameworks, but I will not go into much details, because there are actually at least 10 different frameworks that are currently discussed in the last two years. So one of them is called maximal incoherent operations. This was introduced by Johann Oberk more than 10 years ago already. And this is actually the most general set. These are quantum transformations which cannot create coherence in general. So you apply quantum transformation onto a diagonal state, and it gives you a diagonal state. This is the most general set of transformations that you can have in any reasonable resource theory of coherence. Because if you have a state which if you have a transformation which does not even fulfill this, this would mean that you can create coherence out of a diagonal state which doesn't make much sense if you talk about resource theory. There is a second approach called strict incoherent operations which goes back to Andreas Winter and independently to the group of Vlad Covedral. So these are operations for which the crowd operator daggers are also incoherent. This sounds a bit mathematical, but it turns out that this definition leads to some nice properties which is not represented by the standard framework of multi-planet. The third approach are called translationally invariant operations. And these are quantum transformations which commute with time translations, meaning you have such a time translation here. You apply it after your quantum operation lambda and it commutes, meaning that if you first apply the time translation and then lambda, it gives you the same result. And this is just a nice definition by symmetry. So you have some symmetry here. And it turns out that such transformations are closely related to the framework of coherence. For example, such transformations cannot create all diagonal elements in density matrix. Another nice definition is defacing covariant operations. These are quantum operations which commute with defacing. So delta here is a defacing operation, meaning it removes all of diagonal elements of your density matrix. And it doesn't matter if you apply delta before or after the lambda. Then in this case, then lambda is defacing covariant. And again, if you look at these two things, they are these are definitions by symmetry, as I would call them. And also, this is somehow, these operations can also not create coherence. So these are, they are closely related to the other frameworks of coherence that I have presented here. But this is just to give you a short overview because there are at least four or five more definitions that are discussed in the literature, but we don't have time to go through all of them. So it's just to give you a flavor of what is going on in this research area at the moment. So the, once you have defined your appropriate framework of coherence, you would like to know how to quantify coherence. So if I give you a state, what, yeah, we would like to have a number as, as we have heard today in the morning, we would like to have a number that labels that quantifies the amount of coherence in a state. And the, so this number C should have the following properties. It should be greater equal than zero and it should be zero if and only if your state is diagonal and the number should not increase under incoherent operations. So the, if you apply an incoherent operation to your state and calculate the coherence, then it should not be larger than the coherence of flow. Yeah, yeah, this is a good question. That's right. So usually in most frameworks or in all frameworks that we know today, there is no bulk coherence. Yeah, so thank you. The many coherence measures are also monotonically non-increasing on average under selective incoherent operations. So meaning that you perform your measurement with individual cross operators, not creating coherence and give, you get some probabilities QI for the outcome I and some post measurement state, sigma I here. And then if you average the coherence of these outcomes, this average is not larger than the coherence of the initial state. This was also mentioned today in the morning. And this is actually closely related to the framework of entanglement quantifiers. So if you, where you had, yeah, where this would be E instead of C, so quantify entanglement, you would find very similar conditions there. So, and this was a very abstract notion of coherence quantifiers, but what are the actual examples? So there are two important examples for coherence quantifiers. This is the first one is called coherence cost. And coherence cost quantifies the asymptotic rate of maximally coherent states. So these are maximally coherent states here, which is required to create a state rule via incoherent operations in the asymptotic limit. So for those who are familiar with entanglement theory, this would be entanglement cost and entanglement theory. And we can actually write a formula for that. So the coherence cost is actually equal to the coherence of formation. And the coherence of formation is the minimal average coherence of a state. So this is a so-called convex roof function. So this is sum over IPI. And here we have C of psi I. And this CR will be defined on the next slide. Actually, this is the relative entropy of coherence. It's actually just the entropy of the diagonal elements of this guy. So this is coherence of formation. And it turns out that it's equal to the coherence cost. The second important quantifier is distillable coherence, which is kind of dual to coherence cost. So the distillable coherence quantifies the maximal rate for extracting maximally coherent states via incoherent operations in the asymptotic limit. And for this quantity, we actually have a closed expression, which is a bit surprising because the definition looks not so easy, but we have a closed expression for that. And the closed expression is that it's the entropy of the diagonal elements of rho minus the entropy of the whole state. So delta here is the defacing operation, meaning removing of diagonal elements of your matrix. And these results here go back to Andreas Winter and Dongyang from last year. Okay, and of course, this quantity is also dependent on your framework. So here the formulas for coherence cost and distillable coherence apply only for the incoherent operations of Martin Plano and his group. But if you apply a different notion of coherence, as I have discussed on the previous slide, then the amount of the distillable coherence and the amount of coherence cost change, or can change in general. Okay, there are also other coherence quantifiers that also have also interesting features. One is the relative entropy of coherence, which is just the relative entropy between rho and the set of diagonal states. And it turns out that this quantity also has a closed expression and coincides with distillable coherence. So it's just, again, the entropy of diagonal elements of rho minus the entropy of rho. And last but not least, I would like to mention the robustness of coherence, which goes back to Gerardo Adesso and his collaborators. So this is defined as the minimum over tau and you minimize the quantity s under constraint that is larger than zero, and you have the additional constraint that they are plus s tau divided by one plus s. So this is just kind of normalization. So you normalize this whole thing so that it's trace one. And you require that this guy here is incoherent or is diagonal. So the minimal s for which this occurs is called your robustness of coherence. And it has actually operational interpretation via interferometric visibility, as demonstrated recently here by Andreas Winter and collaborators in this publication. And this quantity or a similar quantity is also known in entanglement theory under the name robustness of entanglement. Okay, so I hope I have given you an introduction to the research theory of coherence. And now we go to the next level, we go to coherence in distributed scenarios. And what does it mean? We remember LOCC operations from entanglement theory. So these are local operations and classical communication as we heard also today in the morning talk. So, and similar to these LOCC operations, we can introduce local quantum incoherent operations and classical communication. And this means that instead of a general quantum operations locally we require that we only perform incoherent operations locally. And apart from it, we also have a classical telephone. And now I will give you some features of this operation or some properties of them. So these LQIC operations, they preserve the set of quantum incoherent states. So you have such a state which is called quantum incoherent and it has this form. So it's a convex combination here of state sigma i on Alice side and i on Bob's side. So these states i are incoherent. They are diagonal. So this is the important difference to entanglement theory. In entanglement theory you would have a general state here on Bob's side and then you would call the state separable. But here we have only diagonal states on Bob's side. And a classical, or here a quantum classical state would be, would allow for a unitary rotation of the basis additionally here. You would allow to have any basis. But in our case, in these states we have a fixed basis because we talk about coherence theory where we always have a fixed basis. So this basis is fixed through the entire discussion. And so this, to be more precise, this set is a subset of quantum classical states where you would allow additional unitary rotation here. Thanks for the question. And the second feature of these LQIC operations is that they cannot increase the QI relative entropy. So the QI relative entropy is a new quantity here. It's defined as the minimal relative entropy between a row and such states. And it turns out that it also has a closed expression which is it is the entropy, the von Neumann entropy of row IB after performing defacing on Bob's part minus the von Neumann entropy of the overall state row AB. And oh, this quantity is important because it's a monotone under this LQIC operation so it cannot increase under such operations. And it has found many applications in the tasks that I will discuss over the next, yeah, over the rest of the talk. And also this quantity is actually additive in the input state. So which makes many, many things easier to evaluate. Actually, yeah, so this quantum coherence and distributed scenarios has been introduced by myself and co-authors in these two papers and also independently by the group of Vlad Covedral here. Okay, let us now present, talk about an application of such a framework. So the first application is called assisted distillation of quantum coherence. This is the task where we have two parties, Alice and Bob and so who share a bipartite state row AB and the aim is to asymptotically distill a maximally coherent state on Bob's side via the operations that I have discussed before. So via LQIC operations. And so we would like to know how much of the states can Bob distill asymptotically and how is the difference to the other, like to the standard distillation procedures in coherence theory. So and the figure of Meret here is the distillable coherence of collaboration. So it's called distillable coherence of collaboration because you can see this task as a task where Bob aims to distill coherence while getting help from Alice. So that's why collaboration. So we have two parties who are collaborating and yeah, one of the parties aims to distill, to increase his amount of coherence locally. And it turns out that the distillable coherence of collaboration, so here, this is the distillable coherence of collaboration is bounded above by relative entropy, by the XQI relative entropy, which had a close expression as I have discussed on the previous slide. So it's von Neumann entropy of this guy, minus the von Neumann entropy of the total state. And even more interesting for pure state, so if you have a pure state here, then it's actually equality, meaning that we have solved this problem for pure state. So you have a pure state that distillable coherence of collaboration is equal to the QI relative entropy, when again, which is equal just to the von Neumann entropy of the diagonal elements of Bob's local state. So meaning that for pure states, we have solved this problem of assisted coherence distillation. And we would like now to compare how this quantity differs from the case where Bob just distills coherence locally without getting assistance from Alice. And so if Bob just runs the standard distillation procedure locally, then he can distill coherence at the following rate. It will be the von Neumann entropy of his local state, the diagonal elements, minus the von Neumann entropy of the state. And if you compare this to quantities, you see that here, we subtract the von Neumann entropy of Bob's state, where here we don't do it. So meaning that this quantity is larger, this means that by getting assistance from Alice, Bob can increase his distillation rate exactly by the local von Neumann entropy of his state. Which is surprisingly, this quantity is basis independent, meaning that the improvement of the protocol is basis independent, although the framework of coherence is basis dependent. And last but not least, I would like to say that this has been, so there has been an experiment performed on the assistance coherence distillation here by a group of Guansanguo in China. Okay, let us now come to quantum state merging and to, so where we will also apply the results of LQCC operations and the overall framework who will also be applied to this task. But before we do it, let us first talk about standard quantum state merging. So standard quantum state merging was introduced here by Michal Horodetzky, Jonathan Oppenheim and Andreas Winter in 2005. And it can be seen as a game between three parties, between Alice, Bob and the referee. They share a joint state. And yeah, they share many copies of a joint state psi. And the aim of the procedure is to give Alice part of the system to Bob while preserving the total state. So this means that the total stage psi RBB prime should be the same as psi RB up to relabeling A and B prime. So in other words, you'd like to give Alice part of the system to Bob while preserving the correlations with the referee. And so in order to be able to do this, Alice and Bob need to have entanglement. And if they have entanglement, they can do teleportation for doing it, but without entanglement, they will not be able to do it. And in this paper, the authors studied how much entanglement is actually needed for this procedure. And they found the minimal number of singlet rates needed for this procedure. And the minimal singlet rate is given by the conditional entropy. So the conditional entropy between A, Alice and Bob is given by the von Neumann entropy of Alice and Bob minus the von Neumann entropy of Bob's local state. And this quantity can be positive or negative in quantum theory. And if this quantity is positive, then Alice and Bob can achieve quantum state merging with singlets at this rate. And if they have singlets with, if they have less singlets, they cannot do quantum state merging. And if this quantity is negative, then merging is possible without singlets. So Alice and Bob can do quantum state merging just by local operations and classical communication. And additionally, they will get singlets at a rate which is given by the minus of this quantity. So in other words, this quantum state merging gives an operational interpretation for the conditional entropy no matter if it's positive or negative. Sorry? Singlets are with two parties. It's a bipartite. So singlets are shared by Alice and Bob. Yeah, yeah. It's quantum state merging from Alice to Bob, yeah. So the merging procedure merges to Bob. It's asymmetric, yeah. No, no, singlets. If I say singlets, I mean between Alice and Bob because the local singlets do not have any meaning in entanglement theory. So it's between singlets between Alice and Bob. Okay, so now we will discuss the difference of standard quantum state merging and incoherent quantum state merging which we introduced here last year. So in standard quantum state merging, we talk about shared entanglement. So this refers to your question, so shared singlets and they are considered as an expensive resource. In standard quantum state merging, local coherence was available at no cost. So the parties could produce local coherence arbitrarily and not take it into account anyhow. So but in incoherent quantum state merging, as we presented here, we also take Bob's local coherence into account. So we don't assume that Bob can produce arbitrary amounts of coherence locally, but take it into account explicitly. And so in more mathematical terms, we have a state, RAB, shared between Alice, Bob and the referee. And we consider quantum state merging via LQUICC operations and where additional singlets and maximally coherent states on Bob's side are provided at rates E and C. So E is the singlet rate between Alice and Bob and C is the coherence rate that Bob has locally. And so we are interested in optimal entanglement coherence pairs. So these are pairs of entanglement and coherence for which merging is possible, but where E and C cannot be reduced. So this means if you have such a pair, you know that you can do quantum state merging with entanglement at this rate and coherence at this rate. And the main problem is to determine all such optimal pairs for a given state. So we have given a quantum state and we want to know for which entanglement and coherence pairs we can look one of state merging. So it's a difficult question, but this is what we have attacked here, what we have tried to solve. So one of the main results of this paper is the following theorem. So given a quantum state or RAB, any achievable state, any achievable pair must fulfill the following inequality. So entanglement plus coherence must be greater equal than the von Neumann entropy of this quantity. So this is the state row RAB, so the overall state. And here you apply a defacing operation only on Alice and Bob's part. So you deface Alice and Bob's part, but you don't deface the referee. And the rates. So this is the rates, the required rates. So meaning that, so this is the asymptotic number of two qubit singlets per copy of the state. So you have many copies and per copy of the state, this is the number of two qubit singlets. And this is the number of single qubit maximal cohesion states per copy asymptotically. And these are the rates. And they are greater equal than the von Neumann entropy of this quantity minus the von Neumann entropy of this state, again, with the defacing operation but only applied on Bob's part. And so again, here a bit more explanation. As is the von Neumann entropy. And here this delta X denotes full decoherence or full defacing on some subsystem X. So this is how you would apply it, let's say, to be a bit more precise. So you apply projectors here, IX, that operate and roll, and again, here IX. What can we conclude from this result? So after looking at it for a few minutes, you would realize that this quantity is actually non-negative. So this right-hand side is never negative. And because of this, so the sum of entanglement and coherence must also be non-negative. And this means that no merging procedure can gain entanglement and coherence at the same time. Why is it like that? It is because if entanglement is negative, this means that you can do quantum state merging, gaining entanglement. If coherence is negative, this means you can do quantum state merging, gaining coherence, but they cannot be both negative at the same time. So you cannot run quantum state merging and gain both entanglement and coherence. If you gain entanglement, you have to provide coherence and if you gain coherence, you have to provide entanglement. Okay, if we have a pure state, we have the following bounds. We have here, the entanglement is bounded above, bounded below by the conditional entropy. This is the result of Horodetsky-Oppenheiminder, and this is our result. So the entanglement plus coherence is bounded above. But actually, this is also the conditional entropy. If you look at these two quantities, they are basically the same up to the fact that we have defacing here. So this is the conditional entropy of the defaced state. Yeah, oh, yeah. And this bound is achievable for pure states, meaning that if we have a pure state, then quantum state merging can be done without coherence by using entanglement at exactly this rate. So C0 and E is exactly this then. Okay. So this is the plot of achievable region. Here E0 is the amount of entanglement that you use for merging without coherence. So E min is the Horodetsky-Oppenheiminder bound. That means that if you have entanglement less than E min, you cannot merge no matter how much coherence you are provided. And here, what we still don't know and what would be very interesting is C max. C max is the amount of coherence that you need for merging at the minimal entanglement rate. And because we don't know this, we also don't know the region here in between. So yeah, this would also be very interesting, but we just don't know how it looks at the moment. For some mixed states, we can actually solve the full problem. So for example, if you have such a state, you can find all optimal entanglement coherence pairs, but because I don't have so much time, I would like to jump over it and go to the next example because we have a very nice open question that I would like to present. And so there is some evidence that a large amount of local coherence can be saved by using little entanglement. That means that imagine you have a state where you can do quantum state merging with some entanglement and a large amount of local coherence. And we have some evidence that also instead of this large amount of local coherence, you could just increase your entanglement just a little bit by some epsilon and this would reduce your coherence dramatically. And so one example where I believe that such a phenomenon can occur is this state. So you have here a referee, which is just given by some flags. This DB is a very large number, is a very large dimension of Bob and Phi IA are single qubit states of Alice. And here we have mutually orthogonal maximally coherent states on Bob's side. And if you think about it a bit more, this state can be actually merged without an entanglement if Bob just applies a local measurement, but for this he would need a large amount of coherence. On the other hand, this state can be merged without any coherence on Bob's side by just teleporting Alice's system to Bob and this is one qubit. So this can be done by just using one singlet or alternatively no singlets but a large amount of local coherence. So I think this state is a good example for demonstrate such a fact, but this is just a hand-waving argument we cannot prove that the procedures I talked, I told you are optimal. Okay, let me now summarize. So we have introduced the task of incoherent quantum state merging in which entanglement and local coherence are resource. Oh, we have showed that entanglement, we have showed that entanglement and coherent sum in this procedure is bounded below by this quantity and this implies that no merging procedure can gain entanglement and coherence at the same time because this quantity is non-negative. So the bound is tied for all pure states. Any pure state can be merged without local coherence by using entanglement at exactly this rate and these results imply an incoherent version of Schumacher compression. So this is a point that I didn't mention during my talk so far. So it means that the von Neumann entropy of the diagonal elements of rho is the optimal compression rate for a quantum state under the assumption that the decompression has to be performed via incoherent operations. So if you look at this, this entropy is larger than the standard compression rate. The standard Schumacher compression rate is just the entropy of rho and this quantity is larger and this is because of the restriction that we assume that the decompression has to be performed in the incoherent way. That means that in general at the decompression level, you need coherence. So the results I was talking to you about published in this paper and yeah, as I also said, there is a recent review on quantum coherence as a resource that will appear in the ref mod fees probably next week or even this week, we'll see. Okay, thank you very much. Thank you.