 record button okay and so people I hope can see that recording is on um let's just wait I guess one more minute oh Luxembourg yes do you have any interactions with Massimiliano there um or his whole uh I know him we are uh I'm in the group of Adolfo del Campo so we are next door basically with Massimiliano yeah yeah I've had a lot of interactions with Massi over the years okay I'm visited Luxembourg two just before the pandemic I guess okay all right um let's get going then so hello everybody um um so this uh this is afternoon for you it's still way early for me um this time of the rotation of the planet um we will start with a presentation by Nicoleta Caraba up at University of Luxembourg and um I take it away on Nicoleta it's all yours okay thank you I share the screen let's see if everything goes sorry I have to I have to I have to uh I was is asking me to unlock some uh screen uh yeah your co-host so yes I think it's something with the um I'm sorry I think it's I think it's something with the with the with the yes with my computer um all of your computer okay sorry share the screen because I've never uh maybe the desktop let's see if the desk opens I know it's it's it's my yes it's my zoom um I need two preferences uh share screen um share desktop uh I don't know why it does not allow me okay I just ask support so let's see okay let's I will try again let's see the message that I sorry the open system preferences I open system preferences okay unfortunately the um system support at the ICTP is now left because I know I also sent yes I sent them the slides I don't know if there's have you given a zoom talk before no oh dear oh dear um um okay yeah um system what kind of a system are you on a Mac yes and um under system preferences you might need to give zoom access might under accessibility okay service service preferences if anybody considers themselves a Mac expert with zoom please chime in I am certainly not one so somebody wrote you may need to change the permissions and restart zoom uh under permissions in the accessibility you got to add zoom accessibility of zoom I have uh so it's in preferences access accessibility okay yes and then there you need to add zoom uh but here I am the system preferences of of zoom but I okay I should go on the one of the Mac oh yeah you go on the system and uh and zoom's preferences are not called systems preferences at all okay uh system preferences and and and then I I don't find the I don't find uh zoom between no yeah you need to add it you need to add it so it's a little plus button or at least my I did not upgrade to the most recent mac os and sorry but uh I cannot see how to add it uh should be a little plus button I think you should add it in the security and uh privacy section of uh system preferences okay yes let's see and uh and here I don't see there is no zoom here is there an add button a plus button in the bottom that should be yeah exactly no there is there is about location and service preferences okay but there is no add the at least I don't see a little plus a little plus yes I don't see any plus I think too much okay there is a lock maybe okay yeah yeah you need to do the lock to be able to get into change anything unlock then use your password yes but okay it's still in enable locations I don't have zoom listed I have Microsoft Teams I have yeah yeah it's not it's not location it's go to a screen recording uh yeah uh yeah unlock it again you should push the unlock button green recording and okay you need to unlock the unlock button in the bottom okay okay I think I made it and I have to leave and join again I think okay okay thanks for sure of course I'm hoping that in GPT 6.5 will actually be able to help us do audio visual conferences successfully that's how we'll know which is a true super 8 super human intelligence AI we've got her screen now we just need her in your zoom make sure that you've got your voice and image turned on we cannot see or hear you sorry sorry so now go to share screen that was working yes yes here okay and you should also see me but yes we do you're a person off to the side okay don't mind when the Nobel committee is deciding whether to give you the prize or not they won't really pay too much attention to this particular problem okay I'm really sorry I'm really sorry for this it's the first time that I share my screen on zoom and okay and okay so um the floor is yours we've eaten up a bit of time so let's actually try to keep it on schedule would it be okay for you if yes yes it would be okay maybe maybe um I will uh yes maybe I will leave five minutes for questions something like this would be fine okay okay well I will try to be as quick as possible maybe skipping something and so um good afternoon everyone I just would like to thank the organizer for giving me the opportunity to give this talk instead of my supervisor Adolfo del Campo who unfortunately could not attend I'm Nicoletta Carrabba his PhD student and this is our group at the University of Luxembourg and I will be presenting some recent work made with Adolfo that tries to answer these general questions so first what is the fundamental time scale of a physical process and this question can be addressed rigorously in the framework of quantum speed limit that will be the main object of this of this presentation and then also how complex is a given quantum evolution and in particular how do we define complexity there are several approaches to quantum complexity one could for example look at the evolution of quantum states but here we would rather focus on the evolution of observables and this this approach is um is known as operator growth so this is the outline of the presentation uh first I will present our two first quantum speed limit for operators and discuss applications to correlation functions, dynamical susceptibilities and feature information and then I will present a different kind of limit that is called a dispersion bound on on krill of complexity which is a measure of operator growth and I will define later and finally we'll present an additional quantum speed limit for operators uh that is um that has a geometric interpretation and I will also make a comparison with the dispersion bound on krill of complexity so first a brief introduction to quantum speed limits they um they were first introduced by Mandelstam and Tham in 1945 and uh first for orthogonal evolution so a quantum speed limit is a lower bound on the time needed for for an evolution to occur from a given initial state to a final state and by giving this bound one provides also a new understanding of the energy time uncertainty principle so according to this view the this delta t involved in the uncertainty principle rather than being an uncertainty over a measurement of time is really the interval of time uh in which the this evolution occurs and uh this this time is bounded in terms of the energy fluctuations so this is the Mandelstam-Tham quantum speed limit for arbitrary angles between the initial and final state a second kind of uh quantum speed limit was introduced by Margules and Levitin and instead of at a time scale of the dynamics instead of the energy fluctuation it it it contains the energy mean now in our first work we have extended this this results to the evolution of operators in the eithemer picture and these are our two results and in particular the role of of the Hamiltonian here is taken by the Ljubljana that is defined as the commutator with the Hamiltonian and generates the evolution of the operators here we use a formalism in which we vectorize operators and the inner product between the two vectorized operators is given by the Hilbert Schmidt inner product so here we see that we have two different characteristic time scales identified by variance of the Ljubljana and the mean and so we refer to them as Mandelstam-Tham and Margules Levitin type of quantum speed limit for operators and these two results can be applied to the study of autocorrelation functions of a so two-point correlation function of an operator O and in particular these two results that I presented identify a crossover in the decay of the symmetric part of the autocorrelation function so here we illustrate this crossover for a two-level system and random matrix so in a first in a first regime we have a quadratic decay well captured by the Mandelstam-Tham type of bound and subsequently after after a crossover time there is a time window in which the decay becomes linear in time and it's better described by the Margules Levitin type of quantum speed limit and finally in the lower panel you can see also some results on the on the initial onset of the imaginary part of the autocorrelation function which is zero at time equal to zero and it's the anti-symmetric contribution to the autocorrelation function and this can be bounded in this case upper bounded by Margules Levitin type of bound and this is particularly useful because it allows also to connect with the dynamical susceptibilities in fact this correlation function enters the linear response of a system under a perturbation so in this setting an initial Hamiltonian is perturbed with a time-dependent driving and to characterize the response of the system one can look at the shift in the expectation value caused by the perturbation and at the linear order in the perturbation this is this can be expressed in terms of the so-called dynamical susceptibility which is proportional to the anti-symmetric correlation function between the observable of interest and the perturbation operator computed at equilibrium we have provided three bounds on this quantity the first two the shant from a standard operator inequality such as the isomeric uncertainty relation and the bugolub of inequality when rho is a thermal dip state and but also in the case in which a is equal to v so we studied the response in the perturbation operator itself we can also apply the quantum speed limit approach to derive an additional bound and the important difference is that now we have a non-trivial dependence on the time because here you can see that the bound behave like a theta function while in the quantum speed limit bound we have a linear time dependence that expresses the fact that the response occurs with a certain delay with respect to the perturbation and also makes makes these results tighter at early times so this this bound we have computed them in two different systems to give an example a system of particles in an electric field that is externally perturbed and also a spin system in a magnetic field so these are two typical examples of quantum linear response theory and the corresponding susceptibilities the electrical conductivity and the magnetic susceptibility can be constrained by by these bounds right that I shown also these bounds can be used to give a constraint on fish information fish information quantifies the state distinguish distinguishability under the transformation generated by a certain observable o and it is important in meteorology because it constraints the precision with which the parameter theta of the of this sort of this transformation can be measured and this constraint is the quantum version of the Kramel Rau bound and here M is the the number of independent measurements now this important quantity fish information was found to be related but to them to the Fourier transform of the of the dynamical susceptibility of the corresponding operator O in a paper by out in 2016 so we can use our bounds to try to provide a constraint on the efficient information at that one is when the the system is at thermal equilibrium but after performing the integral over omega we find that this this this integral has a singular behavior near time equal to zero and and so in particular this function diverges like one over T and therefore the Isenberg and Boguliubov bounds on on the antisymmetric correlation function provide a divergent result while the quantum speed limit approach by introducing linear time dependence remove removes this divergence and so we we obtain a finite constraint on the thermal fish information in terms of the temperature of the system and the this characteristic margulis levitin in time scale so this was the first part now let me move on to this second part of the presentation where I will discuss operator growth in kilo space and a bound on complexity now again we have in mind the evolution of unobservable in the Isenberg picture and by formally solving the Isenberg equation one can realize that the the the evolution of the operator can be expanded in terms of the powers of the new billion that are the nested commutators with the Hamiltonian so the the span of this set of of nested commutators is what we call the krill of space so it's the minimal subspace that contains the full unitary evolution of the operator all now in general from from this infinite set one can extract an orthonormal basis which can also be finite and typically it is finite in a finite system through a procedure that is called a lancso's algorithm and it's a it's a generalization of ground-smith procedure so with this algorithm we we can extract an orthonormal basis and the important property of this procedure is that at each step one generates the next most important vectors only by using the two preceding elements of the basis then one have to normalize the new vector and the normalization factors that one obtains are known as lancso's coefficients and are important to characterize the the complexity of the evolution so after defining this krill of basis what we discover is that the operator growth has been mapped to a hopping problem on a one-dimensional chain where each site of the chain corresponds to one element of the krill of basis the effective Hamiltonian of this problem is the the luvilla and takes this characteristic tree diagonal form where we have the lancso's coefficients on the upper and lower diagonals and therefore they play the role of hopping a parameter in this on this between adhesion sites so from here you can see that the krill of basis is actually ordered in in complexity in the sense that by by acting with the more and more commutator with the Hamiltonian one increases the the complexity of the initial operator for example if you think about the lattice you increase the size of the operator by acting with a commutator with an Hamiltonian therefore what what we have is that the farther we we go from from the origin the the higher the complexity in some sense intuitively and so one defines krill of complexity as the the mean position of the operator over this chain and so the formally is the expectation value with respect to the evolving operator operator evolving observable of a super operator in krill of space that that is the position in krill of on this krill of chain now this definition was proposed by in this work by Parker in 2019 and then many many many works on krill of complexity came out also motivated by by the hope that krill of complexity could provide a new understanding of quantum chaos thanks to this conjecture presented in this paper that in maximally chaotic system that we have a maximal growth of complexity and i will comment on this later now another work that is important for what follows is this work by caputa and others geometry of krill of complexity that contains the observation that this this luvillian can be written as a as a sum of larger operators on the krill of chain so we have a raising operator that has zero on this on the lower part and and a lower lowering operator and they respectively allow to move farther or backwards in the krill of chain and this reminds a lot about the representation group representation in standard quantum mechanics such as the harmonic oscillator so one can wonder when in which circumstances these latter operators form an algebra that is called complexity algebra what happens if this operator closed an algebra is that the krill of basis is it becomes a representation of the complexity algebra and the the evolving the evolving observer follows the trajectory of a generalized coherent state of the algebra because the the unitary time evolution is formally equivalent to the displacement operator that generates coherent states and coherent states are typically known to to satisfy minimum uncertainty relation and we we we kind of confirm this intuition by formulating bound on the complexity rate this that we call the dispersion bound that comes from arises from nice america robertson uncertainty relation so from this uncertainty relation one can establish this upper bound on the rate of growth in terms of the variance of the krill of complexity and the variance of the luvillian which reduces to this b1 and what we have approved is that in the case of complexity algebra so when there is this coherent state evolution this bound is saturated so complexity grows at the maximal speed limit instead in the absence of this of this symmetric structure of krill of space we observe an deviation from from this speed limit in particular in numerical simulations with random matrix theory so this is this is the explicit form of the algebra the generators of the algebra are so that the luvillian this operator b and the commutator between the two of them and so basically we have we have a this is the possible closure of the algebra and then the explicit form depend on on the sign of this parameter alpha and so we have three different scenarios of saturation of the speed limit and i just want to observe that for positive alpha the complexity algebra reduces to the one of the special linear group and one can compute that complexity grows exponentially and this is the case of the maximal chaotic systems of this conjecture while in the case of negative alpha we have a periodic oscillation of complexity this is because the krill of lattice is finite in this case so we cannot have an indefinite growth and and i just want to to mention before moving on that this complexity algebra can be realized even by integrable systems then for example by by acrobit so this is to to say that in order to saturate this speed limit on complexity we don't we don't need we don't need the the presence of chaos so now let me move on to the final part of the presentation where i would like to present this geometric quantum speed limit for operators so the the geometric argument that allows to derive this speed limit is similar to the one used for standard quantum speed limits and basically the operator since the the evolution is unitary it lies on the the trajectory lies on the surface of a sphere in operator space and therefore one one can see that the length of the trajectory connecting the initial and final operator must be greater or equal to the geodesic curve that connects the two points and on the sphere the geodesic is given by the the great circle and so by rearranging this inequality one gets a quantum speed limit a geometric quantum speed limit on the operator revolution and this this quantum speed limit has at the numerator the geodesic and at the denominator the average velocity of the operator flow which is defined in terms of the variance of the uvlian so we can we can talk about a man dressed on type of a quantum speed limit and we need a finite dimensional eberspace to apply this argument so now let me discuss a simple example of saturation that is the case of a two-level operator interpolating between a two energy level of our time independent ameltonian so in this case one can compute everything exactly and one discovers that this inequality reduces to an identity at every time so the the operator quantum speed limit is identically saturated and this example contains the intuition that the operator dynamics is geodesic so it's saturated the quantum speed limit when confined to two energy spaces and this can be shown rigorously and we have we have shown it we have proved it in this work that after that if the operator has support in only two energy spaces after subtracting the stationary component that I will now define we get a saturation so the stationary component is the the component of the operator along the zero eigen space of the uvlian because what happens is that the operator this trajectory is contained in three uvlian spaces corresponding to eigen value plus and minus omega where omega is the energy cap of the two energy levels and zero is yes the zero eigen value now the the component along this the kernel of the uvlian is not evolving and so it but it does not contribute dynamically to the flow and actually it prevents from following the geodesic but after removing this this component we get saturation so I think I will skip this example here I was giving an example of saturation of of this quantum speed limit by by an operator about by an a miltonian flow so a flow of of a miltonians and one can construct an explicit solution to the flow that satisfies them the the quantum speed limit so this was to give an example beyond the the time independent the case of time independent generator but just so you know nicolata you got 15 minutes so if you want to okay some of this you have the time to right now there's only one question pending in the chat so up to you okay maybe I will try just yes to to mention quickly yes that one one one relevant application is given by the case of a miltonian flows for example the one proposed by wegner in 1994 as a method of diagonalization of of the of the miltonian so in this in this case we are looking at the operator that is undergoing the unitary flow is the miltonian itself of the system and the parameter is not time it's a it's an arbitrary parameter that we call L and at the end what one can define a unitary flow where at the end of the flow the miltonian is diagonalized so the off diagonal component is unitary suppressed them and an example of of this of this flow is given by this choice of the generator which is different from from the one proposed initially proposed by wegner but this is the one that we studied and here we have with this choice of the generator we have that by taking for example the miltonian of an x y spin model the couplings of the model obey total equations and we have used these equations to construct an explicit solution of the flow that saturates the the quantum speed limit so the evolution of the miltonian is is geodesic and and this this of course with by choosing specifically the initial conditions of the miltonian and this is this so provides an example in which the the parameter of sorry then the generator of the flow explicitly depends on the on the parameter but still we can we can reach saturation so as as a last application of the quantum speed limit i would like to consider again krill of complexity so in order to apply the the the operator quantum speed limit to krill of complexity we need to switch again representation so before i define krill of complexity as the expectation value with a krill of with of a krill of complexity super operator with respect to the evolving observable now i switch representation and i keep the observable fix while evolving only the the the super operator and we call this super isomer picture and the unitary flow of this operator in flow space is generated by the commutator with the uvlion now one can apply the operator quantum speed limit to this unitary flow and the main message here is that in the case of complexity algebra so just to we have these commutation relations what happens is that the dynamics is extremely simplified by the presence of these commutation relations because every time they take a commutator with the uvlion i can apply the commutation relations and so the the infinite set of the nested commutator with the uvlion reduces to only three super operator the identity the krill of the initial krill of complexity and that this operator be that generates the complexity algebra so this means that the evolution is contained into a three-dimensional space and since the identity is among these these operators there is a stationary component and this means that after subtracting the stationary component we must obtain saturation we must obtain a geodesic flow and this we checked it explicitly first by computing the the quantum speed limit without subtracting any stationary component so here are the details of the computations so here you can see explicitly that these nested commutators these powers of the generator s reduces only to three super operators and then in the case of the s u2 complexity algebra which is the the only final dimensional case we can compute the the quantum speed limit explicitly and what we find is that as i anticipated there is a there is a deviation from the from the geodesic trajectory from the quantum speed limit so this is the autocorrelation function here and it's compared to a geodesic trajectory and here is the you can see the deviation from the quantum speed limit and this is due to the presence of a stationary component which is necessarily non-zero because because the trace is non-zero so there is a non-zero component along the identity so in principle there could be also more stationary component but what we find is that by subtracting the component along the identity we actually obtain a fine i'm sorry a geodesic trajectory so we obtain saturation of the quantum speed limit and this means that the stationary component is is minimally in some sense it's given by the component along the identity and it confirms the the the argument that i gave at the beginning that the the evolution is geodesic so both the quantum speed limit and the dispersion bound are saturated at the same time so now here are the conclusions i i presented two different kinds of operator quantum speed limits of a Mandelstam-Tamm and Margulis Levitin type respectively that identify a crossover in the initial decay of symmetric autocorrelation functions and thanks to the dispensable limit approach i also provided bounds on dynamical susceptibilities and fissure information and finally i discussed operator growth in kilo space and the saturation of the dispersion bound on the on the rate of complexity in the case of complexity algebra and and also i discussed this additional quantum speed limit for operators with a geometric interpretation and i discussed the equivalence between the saturation of this quantum speed limit and the saturation of the dispersion bound on a kilo of complexity thank you for your attention thank you very much we have a little bit of time left thanks for the dealing with all the technical problems especially Nicoleta um so the main thing we have so far is in the chat Vladimir Vyegas um are you if you're still there Vladimir um could you uh yeah my question is that uh did i get it right the the upper limit of or rather the limit of these things are due to a certain for something that is parallel to the uncertainty principle and i getting it right for example the conductivity that's why it has a limit because it follows something like uncertainty principle yes because uh yes yes yes yes actually uh the among for example on the transport coefficients yes among the the three bounds one was really coming from uncertainty principle because um i didn't try the uncertainty principle actually but here you have this the expectation value of of this commutator and this and this is bounded by by the variance of the two operators and since they generalize uncertainty principle so if you take position and momentum you you get the standard uncertainty principle but so you can apply a certain principle to to to bound correlation out correlation functions between operators uh if that is then well for example if position is the momentum then what is like the pair of the conductivity um so here uh conductivity is defined for for a specific choice of the operators so here in this example for example the external perturbation couples with the dipole moment uh this r and and the observable of interest so what we call here a is is the car the current and for this choice so for this choice of operators the dynamical susceptibility is is is what we define the electrical conductivity so uh so i i don't know if this clarifies the the issue yeah i got that thank you thank you thank you for the question okay next up um oh god you've got your hand up yes uh thank you nicole for the great talk um so i have a sort of naive question you were introducing these two the first two speed limits and there is these original ones that are if i understand correctly one is like sort of in a schrodinger picture and one is sort of in the heisenberg picture so are they equivalent can you go from one to the other or are they like sort of different bounds and they they are not equivalent uh but they um sorry what it is so um one can see that for example if you if you as an operator if you choose a projector of a of a pure state then actually you can recast this this quantum speed limit in the schrodinger picture and um but you find something that is proportional to the original bounds but it's not exactly the same so they are they are not they are not they are not equivalent but uh since um we you you you then you get um you get this the same time scale so you you get the madres sometime and margulis levitt in timescale and when you get um to a quantum speed limit that is proportional to the to the standard one we we called it a generalized we call we call this generalization of madres sometime and margulis levitt in cooperators but it's true that they're not equivalent so have you compared the tightness of each of them in the case that they in when they can be compared yeah so they they are not tight overstates uh since up and they are they are proportional but with um with a smaller than one coefficient and uh yeah so one of the issue with the first work it was really tightness which is kind of solved with the geometric approach uh typically the this bounds gives give a tight description uh at a small times but but not but um as far as we know not not later and so this is also one of the motivation for looking at a different kind of of quantum speed limit so yes this typically described early time dynamics beautiful thank you thank you thank you for the question yeah i've got then if there are no others a quick question we've only got a couple of minutes um obviously you've not yet had time to think about it but i was curious whether any of these kinds of um extension slash modifications of the bottom speed limits if you have any intuition about how that might be relevant for the kinds of speed limits for information processing that were discussed earlier today which is all about changing sets of qubits from one particular state to another one formalizing that as information um processing um is uh anything occur to you or just this is something no well i i don't i don't think i have um i would precise answer yes of course yes i think that uh in general quantum information processing is a as an is an important application of quantum speed limit so um especially especially maybe the the the last one the the one that is tight uh that is can be commissioned to be tight so the geometric quantum speed limit maybe could could say something interesting uh if if one could formulate i'm not sure how one could formally uh these these processes in terms of um operator growth rather than states evolution of states but but uh but i i think that it could be of course an important application but we i i i don't have anything more to say on this okay great um it looks like there are no other questions so thank you very much i'm especially given the technical challenges um putting on the spot yes i'm sorry for the uh for the you have this so i i i stopped the sharing okay yeah and then um uh eric