 OK, so thanks for giving me the opportunity to speak here today. And indeed, I will talk about the resource theory of superposition, which is not the resource theory of all kinds of quantum superpositions, but a special instance of it. And this is work I did in cooperation with Nathan Killaren, Dario Egloff, and Martin Blenio. So let me start with a few words concerning quantum resource theories in general. Well, as you all know, there are protocols and devices which work on the laws of quantum mechanics that outperform their classical counterparts. Now, a natural question is why. What in quantum mechanics makes this protocol more efficient than classical protocols? And one way to attack this question is in the framework of resource theories. So the most studied and most established quantum resource theory is the entanglement theory. And it's based on a restriction. All quantum resource theories are built on a restriction that is placed on top of the laws of quantum mechanics. In this case of entanglement theory, the restriction is the restriction to local operations and classical communication. And that is a physically motivated restriction because it's much easier to communicate classically because you can amplify classical information, but you cannot amplify quantum information in general. So then with this restriction, the property under study is entanglement because it cannot be created using only local operations and classical communication, but it allows for tasks which are forbidden otherwise. For example, it allows one to overcome the restriction. The simple example would be I teleport my system from one lab to the other lab, then I do whatever I want and I teleport it back. So with the use of entanglement, I've overcome the restriction to local operations. Now in general, all quantum resource theories are built like this. One puts in a additional restriction which should be modified physically. From these restrictions, there emerge two main ingredients. The first one is the three states. So these are the states that are easy to create or simple. And the second ingredient are the three operations. And the minimal restriction onto the three operations is that at least they should transform the three states into three states. Otherwise, I could create resources for free. Now in the example of entanglement theory, these are the separable states and LOCC. And then all states that are not free, we call them resource states. Now once this basic framework is set, one tries to answer questions like, how can I manipulate my resource? So can I transform one resource state into another one using only the three operations? How can I detect the resource both theoretically and in the lab with measurements? And how can I quantify the resource? So can I introduce an order or a partial order into the resource states? And all this is in mind usually with operational advantages. So one wants to use the resource in order to perform some tasks better than without the resource. And in addition, so that's basically the idea for applications. It helps to study the property by itself from a fundamental point of view. Now if one investigates these questions rigorously, then one hopefully understands the resource better. And so one can use it more efficient in new applications or one even finds out which resource is present in which application. Now let me modify the resource theory I will present to you now. So in principle, all properties of quantum mechanics that you cannot reproduce using classical physics can lead to an operational advantage. So in general, all types of non-classicality are a resource or might at least be a resource. One underlying principle of different quantum properties is the superposition principle. So in coherent theory, one studies the superposition of orthogonal basis states. In entanglement theory, one studies the superposition of separable states. And an basically optical non-classicality is the superposition of global coherent states. Now so it's very natural to investigate non-classicality in terms of superpositions. And that's exactly what was done in this paper here. Now if we want to define entanglement, there's a relatively big agreement on how we do this. But non-classicality in general is not a very well defined quantity. So one idea to quantify non-classicality like mathematically rigors is can we convert it to entanglement? So if we can convert state to an entangled state using a fixed operation, then it's a non-classical one. If we can't, then it's a classical one. And with the theory I will present you here, that's exactly possible. So we can use the theory to define non-classicality via entanglement. Now let me come to the basic framework or the mathematical framework I will use. First, we only consider finite dimensional Hilbert spaces or state spaces. And the pure free states are defined by these CIs. And these CIs, they form a linearly independent, but not necessarily orthogonal basis of the state space under consideration. So in coherent theory, these pure free states, they are orthogonal here or not. And I will tell you later why we made this choice. Now the free states in general are just the statistical mixtures of pure free states. And the free operations, they are defined using free cross operators, the cross operators Kn here. And they are constructed in a way that they map the set of free states into itself. Now then the free operations in general, they are all quantum operations that can be decomposed into these free cross operators. And they should be trace preserving. Now as in coherent theory or in most other resource series, you can define other sets of free operations. And this gives rise to other resource series. So that's not a special feature of this resource theory, and it's not a deficit. For example, in entanglement theory, one could also consider only one way classical communication. And that's for some application, a well-defined setting, which gives rise to another resource theory, but to a meaningful resource theory nevertheless. So I will not comment on alternative definitions of the free operations. I will just say why this one is a good one basically. So if we see this quantum operation as a generalized measurement with outcomes n that are associated with these cross operators Kn, then if we are allowed to sub-select according to this measurement outcomes, we can never deduce that resource states have been created out of free states. So in a sense, this is the maximum set of free operations that if we are allowed to do post-selection cannot create the resource for free. Now let me come back to the relevance of the theory. So if you want to define non-classicality using the principle of superposition, then you can use what is called a classical rank and it has been introduced in these two papers. So if we have a general set of free states, so in this case, not linearly independent, then the classical rank of a pure state is just the minimum number of free states we have to bring into superposition in order to represent the state. So in a sense, this is the analog of the Schmidt rank in bipartite entanglement. And now, having this quantity, this classical rank, we can define faithful conversion operations. So this is a physical operation that transforms all pure states with a given classical rank into an entangled state with the same Schmidt rank. The idea behind this is that we do not create entanglement out of nothing, but we rather transform the resource of non-classicality into the resource of entanglement. Now, these faithful conversions, they allow for the definition of non-classicality because we can just say now we quantify the amount of non-classicality by the amount of entanglement we can create using this fixed conversion operation. Our S is the Schmidt rank. Now, not for all set of free states such faithful conversions exist. And indeed, such faithful conversions, if we are in a finite dimensional Hilbert space and have a countable set of free states, they exist if and only if the free states, the pure free states are linearly independent. So that's why I made at the beginning a choice of linear independent free states. Now, these faithful conversions, we will see later what an example for these conversions is. So in addition, the theory here is a generalization of coherent theory. So coherent theory is the same theory where the free states are orthogonal. And in a sense, this allows us from a conceptual point of view to study the importance of orthogonality versus linear independence in such theories. Well, now for a precise example where we can use the theory is in quantum optical non-classicality. So if we have a finite number of global coherent states, then we can use the theory to study the non-classicality in their super positions because a finite set of optical coherent states is linear independent, but they are not orthogonal. And in this case, the faithful conversion operation can be done using a simple beam splitter where the second input port is in vacuum. So this would be a conversion operation which is really easy to implement in the lab. And maybe most important, this research theory here can be seen as a starting point for more general research theories and I will comment on this later. Okay, I hope I've motivated the work. Now we have defined the three operations. We are the three cross-operators and we would like to characterize them explicitly. In order to do this, we can use the method or the idea of retzy-broken states or vectors which you might know from unambiguous state discrimination. So these are just, since our set of free states is linear independent, we can always define another set of states which are orthogonal to all states but to one. So in 2D real space, we have our two free states and then we can always find states like this. So C1 orthogonal is orthogonal to C2 and this projection onto C1 is one. So they are not really states, but rather vectors because they are not normalized, but yeah. Now if we have these vectors at hand, we can write down the explicit form of the three cross-operators. So these are all cross-operators that have this form, where the fn of k, this is just a function that gives me an index and the Ck and they are complex valued coefficients. Now, another question which is very important if one deals with trace decreasing or trace non-preserving quantum operations is whether we can complete a map which is trace decreasing for free. So for example, if we have a map that we can decompose into three cross-operators but which is not trace-preserving, then we should naturally ask ourselves, well, if we want to implement it physically, we always have an operation which is in total trace-preserving and if we cannot implement the missing part for free, then we shouldn't really call it free because it's just then neglecting the part in which we do something non-free, which we cannot do in the lab. No, it's not valid for every resource theory. So in this theory here, we can whatever trace decreasing map, so it doesn't even have to be decomposable into three cross-operators, we can find three cross-operators that complete it for free. But in entanglement theory, that's not possible because there's the problem of uncompletable matrix product bases, so that's not possible in general. Here it is. And as I already said, that's important if you want to work with trace-decreasing operations, and it was also an open question in coherent theory, but since coherent theory is basically a special case of this theory, where your free states are orthogonal, then that's also valid there. Now, if we want to compare the superposition in different states, we want to put a number on them and say, well, if the number associated with one state is higher than the number associated with another state, we have more superposition. And as an other resource theory, we do this by so-called measures, so this is a function that maps quantum states to the positive real numbers, and it's a measure if it satisfies certain axioms, on which you can also argue, and I will present you one set of axioms, which is motivated physically, as it should always be. So the first axiom is that it should be faithful. So the measure should be zero on all free states, and it should be non-zero on non-free states. So this is really, we have separated our state space into two parts, and now we want basically our measure to represent the separation. Then the second property is that our, well, as we heard in entanglement theory, our measure should be monotonic under the free operations. So if we take a state, act with a free operation on it, then the amount of resource in the output should be at least, well, it shouldn't be, it should be larger or equal to the amount of resource in the input state. Now, the case of monotonicity under selective free measurements on average is basically we do a free operation, we do sub-selection according to the outcomes, and then on average, we shouldn't be allowed to use this method to increase the amount of resource we have. And the fourth condition, which is convexity, basically states if we mix two states, we do not want to use this, to be able to use this procedure to create resource states. Now, if we have properties three and four together, this implies property two, and there are useful measures or functions that only satisfy some of these properties, and for specific applications, they might be useful as well, so that's why I've kept this point two of monotonicity under the free operations. One measure that satisfies all these axioms is what we call the one measure of superposition, and if we have a quantum state, we can always decompose it into its free basis. Since they are linear independent, but form a basis, such a decomposition is always possible, and if we then add up the absolute value of all kinds of off-diagonal elements, then this is a function that satisfies all four axioms I've presented to you before. And the advantage of this measure is that it has a closed form, so there are other measures, but usually they include some minimization, some maximization, and since this is closed, we can calculate it for qubits explicitly, so if we have the free states, which are depicted by this line of the arrow, then the one measure is low in the blue area and high in the red area, so the further we go away from these free states, the more superposition we have. As an all-resource series, an important question is, if we have one-resource states and we use only the free operations, can we transform the state into another resource state or, in general, into another state? Can we do this terministically or if not, can we do it at least with a probability P and if yes, what's the highest probability to do so? Now, first question is, okay, when are probabilistic transformations possible at all? And the answer is in this classical rank, so the number of states we need to bring into a superposition of free states, and whenever the classical rank of the initial state is larger or equal than the classical rank of the final state, then at least a probabilistic transformation is possible and if not, then the probability is strictly zero. Now, if such a probabilistic transformation is possible, we would like to know, well, what's the highest probability P? And the answer to this question is given by the solution of a semi-definite program, so we need to maximize the probability under some constraints and the FNs here, these are cross operators, they are known and they depend on the initial and the final state and using such a description in terms of a semi-definite program is useful because we know how to attack those problems numerically and there are also loads of analytical results concerning the treatment of these problems. Now, for qubits, with the help of the semi-definite program, we can analyze the maximum probability of transforming one state to another one. Again, we have the free states depicted by this tiny line here and our initial state is denoted by this line here and here, so it's the same image just from different points of views and if we take this initial state, then in this color scheme here and the block sphere is encoded, the maximum probability of transforming it to, for example, this state, it's close to one, to this state here it's, the probability is, well, about one half and as you can already see, it's not, at least in the block sphere, it's not really easy to see first to which state we can transform it with certainty and then how we can estimate the probability of maximum transformation probability. If we have one state which we can transform with certainty into all other states, then it's the most useful state, so it's the state with maximum superposition because we can use it for all applications, for all protocols. And so it's often called a golden unit and in the case of qubits, such a state always exists, so if we have again our two pure free states here and here and their overlap A, then the state with maximum superposition is the one on the block sphere which is the furthest away from the set of free states. So this state here, we can use only free operations and transform it into all other qubit states. For higher dimensions, this isn't generally not a case. In the limit of coherent theory, such states exist, but here not. And now I want to give you a rough idea of how we can prove this because it basically uses everything I've talked about so far. So it exists out of two steps. So first we find certain candidates, so states which might in principle be maximally non, or states with maximum superposition and then we show that none of these states indeed can be used to create all other states with certainty. And to do this, we just build a count example in 3D. So we choose a basis, a classic basis or three free states and then we notice, well, if they want to be states with maximum superposition, they have to maximize all possible measures because they cannot increase another free operations. And so in addition, we've already seen that a classical rank can never increase. So we need states that have maximal classical rank and amongst these states, we want the ones which maximize the one measure of superposition. And if one chooses the basis with an asymmetry, one can do this relatively easily analytically and ends up only with a few candidate states. And then one just uses the semi-definite program to show that these candidate states, they cannot be transformed to certain other states with certainty. And this can be done analytically because for the semi-definite programs, we have the property of duality. So we can transform them into another problem, bound to solution of this problem and then we have also bound on the initial problem. Okay, now if we have this resource, what can we do with it? So does access to the resource really grant us with an additional power or with additional possibilities? And indeed, if we have access to the resource, we can overcome the restriction. So if we have a unitary, an arbitrary unitary that acts on qubits for simplicity, then, well, we have for every input state, we act on it like a unitary, a qubit unitary and this is basically equivalent or we can reproduce this operation using only a free operation, Psi, that acts on two qubits. And what it does, it acts on the target qubit on which we want to apply the unitary. And in addition, it acts on another qubit state which is in a maximum superposition state. And the output is separable and we have applied the unitary on our target qubit and we have basically destroyed the superposition in our auxiliary qubit. So we've used the resource, we have consumed it in order to overcome the restriction. And it's maybe important to emphasize that for a given unitary, this Psi here is independent of the input state. So it's really reproduced the operation and we do not only reproduce the action of the operation on a certain state. In addition, we can use superposition as a resource in decision tasks. So decision tasks are, as we heard yesterday, a part of quantum etiology and we can, well, basically describe it as a quantum game and something similar has been done in the case of coherence in this paper. So we have two players, Alice and Bob. Alice might or is allowed to send a quantum state to Bob who applies a quantum operation on the state. And this operation is a selective measurements with D plus one outputs. If the output is zero, then he just says, well, please give me another state. So he asks for a new turn. However, if the output is not zero, he returns the output state of his quantum operation to Alice. Now, the goal of Alice is to guess N. So she wants to find out, basically decide which output N Bob received. And in order to do this, she's allowed to do an arbitrary quantum operation on the state. Now, she has two possibilities. For example, she might do unambiguous state discrimination in order to find out which N Bob obtained. Well, if this fails, then she has the ability to just ask for a new turn. So she can hand in another state, a new state and try again. However, she can also make a guess. And if she's correct, she wins. If she's wrong, she loses. Now, if we use a certain quantum operation here, so a free quantum operation, then for N not equal zero, what the operation does is it basically imprints a phase which is dependent on N onto the free states. In the case of N equal to zero, this failed and Bob asks for a new turn. Now, if Alice has only access to free states here, then she cannot win the game in the limit of large D because the output row N is independent of N. So whatever free state Alice puts in, row N is the same for all N. And if, well, in the limit of large D, we have, and in addition, the probability of output N is the same for all N. So the best thing Alice can do is make a random guess. And if, well, if we have a high enough dimensional Hilbert space, the probability of making the right guess goes to zero. However, if she has access to certain superposition states, these output states are different. And in addition, they are linear independence. So she can do indeed an ambiguous state discrimination when it fails. So when she gets an indecisive answer, she just tries again. And if she gets an decisive answer, well, she can guess with certainty. So she can give the correct answer for sure and thus win the game. So access to the superposition states basically turns certain loss into certain win. Now let me come to my conclusions. So I hope I have convinced you about the relevance of this research theory. So we can use it to define the a bit vague definition of non-classicality using entanglement. Then it's a generalization of coherent theory. And most importantly, it's a step towards a quantification of optical non-classicality. So we've spoken a bit about the mathematical structures, for example, of the three maps and the completion of three maps. I've introduced to you the concept of superposition measures and we've spoken about superposition manipulation in the form of a semi-definite program. And we've investigated states with maximum superposition. And we also noticed that there are advantages in superposition in having access to superposition states. Now, of course, that's far from being completed the theory. So we haven't said anything about mixed state transformations. And what is also very interesting concept is the concept of approximate transformations. So we want to transform an initial state, not exactly to a target state, but only approximately to this target state. Also, we have only considered transformations in the asymptotic limit. No, we have not considered transformations in the asymptotic limit. So we have only focused on single copy transformations. So I have only one copy of the initial state and not access to other copies. Well, and all this state transformations, it would be very nice to find a closed condition when we can transform one pure state into another pure state with certainty. Because at the moment, that's still the solution of a semi-definite program which you have to solve. Well, as an other resource series, or especially in a resource series of coherence, we can combine this restriction with further restriction. So one can think about local superposition-free operations. So for example, if I have some specially separated labs, I should in a sense include the idea of spatial separation and of classical operations on the systems in the lab itself. And for mesoscopic or macroscopic systems, one would also like to include energy conservation, for example. But the most important next step in my opinion is that one works on generalizations of the series. So it's still very restricted because we have linear independence, we are in finer dimensional systems. And if one would like to quantify optical non-classicality, then one has to go to infinite dimensional Hilbert space. Probably one would like to choose for the free states the global coherent states because they are most classical. And they are continuous. So they are not only not linear independence, but they are also not countable, which makes the series more difficult. But if one manages this generalizations, I would say this is a step towards maybe a full description of optical non-classicality and that's what we are working on in the moment. So thanks for your attention and I guess we have time for a few quick questions.