 Okay, so we will resume now. We start our afternoon session with Dr. Marcello Dalmonte, who is actually at ICTP, and so it's my pleasure to introduce him to you, to the audience. Thank you, Marcello. Good afternoon, everybody. Can you hear me? Not too loud? Yes, okay. Otherwise, switch off the microphone. Okay, so let me start by thank you for coming and thank you the organizer, Fabrizio, Sarro, Martin, and all of them for setting up this very nice meeting. What I will be telling you about today is a work that we have been recently done with Benua Fermes, Shempeter Solar in Eastbrook, and it's about the quantum simulation and spectroscopy of objects which are called entanglement Hamiltonian. Okay, let me immediately go to the main question that we want to address. The question is as follows. We want to develop experimental protocols, I mean find devices at the zero level experimental protocols that are able to measure entanglement spectra in many body quantum systems, being that ions, circuit QD, atoms, and so on and so forth. And the figure of merit in terms of many body physics for this entanglement spectra is as follows. Suppose that we have a box and we have a system inside here which is defined by a certain Hamiltonian H. H can be divide, we can divide this box into two parts, A and B, and with the Hamiltonian, the result in Hamiltonian we can write as HA plus HB plus the interaction, the surface, the surface Hamiltonian. What we're interested in is the reduced density matrix rho A obtained for instance from the ground state when we trace out B. Okay, and typically you see that this is written as the second, as the right most element of this equation. This is the sum of all eigenvalues and eigenstates of this operator rho A. This is called entanglement spectrum, this lambda alpha, but in particular you can also write this guy as an operator, in a full operator form as an exponential of some H tilde. Okay, this is already very tempting because it's telling us, okay, we can try to understand reduced density matrices as partition functions or something like that. Okay, and this object is called entanglement Hamiltonian. What we're interested in is these guys, this lambda alpha, the entanglement spectrum. Okay, and this is the modular or entanglement Hamiltonian. So how can we measure this in an experiment? Okay, the obvious way is to do full state tomography, but we know that this is extremely challenging, is exponentially hard in the number of degrees of freedom that one has to probe, so we would like to devise something different. And before telling you what the main result is in terms of physics, I would like to actually use a simple figure of merit. What we want to do is we want to completely shift the paradigm from probing density matrices to directly engineer these entanglement Hamiltonians. And this is a bit like, I mean, if you want to understand what are the ingredients of a cake, how do we do that? Okay, in principle, we can try to measure the cake many, many times and try to guess how many grams of eggs we have used to build this cake. But what we can do, alternatively, is to do something different. We can look at the shop bag and inspect it. Okay? And if we do that, for example, we can understand how many carrots I've used to do this cream cake, obviously not many, and instead of how much butter and how many eggs, many. Okay? So the idea is really to completely decouple from the original problem as formulated, so building a cake, building a density matrix and probing it, but instead mapping this to another problem which is much easier to solve. Okay? Because here I can tell you how many ingredients you have used and how much of them. Okay, this is the figure of merit. Now, let me show you, for our purpose, is what this will be about. We come back to our system and we want to do, we want to directly engineer this object, this H tilde, this entanglement Hamiltonian, because then if we perform conventional spectroscopy, something which is, I mean, done extremely well in a series of experimental setup, we get the entanglement spectrum we are interested in. Okay? And what I will be showing you in the following is a specific recipe, which is not generic, but yet is applicable to many, many problems, to most quantum field theories, to topological, quantum, many quantum critical points and so on and so forth. One, two, three, no problem. It applies on the lattice, the lattice problem also in the continuum, does not need copies, does not need in situ imaging. The only thing that needs is spectroscopy and it allows us to get all the important information that is contained in this entanglement spectrum. Okay? And what will be the final ingredients just to give you a tweezer, the only thing that will be really needed are light-induced interactions and this we know very well how to do in many systems like trapdions, circuit QED, read bar atoms and so on and so forth. Okay? And for instance, this is an example that I will illustrate later on. What we want to do, for example, if you have a spin chain, we want to be able to engineer this interaction locally and this way we get entanglement spectrum if we do it in a particular way. Okay? So that's a teaser. Now the outline, the first thing I want to show you is, I mean, the entanglement spectrum in a bit more detail what it is and why it's useful. Many of you probably know better than me, but this just to give a general understanding of the problem. I will tell you something about the entanglement Hamiltonians and I will show you that there is a deep connection between actually entanglement theory and many body theory and a field of math which goes under the name of axiomatic quantum field theory. And this connection will be the one that we exploit in order to find this experimental recipe. Math and experiments in the same sentence, I know it can be kind of funny. And at the very end, I will show you some concrete proposals and some numerical simulation that illustrate the recipe that we want to propose in terms of quantum engineering of this entanglement Hamiltonians. So the entanglement spectrum, this object here, why it is useful? Okay, for what it is useful for? And I think there are at least four reasons for that. Okay, the first one is that if I know the entanglement spectrum, I obviously have assessed too many informations in terms of properties of the system. For instance, I have direct access to the phonon monanthropy, entanglement entropies. Okay? The second reason is that this object has a paramount importance in topological phases. This was first introduced by Lee and Alden, this connection in 2008. The second point is that since it's a spectrum, it contains more informations, in principle, infinity more information than its average quantities. And in particular, the entanglement spectrum contains much more information, quantitative and qualitative than entanglement entropies. And the first motivation, which is somehow, well, it's not the direct motivation, but it's somehow funny, is that it's very hard to get this stuff in theory. Okay? It's very hard to calculate it unless you have a wave function based method. Okay? So why it is useful? I've already said, I mean, this many of you already know. I mean, in principle, you get most entanglement measures, and they are good for diagnose in topological order, classify quantum field theories and measure entanglement, at least at the level of pure states. And this is the typical example. You have direct access. If you know this sector of eigenvalues, you can calculate entanglement entropies, in particular, in this case, the phonome and entropy. That's the first basic motivation. The second motivation, as I was mentioning before, is that this is very important to diagnose topological matter. This is the concept introduced in this paper by Ilien Aldein 2008. And let me illustrate this with a figure of merit. Okay? This is a Coulomb gas on a sphere under a certain applied magnetic field, and this is actually described by some quantum wave function. And this stuff that you see here is not a physical spectrum, it's entanglement spectrum. Okay? So basically what these people did, they calculate the system wave function, they did the bipartition, and they calculate this spectrum out of the wave function. And you see, it seems to have a certain structure. Okay? And in particular, there are two features that these people emphasize, is that if you look at the sectors in the same number of particle or angular momentum, there is a gap. Okay? And the second feature is that these states here, which have different angular momentum, they seem to follow a pattern. In particular, they follow this what is so-called edge state counting. Here you have one state, here you have one, you have two, three, five, and I think the next one is eight, and then the next one is eleven. It's the partition of real numbers. Okay? And these two features indicate that this wave function describe a fraction of quantum state. Okay? So this is very important for diagnostic. And if you're interested in reviews on entanglement spectrum in general, I think there is this very beautiful paper by Renaud that is worth looking at. Okay? So and then let me go directly to the last motivation. So what we do in terms of classical simulation to estimate this entanglement spectrum, this is typically very hard. Okay? It's much harder than entropies. Entropies are relatively easy. For example, if you have a Monte Carlo method, the only thing you do is to calculate different partition functions and kind of swap operators. But for entanglement spectrum, this is very hard. Actually, this is not possible. What you do typically, you get many rainy entropies and you try to reconstruct from the momentum of a distribution to distribution itself. A problem that we already know from our first course in math is extremely challenging. Okay? What can one do? I can measure this stuff with the when there is an exact knowledge of the wave function like in an exact diagonalization of the MRG. But this is obviously limited in terms of applicability. Okay? So we have understood, I mean, the message is this entanglement spectrum is useful for many things. In particular, it's a paradigmatic quantity in many contexts of many body theory. I just illustrated the topological aspect. There are many more. The challenge is now how we want to measure it. Is it measurable at all? This is non-trivial statement. Can we find a way of measuring it? Okay? Indeed, there are. There is a proposal by Hannes Peekler and collaborators in 2016 where what they do, basically, they exploit the fact that if you have many copies of the same system and you are able to entangle them via additional controlled degrees of freedom, in particular, in this case what they use, they use control atoms, is in principle possible to actually extract entanglement spectra or at least few eigenvalues of the entanglement spectrum for arbitrary difficult density matrices. However, this scheme is very general and requires a lot of resources. Okay? It's extremely expensive. For example, just for Alden chain of 10 sites, they need something like under 50 copies, very long experimental runs and so on and so forth. So it's general, it's beautiful, but it's very challenging. Okay? Our goal is to do something different. It's actually to do something which is easily scalable, does not require copies, can be applied in any dimensionality and so on and so forth, does not require copies of single set addressing at all and is applicable to a broader class of problems. Maybe not to all the possible reduced density matrices, but to a large class of those, in particular for ground state problems. And this is exactly what I will show you in the following. We focus directly on entanglement amyctonians. We will discuss complete experimental protocols to measure this entanglement spectrum at the very end of the day. If you are more theory minded then you can see this as an over theoretical route to tackle entanglement or at least bipartite entanglement in many body systems. We're introducing the concept of entanglement field theory. So the problem is the following. I mean I told you this, this we one can almost always do, so writing the reduced density matrices as a financial of an operator, the point is what is this operator? Okay? And typically the first time I saw this equation I said okay it's impossible. Okay? This operator, it's still there will be no local, will be made out of many body crazy interactions. This is completely, I mean it's completely nuts to do that. So no local many body interaction. This is not good. Okay? It will be very funky amyctonian. There is no way of realizing it. However this is actually not the case. Okay? It's not the case at all. In principle also if it's funky we don't know how to get it. In principle there are results from this field of axiomatic field theory that saw this and this go under the name of Bisniano-Wittman theorem. And let me spend you one slide which I try to keep as less stating and as possible to explain you what the idea behind this is. Okay? So imagine that now we want to study, these are results are actually from, you see the age, the years 1975. Okay? 1975, 76. They were not discussing entanglement. They were discussing modular amyctonians and so on and so forth. But what they indeed the object that we're focusing on are exactly this entanglement amyctonians. So for our purposes we look at a system which is described by a amyctonian which is translation invariant and as a certain amyctonian density h of x. And since they were dealing with field theories the amyctonian density was also, I mean the system has to be Lorentz invariant. Now if I give you a bipartition here you see this axis, this axis parametrizes the distance from the boundary x. x is equal to zero it means that we are at the boundary x larger than zero this means that we are inside the bulk of this region A. This theorem tells us that the entanglement amyctonian of our reduced density matrix of the ground state of the system is actually defined as something like this. It's the integral of a dx not of h of x so we will not have the same amyctonian density but we will have an amyctonian density that actually increases as a function of distance from the boundary. This means that close to the boundary we will have amyctonian couples which are very small and far from the boundary we will have amyctonian couples which are very large. Now what is red and blue? What are red and blue have to do with the fact that couples are large or small? Well one of the first interpretation of these results was in terms of entanglement temperatures. Now imagine that this now goes as an exponential of h tilde and we can interpret this x as an inverse temperature as a beta. So beta is very small close to the boundary, beta small means very high temperature that's why it's here it's red and instead very large are far from the boundary in the bulk. This means that it's very cold. This is a very simple intuition that tells us that we can understand entanglement by just saying that close to the boundary where the system is more entanglement it's amyctonian its entanglement amyctonian is actually effectively an infinite temperatures describe an infinite temperature system while far from the boundary the system is essentially not entangled anymore with the rest so and it's like being essentially at zero temperature. But there is another important feature which is actually quite fundamental for our purpose is that this amyctonian has the same structure of the original amyctonian. So if the original amyctonian was local this is local. If the original amyctonian was too body this is too body. The only thing that changes is that the couplings are not homogeneous in space but they have a specific spatial dependence this x. Okay? So this is this result this is a Bisonier-Wickman theorem and now this fundamental tells you what it is. We have a given problem the first thing we have to find is the entanglement amyctonian but we already have the tool for doing that. The second thing is that we have to devise a protocol protocol to actually realize it and at the third stage is that we have to do spectroscopy and if we do spectroscopy we get this epsilon alphas the eigenvalues of these operators and they are immediately related to this lambda alpha okay there is no ambiguity. There is a however a catch. Okay? These results since they were somehow derived in field theory they rely on set of assumptions like this Lorentz invariance I was mentioning that the system is infinite and so on and so forth. The question is can we actually relax some of those and in particular in terms of cold atoms or trapdion systems? Can we apply these results to finite lattice systems? Do they hold? This is one thing that at the beginning when I saw this result first I did not believe it. Okay? I think it's not possible yeah? And the first thing I did is okay let us look at some models and see if this is working okay? And what I will present you is a follow this is a list of checks that we have performed both analytical and numerical and I will show you some of those in order to try to convince you that this is indeed working. Now let's start with the please. I'm starting with the easing model. At the end of the slide it's this is actually open point. What is true for integrable models is that the retransfer matrix commutes with this with the modular Hamiltonian but they are not always exactly the same and there are examples. Okay? There are thermodynamic examples. I have to say that for the spin models that I know they seem to be always equivalent like for the XXZ and also for this model but for thermodynamics model it's not the case. Okay? So the first thing that you do when you want to do a check you do it on the simplest model that you know this is the easing model. Okay? We take a chain which has this Hamiltonian as couplings that are now homogeneous and we take for example a chain in a box. Okay? And this is the Hamiltonian. Sigma z plus lambda sigma x sigma x. So what is its guest entanglement Hamiltonian the Byzantium vichyma? Well this will be just a Hamiltonian that has n in front. Okay? So where the coupling will increase as a function of distance from the boundary not only the spin coupling but also the coupling of the magnetic field because all the Hamiltonian is multiplied by the distance. Okay? So the question is if I calculate the entanglement spectrum of this guy and I calculate the physical spectrum of this entanglement Hamiltonian do they match? This is a simple result is the first thing that we check is that for a very small system like 20 sites by partition of 10 sites here we compare some universal ratio of this entanglement eigenvalues just to avoid to rescale stuff and so on and so forth as a function of delta over j I apologize this delta is actually is one over lambda okay as a function of the magnetic field strength and the blue and red curve the red one is the entanglement exact entanglement spectrum and the blue one is the physical spectrum of this entanglement Hamiltonian please. This one? Okay. The point is for the easy model this entanglement Hamiltonian can be calculated analytically and it matches the results in 1975 of Bisgnan and Vikvan but Pechel was not aware of this not for easing not for easing for easing this is exact for the fermions there will be long range interaction that's that's why okay but I mean I didn't know about these papers okay because at the beginning I really did not believe I mean these results and I want to check and the check is actually quite good this universal ratio already for me I mean the by partition is 10 sites so it's extremely small you expect that this is rubbish you get precision at the level of 10 to the minus 5 okay but obviously the easy model is not a great test okay easy model even mean field works okay we want to do something which is a bit harder okay and the second thing we look at are Lattinger liquids both free fermions and XXZ spring chains and these are some I mean this is a small system of 32 sites this is a bit larger system XXZ and here we are always comparing this K alpha and this K alpha these are this blue and red curves that I was showing you before just with universal rations and they are matching essentially perfectly and notice that this is actually this is entanglement scales so this is an exponential okay in order to see the script and see in one of these eigenvalues at the level of 10 to the minus 10 to the minus 10 you have to go up here so this means that this is sorry it's not up here because this is E scale but this means that at least at the level 10 to the minus 5 the entanglement spectra are actually equivalent okay there is no difference for the spring chain here you wonder why there are no there are no more points here and indeed there are but this was just a preliminary calculation and calculating this red stuff with the energy is not so easy because you do have to do a technique which is multi-targeting but we actually have to I mean fill this up almost until the end but still you can wonder this is CFTs these are very peculiar critical points both these models are integrable so maybe we are very lucky and then we did something a bit different we check for you we we check what is so called al dain chain spin one isenberg model which is known to support topological phases and when there are also interesting features in particular for this topological phases in 1D you expect entanglement spectrum to be at least two fold degenerate for the particular points we were looking at this has to be four fold degenerate here I don't show you plot because plots of the genesis are not good I show you directly the table and these are the these universal entanglement ratios the first three have to be degenerate because they are referring to the first initial state the same year and the same the fourth the four guys here and here we see essentially except the generacy at level 10 to the minus 6 okay so and this I mean with this check we were kind of convinced that at least in one dimension this is working extremely well okay sorry and we are able to grab not only the the numbers but also the features of this entanglement spectrum for concrete systems but again you can wonder 1D let's try to do 2D in 2D you have to be a bit careful because when you put a quantum field theory on the lattice actually the the angles can create can create probably the corners okay so what we did we did the trick we exploit the conformal mapping even though actually our theories are not always conformal in variant and we did a few checks this is a check done for free fermions at quarter filling and in a lattice which is infinite and with a bipartition is 696 and here we see relatively good agreement for the first thousand eigenvalues actually if you go a bit up here you start seeing discrepancies at the level of 20-30% so you will not you will stop trusting this technique okay but at least for example for the first 250 eigenvalues more or less you have 10 to the minus 6 precision so they're very they match very very well and the second thing we did we check a model for a topological insulator it's the so-called two-dimensional Dirac model and here we also see not only good agreement between the two techniques but we are also able to see in this entanglement Hamiltonian spectrum the edge modes that you will have in a true topological insulator which are these ones they actually show up extremely well in this entanglement spectrum even better than in entanglement spectrum that you actually compute from the discs this was a bit surprising but that's it and in particular what we also did for this massive Dirac model we look at the entanglement spectrum number resolved and for topological phase this has to satisfy certain counting and this was working quite well and here what we are actually comparing are the exact entanglement spectrum these are these black circles with different mappings of a square to a sphere and in particular the conformal mapping which is the blue one typically is the one that performs better there are also limitations because as I told you at the very beginning well not at the very beginning but when I discussed this being a big frontier there is this Lorentz invariance so I mean what what means Lorentz invariance if you are looking at ground states of physical system means that the spectrum has to be a pro... well it does not mean that but typically it is realized if the spectrum is approximately linear so we want to check a case in which our technique actually does not work and it's simple to do for example we can take three fermions and we can take very very low densities at very low densities the spectrum of three fermions is not linear anymore it's essentially quadratic and indeed we see that for example if you look at the 1D model 32 sites for our feeling we get perfect agreement if you go feeling 1 over 16 we start seeing consistent disagreement and at some point actually two lines completely coupled okay so in principle one has to be careful in applying this technique there are some limitations related to the microscopic theory well in our experimental strategy the first point is done okay we have we found the entanglement aminternium we know that this is actually working okay the second point is we want to have a protocol to devise we want to have a protocol to realize this in experiments and let me illustrate in a minute an idea I mean there are many potential platforms where you can do it atoms, circuit QED, ions and so on and so forth and in this example I'm discussing river dressed atoms in particular I mean if you put them on a line you want to realize a spin chain like one of the ones we were discussing before and the only thing you need to do is to modulate the coupling locally okay and now you do that well the only thing that you really need are is a local control over the Rabi frequencies because this gives rise to the right modulations of the coupling strength that we are searching for okay then the last step you know the Hamiltonian you realize the model how do you probe it you do conventional spectroscopy okay the hepatic spectroscopy so you can do the spectroscopy that you wish there is a question is this entanglement spectroscopy more sensitive to I don't know defects or the coherence that the conventional spectroscopy is what we did is we performed concrete simulation this is simulation for a lazy model it's six sides this is just to show the agreement of the entanglement spectrum between the exact and what we call business in your weak one spectrum there are some discrepancy for such a small system but they're not very large and here instead is the is the spectroscopy which include obviously the finite size effects because we are in a finite system noise in the state preparation both in the hepatic state separation and during the spectroscopy and the only thing you notice as in every spectroscopic means that your signal just gets broader because you have noise but nothing more so it's essentially the same stuff as in a conventional spectroscopic experiment and here the green line is the exact result so it's essentially as a resilient to imperfect as convention spectroscopy okay with this I'm done let me draw the conclusions I mean the first message which is not a result that we found but these are results that we somehow discover in the sense we discover in the math literature is that entanglement Hamiltonians typically are local and few body and can be written in cross forms for a very broad class of problems okay and it's also interesting that this business week my result allow us to recover a lot of results which are which have been derived in connection matter theory by different means in few lines okay and in particular it has nice connection with the the tensor networks that was discussed yesterday then what we have proposed is we want to use synthetic quantum system where we have a lot of interaction engineering to actually probe this entanglement spectra and the only thing that we really need is just locally tailored interaction plus spectroscopy it's robust to imperfection and is in principle adaptable to many platforms where you have where you can control the interaction locally even if you don't have imaging in situ but you just can modulate your interaction in space now in terms of outlook I think there are some interesting things in particular I mean this this kind of connection between density matrix as entanglement Hamiltonians strongly suggest that one has to use one can use potentially a new language for understanding entanglement in many body systems in the concept of entanglement field theories and we are currently looking at some of these things in real-time dynamics for topological order for 2D interacting system this is actually I have to say a bit hard but maybe for for certain models like lattice gauge theories become simpler and then the natural question is can we extend some of this understanding to entanglement not in bipartite setting but beyond bipartite and the bottom line is this entanglement field theories can really offer us a brand new look to look at entanglement in many body systems with that I'm done I think I mean this is this is the archive if you want to have a look at the manuscript and thank you for your attention