 In so hr, da se vse, zelo bi prejel vsočit, mihrožitev za vzgledanje in vzgledanje, vzgledaj za vzgledanje in vzgledanje, vzgledaj za vzgledanje, vzgledaj za vzgledanje. In tako, sem tko, da je z radiobaz, vzgledaj za domenje, vzgledaj za domenje, za tudi, kako jaz radiobaz, za tudi, kako jaz radiobaz. In tako, da je zdaj, Zigen. So now I show you where I work. So here I work in Bilbao, which is in the bus country, so in north of North Spain. And these pictures are from Bilbao, and this is from the sea next. So why is this multipartite entanglement topic is important? Because full tomography is not possible. So you have to do something else to qualitatively karakterize the quantum state. Of course, you can say entangled, but this is not sufficient because for a many particle state it does not give enough information, so you should say something about how entangled the state is and what it is good for. So can you hear me? Okay. So let's see first what entanglement is. So simply a state is separable if it can be written as the mixture of product states. And if a state cannot be written like this, then it is entangled. So the definition is very simple, but again the problem is that if you have one million particles and two of them are entangled, then the entire one million particle state is entangled. So we have to find some other definitions. And here a k-producible state is the following. It's a state which is a tens of product of at most k-qubit units. And the mixed k-producible state is simply a mixture of pure k-producible states. And if a state cannot be written like this, then we say it's k plus one particle entangled. So for example, this state is two-producible because you can put it together from two-qubit units, then don't do anything with each other. And here this is k-producible because you can put it together from units that three-qubit units that don't interact with each other. So, and now we can also think on what can be measured in a many particle system. We cannot just measure anything because we cannot access the particles individually. So what we can measure is the collective angular momentum components. So that we can measure their expectation values and we can measure their variances. Basically these are the typical quantities. You can measure in an ensemble of one million particles of spin half atoms. And so in this context, they developed this so-called spin squeezing criterion that detects entanglement in a very simple way. So there is the spin squeezing parameter defined with these operators. And the statement is that for every separable state it is equal or larger than one. And if it is smaller than one, then the state is entangled. So what kind of states give you a value smaller than one? Of course a state for which this is large and this is small. So it means that the state has to have a large spin polarization in one direction. And in an orthogonal direction, it has to have a small variance. And later it will be clear why is it good for metrology. Generalized spin squeezing criteria have also been developed. So for certain cases, actually we have even a full set. So if you have only these input parameters in a multi-particle system, then there is a criterion that detects all entanglement that you can detect based on these. So this is so far just entanglement. So we still have this question that if you have one million particles and you say entangled, what does it mean? But you can also manage to detect multi-partite entanglement, which also tell you a number, let's say five particle entangled. That is the work of Sorensen and Maulmer, which detects such entanglement close to spin squeeze states. So the states that I mentioned is highly polarized states, which have a small variance in some direction. And there are also other criteria for unpolarized states, which are actually sort of, they contain the first criterion as a special case. And there is no more as far as I know. So with collective observables, these are the only criteria that can tell you 100-particle entangled or 50-particle entangled. So this is now a problem that we would like to solve. So far this was the summary of my, so my summary about spin squeezing and entanglement. And now I would like to talk about metrology. So why is this metrology interesting? Because, so first, we would like to detect multi-partite entanglement that is surely good for something. So it's not only mathematically multi-partite entanglement, but it is useful for some metrological procedure. And the second point, and we would like to have a method that does not have to be, let's say, tailored or made for a certain state, but we can detect entanglement around various states. And we just plug in certain data, and then we get a method that can detect entanglement in various experiments. So now I will attempt to show such a method. But let's see first what is quantum metrology. So fundamental task in metrology is that you have a unitary dynamics where a density matrix enters, it suffers a certain unitary with a parameter theta, and then you get this rho theta as the output. And what you would like to do is you would like to estimate this theta based on measurements on the rho theta. And typically, they take not just any dynamics, but when the Hamiltonian operator is the angular momentum component, because then this is just a sum of single spin terms. So it does not contain an interaction, and it's much easier to create such Hamiltonians in a controlled way than creating interactions in a controlled way. So now let's see that, so how can I compute the variance of the parameter estimation? Let us imagine that at the output, so here we measure m, we measure the operator m. And we would like to, or the expectation value of operator m, and we would like to estimate theta based on this. Then the variance of this estimation depends on how big is the variance of m and how much the expectation value of m depends on theta. Of course, if it depends on theta very much, then the variance will be small. On the other hand, if m has a large variance, so the operator that would tell us something about this parameter has a large variance, then the variance of theta will be small, that will be big. So this is just a qualitative explanation, but this is a standard expression called the error propagation formula. So now there is a statement that the parameter estimation variance is always bigger than one over the quantum fissure information, which is just a quantity that depends on the density matrix and the operator a, in our case jl. And this is just a formula. So it does not look so nice, but simply is the function of rho and a, and what it does is just tells you a lower bound on the parameter estimation. So you notice that here, this variance depended on m, depended on n, so the operator what we measured. And here you don't see an m on the right hand side because this is a lower bound for any measurement. So it sort of tells you what is the best precision you can achieve if you are able to choose any m. So now there is a statement that for separable states, this quantum fissure information is smaller than n. So it's a very interesting finding because it's a general relation between the quantum fissure information and separability. So for any state where you find that the quantum fissure information is larger than n, then you know that the state is entangled. So this relates the metrogika usefulness of the state to entanglement. And there is another related finding that actually if you have k-particle entanglement only, then this thing is smaller than kn. Of course, n-particle entanglement would correspond to n square, which is the maximum of this quantum fissure information. So it means then if you measure this quantum fissure information, then you can get indications of five-particle, ten-particle entanglement quite straightforwardly. And n is the number of particles, and this is for spin-half. So this is for spin-half particles, and this is a component of the collective angular momentum. So you can easily change it to q-dits, but this is just q-bits, but then there is an additional constant if it is q-dits. And since I said that the quantum fissure information gives you a bound on this parameter estimation variance, you can have similar relations for the parameter estimation variance, so for separable states, this holds, and for states with a k-particle entanglement, this holds, and state with n-particle entanglement, so basically any quantum state holds that variance of theta is bigger than 1 over n square. And in metrology, one would call this a short noise limit, and one would call this the Heisenberg limit, and what is interesting that these things can be derived in this framework in two lines, and before there was a lot of literature on how to show these bounds in various setups. And the idea is that any state that violates this has k-plus one-particle entanglement, so there is an easy way to detect multipartite entanglement, we just look how well we estimate a parameter, and if we estimate it better than a bound, then we know that the state is entangled or multipartite entangled. So now I go to the next idea, so what does it mean now witnessing metrological usefulness? So one way to detect entanglement or metrologically useful entanglement is what I explained, that we just measured the variance of theta, and if it is, we know that the quantum fissure information is larger than this quantity, and so if it is smaller than a constant, then we know that the quantum fissure is larger than this. So the idea is we just make the metrological process, we look at how well we can estimate the parameter theta, and from this we can get bounds for the quantum fissure information. And so this has been done in several experiments, and what is the difficulty in such experiments that you have to produce the dynamics. So you produce the quantum state, you produce the dynamics, and afterwards you measure something in order to estimate this quantum fissure information, and also one has to remember that the dynamics might be noisy in this way you underestimate the quantum fissure information. And what we would like to do is something else, because we would like to estimate how would the precision work if we did the metrological process, and we assume perfect metrological process, and this way the dynamics does not play a role, we just characterize the state only. So the point is that we use the same ideas with entanglement witnesses that we measure several things, and then we say the quantum fissure information is at least, I don't know, five, but we don't carry out the dynamics. So let's see. Now I show you a simple example of this. So again about the spin-squeeze states, what I have mentioned. So you see this is the spin-squeeze state, it has a big spin in one direction, and the uncertainty ellipse is deformed for a fully polarized state, it would be a circle or at least a symmetric object, but now in one direction it is squeezed, and in the other direction actually it's increased, and it's clear that as a clock arm you can use it to estimate rotation angles. So if you rotate this spin around this axis, then if you measure JZ, expectation value, then you can estimate the theta angle very well and better than with a fully polarized state. So as I said, there are several experiments with this in quadrasses, and you can even obtain the variance of the parameter estimation with collective quantities, so you don't have to make the estimation, you just measure this quantity, this quantity, and then you get the parameter estimation variance, and from this if you reach various levels you can deduce that the state was two-party entangled, three-party entangled, four-party entangled, you don't have to make the metrology itself. So, and now there is another example with other type of states, so these are now not like the spin-squeeze states, fully polarized, these states don't have any polarization, which means that the Jx, Jy, Jz spin expectation values are zero. On the other hand, they are very interesting because in two directions the variance is large, the spin variance is large, and the third direction is zero. So it's like a pancake. You have a pancake, and then you rotate this pancake of uncertainties, for example, with these dynamics, and you try to estimate the theta parameter, and now you have to measure Jz square, not Jz, because Jz expectation value is zero. As I said, it does not have a spin. You see that this little dot indicates that Jz, Jy, Jx expectation value is zero, and one can again get a closed formula for the variance of the parameter estimation. So you measure these quantities, these are expectation values of the initial state, and then you get the optimal parameter estimation variance without making the metrology. So, this was also tested on experimental data, and you see that these were two examples, where for each setup, somehow a certain formula had to be derived, but if there is another setup, then one needs to look for another formula, so one would like to find a better method or a more general method, and a method that estimates the quantum fissure information rather than the parameter estimation variance. So, what does it mean that we would assume that we can measure anything at the output, on the output state, we can measure anything, and then we would like to know what is the best parameter estimation variance we can achieve. So, in one word, we would like to know how would the state is for quantum metrology, we allow for any operator to be measured, and essentially we want to estimate this quantum fissure information. So, I mentioned that there are works about related results in the literature. So, in the literature there are similar results, where they calculate the quantum fissure information based on measurements for systems in thermal equilibrium, and they need for this so-called dynamic susceptibility, and you see that they need this as a function of a parameter, when afterwards they calculate this integral and they get the quantum fissure information. So, now we would like to have results not for thermal equilibrium, and also we would like to have a method that can estimate the quantum fissure information based on few measurements, so not let's say a continuous measurement or measurement of a function of a continuous real variable. So, for this we use this characteristic of the quantum fissure information, which has been found recently, that it is the convex roof of the variance. So, it's a surprising thing, because convex roofs appear typically in quantum information, and they don't appear in other areas of physics, but it turns out that the quantum fissure information is the convex roof of the variance, where this optimization goes over all possible the compositions of the density matrix. And so, because of that one can try to use similar techniques to estimate it as has been used, as have been used for estimating the entanglement of formation. So, what is the idea? We want to find the lower bound of a function g rho based on the expectation value W of an operator upper case W. And this linear bound typically looks like this, that you have R, which gives you the slope, times W minus a constant. And the constant takes care that the line is always below the function itself. And of course, for every R you would have a different constant. And the textbooks say that actually this constant here is just the so-called legend transform, this quantity, which is obtained as an optimization over density matrices. And then, of course, the best linear lower bound comes from an optimization over the slope R. So, simply you have to calculate this in order to get the particular function of rho based on that we know the expectation value of this upper case W operator. And it turns out that if we have a quantity that is defined as a convex roof, then we need here only an optimization over a pure state. So, you remember that here it was an optimization over a density matrix. It's quite complicated, typically. Now it's much simpler. It's an optimization over a pure state. And this is quite similar to the simplification they found for entanglement measures. But now we can have an even better formula. So this would have been the formula that I mentioned for our case, optimization over a pure state. And after some algebra, it turns out that it's enough to optimize over a single real parameter in order to get this legend transform. So it's very good, because imagine instead of all the elements of a pure state, you have a single real parameter. And because of that, somehow this quantum fish and information looks like optimal or ideal quantity for this legend transform technique. So these are the results. So imagine that you measure an operator and you want to give a lower bound on the quantum fish and information. So, for example, you measure the fidelity with respect to gs state. So essentially you measure the expectation value of the projector to the gs state. And here you get a lower bound on the quantum fish and information normalized by n square. And so if you reach a fidelity larger than 1 half, then it is non-zero and goes up. And it's obtained analytically. So it's very interesting that we could even obtain this analytically. And the similar thing we found for dike states is just another type of highly entangled quantum states that I have already mentioned. So these are the symmetric superposition of all the states where half of the atoms are in zero and half of the atoms are in the one state. And for various n's now they don't collapse even after I normalize dn square. So, why is it good? Because imagine you measure the fidelity and immediately you know what is the metrological usefulness of the state. So you don't have to carry out the metrology. You can imagine what would happen if you carried out. And before making the metrology, you can, for example, prepare a state which is good enough, or you can just use that you know the quantum fissure information to characterize the state. And these fidelity values are known in experiments. You can look them up and calculate what would be the corresponding quantum fissure information. So this we have done for many experiments. So these were small systems with a couple of qubits, ten qubits. But you can now consider very many particle systems and for them this inequality gives a lower bound on the quantum fissure information. I have shown this already. So again, the idea is that you measure this. This is an expectation value. You measure this variance. This is a variance of a collective quantity. These are straightforward to measure and you get a lower bound on how good the state is for metrology. Now we can do this with our method. So again, we measure this jz and the variance of the jx squared. And our method for every point in this figure, for every physical value for these two parameters give you a lower bound. And so this is the scale of the quantum fissure information. And the interesting states here, this is the fully polarized state when the spin is maximal, you see. Then the spins quiz states are below because then you decrease the variance. It becomes smaller. You see it's very red because simply the quantum fissure information increases. This is the fully mixed state. The quantum fissure information is zero because it's useless for metrology and this is the decay state. So our method just gives a result. You plug in these two operators, push the button and then it gives you the values of the quantum fissure information corresponding to those measured expectation values variances. And what is interesting is that in the lower part we get back exactly this Tadze-Smerzi bound. It's very nice because actually it shows that we probably did not make a mistake in the algorithm. Second, it shows that their method is optimal. And we can even add further operators to these two. So for example, if you assume that we know jx to the force, so this other moment, which is a reasonable thing in an experiment, then the original bound, which is here, can be improved. So you see that you get a better bound and we can also assume bosonic symmetry, then you get again a better bound. So it means that you can add as many operators as you want or various characteristics of the state and it will always give you a lower bound on the quantum fissure information. And this is now an experiment with 2,300 atoms. So for this speed squeezing parameter is 0.15. You remember that if it's smaller than one, then the state is entangled and sort of the smaller it is, the better the state is squeezed. The Tadze-Smerzi bound gives you then a quantum fissure information corresponding to this value. And our method gives the same, which is, again, shows that our method is probably correct, shows that in this case, the Tadze-Smerzi bound is optimal, but it also shows that we can calculate sinks for systems of this size and we could also add further parameters and get a better bound. So we don't have to measure only those two sinks that appear here. So we can instead of this and this, of course we can measure, for example, this. And we made similar calculations for the decay states, decay state experiments. So I just mentioned before I finish one of our ongoing works. So we are interested in other bounds for the quantum fissure information and for pure states, the quantum fissure information over four equals the variance. So it's a mathematical property of the quantum fissure information. And for mixed states, there is a difference. So this is always bigger than the other. And we found the upper bound on the difference, which depends on the linear entropy. So essentially the purity of the state. You can see that if the state is pure, then this will be zero. So the two sinks have to be equal, but if it's not pure, then there is a difference. And for example, this makes it possible to estimate the quantum fissure information if you have an idea on the purity and you measure the variance. So I would summarize my talk. So we discussed the very flexible method to detect multi-partite entanglement and meteorological usefulness. The method is general because you can choose any set of operators and the method gives you an optimal lower bound on the quantum fissure information. And these two papers contain mostly what I have been talking about. So I was mostly concentrating actually on the second one. This is about the quantum fissure information and bonding it from below. And the other was the special case of the Dicke state I also mentioned. So thank you very much.