 I was just waiting because there was a bulk of people coming in. So I thought it's, that's it. Oh, the cover fell off. Okay, so welcome back, everyone. Well, I'll continue the introduction with hands on networks and let me just briefly remind you what, where we got to yesterday. So what we talked about was the physics of quantum antibody systems. So specifically systems where the interactions between the individual quantum particles play a strong role. So which we can't understand just by looking at individual particles and kind of looking at the entanglement only in a corrective fashion. These systems were basically governed by some Hamiltonian, which was a local interaction on some kind of lattice, like a 1D or 2D lattice. And then what we realized is that these kind of systems, if we have n particles live in some Hilbert space, cd to the n, which has a very high dimension, namely a dimension which grows exponentially with a number of particles, which was prohibitive for efficient description of a state of such a system if you want to describe the full state. But then what we also saw is that basically in this huge Hilbert space, there was kind of a small corner of physical states, simply from the intuition that we only needed very few parameters to describe the Hamiltonian. And the Hamiltonian term would describe the ground state. So this motivated to ask what's special about these states, can we understand from a point of view of entanglement of quantum information, what makes these states special? And then what I told you is that these states are very special in the sense of their entanglement. So I should indeed maybe stress that what we care for in this corner are things like ground states, since these exhibit the strongest quantum effects. And they're very special in their entanglement, namely when we would take a lattice system, say, and look at the entanglement of a region A with the rest, then the entanglement of A with the rest, which we could compute as the von Neumann entropy of the reduced state only on system A, would scale like the boundary of A rather than the volume. So only very few degrees of freedom, effectively very few degrees of freedom in any region A would be entangled with the outside. So in particular, in one dimension, the boundary of the region, of course, is only on the left and on the right. So in one dimension, this would mean that the entanglement between any region A and the rest would be bounded by a constant, basically. It might go initially as we make a region bigger, but then it will saturate and not go beyond that constant. So that's the starting point for today's lecture. So today I would like to start from this point and motivate how one can write down an explicit way and explicit ansatz of describing many body wave functions, which kind of has this property that entanglement is only located at the boundary, but for any way of partitioning our system and how from there we can actually understand the behavior of quantum many-body systems better. Okay, so this would be part two in the basics of tensor network states in one dimension. So I will start with one dimension because many things are easier there. We can also make stronger statements and later on in one of the subsequent lectures I will also tell you a few things about two dimensions and higher dimensions. So what's the construction, the idea? So what we would like to do is, I mean, if we think about this area law construction one dimension or in two dimensions, what this tells us if we have a spin chain and we look at some region and it has this entanglement, this area law behavior regarding its entanglement with the rest. What this tells us is that we would assume the entanglement is only located at the boundary. So we could say, okay, well, why don't we try to take this particle and entangle it in some way with that particle? And similarly, this particle entangled with that particle. And the rest of the chain we don't really have to care because this kind of has the right behavior, the entanglement is located at the boundary. But of course the problem is this doesn't work if we put our cut somewhere else. Say if we take our cut and we extend it to here, then this behavior is suddenly gone, right? The entanglement is, there's no long entanglement here. So we could of course put some entangled state here, but then still in some other cut, like say in this cut, we would still have the same problem, right? There would still be no entanglement. So that's something which kind of as a cartoon picture might work, but it's kind of not good enough, right? Simply because it doesn't respect the translation symmetry of our lattice, right? If we entangle specific states, if we entangle that state with a one to the left, we can't at the same time entanglement with the state to the right. And that's due to this property known as monogamy of entanglement, right? We cannot have entanglement shared with several different parties at the same time. So if you really want to take this particle entanglement with a one to the left, we will not be able to entanglement with a one to the right. And this doesn't give the right behavior of the entanglement. We want that the entanglement is kind of the same in every cut. Let's say we consider a system which is translation invariant, right, which behaves the same everywhere. We don't want different cuts to behave differently. So that's too easy. So what we will do instead is the following thing. We will start from our spin chain and we'll first say, well, let's for a moment imagine that each of these particles is actually composed of two particles, one kind of left and one right particle. And I'll tell you in a moment how this construction works explicitly. So let's for now just assume this is some kind of very hand waving construction that we take each particle and we replace it by two particles, which together in some way build up the original particle. So it kind of consists of quarks or whatever, right? Don't take this literally. So now what we have at each side, we have two subsystems which make up our system. And now we can indeed do what I try to do here. We can try to start building up entanglement to both sides, simply because well now we can take this particle here and entanglement with a one to the right and the other one and entanglement with a one to the left. So now we can build up a construction like that. And now you certainly see it has this kind of property we wanted that whenever we make a cut somewhere we cut through this entangled pair or this entangled pair and so on. So regardless of what region we cut we will always have the same amount of entanglement of that region with the outside and the entanglement will always be located close at the boundary in some sense. But of course I'm still kind of very sketchy in the way I explained this construction. So what are these states here? Well, let's be a bit more concrete. Let's assume that each of these two systems has a capital D dimensional Hilbert space. So I have a space CD times CD and this state here I say will be a state omega which is a maximally entangled state. So I just maximally entangled these two particles. Of course this means that I have a very specific construction the entanglement of a very specific type but we will add in an extra layer which we'll make this more rich later on. So as I said what we have now is that the state kind of has a right entanglement behavior entanglement sits at the boundary across any cut I can choose. But of course it's not very rich, right? It's basically only one state the only thing I can change is this dimension so I have a discrete set of states. So that's of course not rich enough for what I want so what we will do here is this thing up here. What we will do is we will basically add a construction which will give us back the original particles. And what this construction will do is it will take these two guys here. So what I have is I have a CD dimensional space here and a CD dimensional space here with some capital D. Now I will put these two guys together and I have to describe how I get my original particle back. And the means of doing that is to say well let's take these two particles and apply some linear map which out of this space of a left and a right particle maps out an effective space for the model I want to describe. So it's just some general linear map. So one thing I could for instance do is if this system here is a qubit, a two level system I will necessarily have too many degrees of freedom up here, right? Because let's say even one capital D is two and this one is two, I have a four dimensional space. So to get to a two dimensional space I might want to project onto a specific degree of freedom of that space or well a two dimensional subspace of that space. So a typical choice for this P might actually be a projection which only keeps the relevant degrees of freedom. You could think of this as perturbation theory kind of singling out a low energy space, right? In which kind of the low energy physics takes place and they're very high line excitations which are leading out of that space. You don't have to think of that. It's really more a mathematical construction but you could motivate it from perturbation theory. So this P could be a projection which keeps some interesting degrees of freedom for us but it doesn't have to be a projection. It could be any other linear map as well. So we will do this at every side and this way what we get is a way of obtaining a chain on our original system and as we've seen the following this P actually allows to do many things. So for instance, before we applied this P we had no correlations whatsoever between anything here and anything there, right? Because the only connection was through these entangle states between nearest neighbors but not on a longer distance. But by adding in a projection here for instance we can create longer range entanglement. We can create longer range correlations. So the state gets much, much more rich that way. Now generally we might want that these states, these P's to depend on the site. So it depends what we want to describe. If we want to describe a translation in variant state we might choose all of them the same. If we're not sure if our system is translation in variant or we just want to introduce extra parameters we can make them site dependent so I will put the site as a superscript here in brackets. So to each slide S there corresponds a map P S. Excuse me. Well it goes to a smaller space so I'm not sure. Well if you probably an isometry if you're more strict, if I say projection, right? Because it goes into a smaller space but it can be any linear map in fact, right? It's just that kind of, if you think what would be a natural choice you might say oh well I'm in a big space and I want to single out an effective space and then you would project onto that effective space which is say two dimensional. And actually many canonical examples have a projector as a map P but there's no obligation to choose a projection. So any map will do. Exactly that's why I'm saying if you're strict it should be an isometry. You should call it isometry and not projection, right? Well I should call it. No indeed so. So it's not a projection in the sense that it doesn't map between a space and itself. So in that sense it would be an isometry. Now people tend to refer to this P nevertheless as a projector even if it's neither a projector nor an isometry but just a linear map just out of convention and because several examples have this projection construction. Okay. So we can do this construction for different types of systems. So as I said we can for instance enforce translation in variance. We can do it with open boundary conditions which would mean that on the left we only have one of these entangled pairs. Let me stick to my color scheme. So we might have end points which are different. We can do it for periodic boundaries in which case we would connect the entangled pairs on the two ends. We would do something like pairs and then connecting the last one and the first one again like putting it on a ring. So we can adapt this to different situations and I will change this a bit throughout the next lectures what I consider. Ah no, but that's what I call a site is this right of the original lattice. The original lattice, like that's the green things the green things are sites of the original lattice. It's just when I make this construction I split these sites up in kind of auxiliary step and intermediate step I split the sites up into two except at the boundary which I don't split up. So in that sense once you split up all guys in the middle you will always end up with an even number of these sites. Okay so in principle we can generalize it further, right? So what I said is that these entangled states are all the same, cd, cd. But in principle we could even make these. So in that case this would also be a c capital D. These particles should all be the same usually if you consider a normal spin chain. So that's all small d dimensions. That's capital d dimensions. So which in particular I mean that this map here p1 is different, right? p1 larger p1 only maps a single one of these auxiliary particles to a single one of the physical particles. So it might have a different structure. It might even map from a smaller to a bigger space or things like that. Now more generally if I don't insist on translation in variance I could make everything side depend not only the p but also the dimension of this particle. Now in some senses this doesn't make the end that's more rich because of course I can always choose a maximal dimension and just choose to ignore the extra degrees of freedom. But of course in the sense of having an efficient description it might still make sense to keep that in mind. No problem. The pairs connected with strictly lines are maximally entangled, right? So what I do is I put these two guys into a maximally entangled state. These two guys, right? But then I apply these maps in a displaced fashion. So the idea that these pairs will build up entanglement between this side here and that side here. And while this kind of also builds up entanglement in some sense, well it singles out kind of the degrees of freedom I'm interested in but also starts building up correlations on a longer distance. Well thermodynamic limit is a subtle thing. I would prefer not to talk about the thermodynamic limit in detail because it's getting very mathematical. It's a very interesting question. It can be properly defined here. Well you would indeed think that the thermodynamic limit it doesn't matter so much what your boundary conditions are as long as you only look in the inside. So in that case you would indeed probably, well first of all you want translation in variance that it indeed doesn't matter. But kind of the subtle thing is of course in the thermodynamic limit you only want to look at a smaller region of a long chain, right? Because if the chain is infinite it doesn't make sense to talk of the whole chain. And then indeed it turns out that things converge nicely it doesn't matter what you do at a long distance. So in that context these states are known as finitely correlated states. That's something which has been done in the early 90s by Fannes-Nachterhalle and Werner where they work on the mathematical formalism to define the thermodynamic limit properly for these states. Of what, sorry? It looks like that. Well there's a relation of that to the normalization group. I'm not sure if it's generalization. Well we're talking about spin systems here but you can do the same thing for fermions and you can do it for non-interacting fermions. So it does work. I mean maybe I'm thinking whether I should give one example right now to illustrate the construction. It might actually be a good idea even though the example will only show up later on in more detail but maybe just to give you a kind of more physics intuition how this might look like. Let me give one example which is known as the AKLT state. So what AKLT stands for Affleck, Kennedy, Leib, and Tazaki. Not sure I guess the original paper is something like 88 or 89. Okay so what is a construction? The construction is slightly different. I'm not sure if it's a perfect example but it's a kind of somewhat practical example. So what we do is that our entangled states will consist of two spin one halves. So we have a two level system here, a two level system here. But now it's really a spin one half. It really transforms like a spin one half particle. And this state we now put in a different state. Maybe I shouldn't call it omega, let me call it omega tilde. Namely a singlet state. So zero is spin up, one is spin down. So we put in a singlet state. So it especially means that this state transforms nice. It transforms trivially under spin rotations under any action of the rotation group. So now we make a chain of these guys. So we have all these spin one half particles and now we really think of them as spins. So that's really a spin one half particle. And well now what we want to do is we want to take each two of these guys and look at some effective degree of freedom or so, kind of glue them together in a way which makes up something more interesting than just two spin one halves which don't have anything to do with each other. So what can we do? Well we have two spin one halves. I mean what kind of, what would be a natural way if we want to respect symmetry, spin symmetry? Well we know that two spin one halves can be decomposed as either a spin zero or a spin one. So the total Hilbert space of two spin one half particles has a spin zero subspace and a spin one subspace. So a natural choice with respect to spin rotation symmetry would be to either choose these degrees of freedom or these degrees of freedom. Now choosing the spin zero degree of freedom is not very exciting, because after we project onto the spin zero that's a one dimensional space we're just left with a chain of spin zero particles. So not very exciting. So maybe it's a better idea to keep the spin one degree of freedom. So what we do is we apply a map P which does what? Well P is equal to the projection onto the total spin equal one. But really more in the sense of an isometry, right? There's a map acting from, acting from two spin one half particles onto a spin one particle of three level system, right? So we just, I mean we can write it down explicitly because we know what the spin one states are, right? So we know that basically what we do is we have a total SC equals plus one which corresponds to the zero zero state, the two spins up state. We have an SC equals zero which corresponds to the triplet state, zero one plus one zero divided by square root of two. And we have an SC equal minus one which corresponds to both spins pointing down. And if we add the three that's P and we apply this everywhere then. So what we get this way down here is a spin one chain. And so we'll get back to this construction later on but it has a number of nice features. So in particular one feature we get by this construction that it's not just a spin one chain in the sense of having three level system but it actually transforms nicely under the spin rotation. And the reason that we start from singlets which are invariant under rotating everywhere, right? If we apply a rotation everywhere about any axis the state will be invariant because it's just composed of singlets. And the second thing is if I apply any rotation here whether I apply a rotation before projecting or after projecting onto the spin one space doesn't matter, right? The rotation of the two spin one halves is the same as a rotation on this space and on that space. So if I just keep this space whether I rotate here or here doesn't matter. So rotating here is the same as rotating here on these auxiliary particles I used in the construction. So I obtain a symmetric wave function. And where this wave function has a number of other nice features. It has non-trivial correlations. It's a unique ground set of a local Hamiltonian. That's all things which I, well, hopefully we'll mention in one of the next lectures. Well, that's one way how we can think of the construction I guess, that's saying I want to spin one chain. Let's try to think of it as an effective. I mean, you can see that you could get this in some kind of perturbation theory. If you say you actually have a spin one half chain like this, but here you have a Hamiltonian which is very strong and which forces these guys to be in the joint spin one state. So then kind of the effectively accessible space at low energies will only be the spin one space. So you put a high energy on the singlet. And then you could say these singlets can establish by having a weaker interaction which acts as a perturbation between these guys trying to put them in a singlet. And then if you do perturbation theory you just get the projection of the, well, the interaction which one singlets on this effective subspace, the spin one subspace. So if you want, you can think of this as an effective low energy theory. It's not necessary, but it's certainly possible. Okay. So okay, maybe I put this back here. Let me just fix a bit nomenclature because I will keep using that. So the red particles we will call virtual or auxiliary particles or systems or whatever. The green ones physical, I guess that's more natural. The entangled states, the entangled pairs will also be denoted as bonds because they create some kind of entanglement, some correlation between different particles. And well, since I will probably use it anyway, let me already tell you that this is what is called a matrix product state. Let me put this over there, maybe. Can I raise this? Okay, so we have that construction. We could also write it as a formula, right? So rather than drawing a picture we can also write formulas, obviously. So what do we have to do? We have to start from a state omega acting between n sides. And then what we have to do is we have to apply p1 times p2. Let's say we look at periodic boundary conditions. So basically it's just a number of these maps p applied to a number of entangled states. Of course this is kind of misleading if I write it like that because the real point is of course that the partition in which these maps act is shifted with respect to the partition in which I put my states. Otherwise I wouldn't get something very interesting. So one can write it obviously in this form but one has to keep in mind that the systems on which the individual things act are kind of shifted. So if you label the subsystems, you could say this is one left, one A, one B, two A, two B, but then the entangled states would be between one B, two A then between two B, three A and so on, right? So that's some shifting in the index. Okay, these states are also known as matrix product states. And let me see, I actually want to say something else first but maybe I rearrange things and first explain why they're called, yes. So let me rearrange things first and let me first explain why they're called matrix product states and then tell you a few things about their properties. Okay, so let me just have another look at this formula I just wrote, kind of the explicit expression for the state. So we have that we can think of this state as being created, say, on periodic boundaries by acting with all these piece on our entangled states. And now we can ask, well, can we maybe derive a more explicit expression for the form of this psi? So what we also know is that obviously we can expand psi on some basis of our space. We had this expression last time already when we talked about the exponentially big Hilbert space. So we could ask, can we derive an explicit expression maybe for the C I want to IM? In terms of the map P and the state omega. Now the state omega we know, right? So what do we have? We have omega is this maximally entangled state. And well, we have the map P at some side S can be expanded in a basis. So we have that alpha and beta go from one to D and I goes from one to small D. So I just basically express this map P in a basis where I have a three index object, A, I, alpha, beta for each side S, right? So what this does is it takes the auxiliary particle in state alpha and beta and maps it to state I. So now let's try to see if we can understand what this expression should evaluate to if we want to get something of that form. And of course we don't want to write everything at once, right? This is a very long expression. So it might make more sense to go step by step. So what we would like to do is we would like to start from kind of the simplest building block. Well, the initial thing is that we have a map acting from two systems to one, right? We have this map acting from two systems to one. And now we can try to do some kind of first step and as a first step in constructing this state, we will attach an entangled state and a second side. So we will take this entangled state and we will append it. So we would have P one and P two. So you see this forms an inductive step, right? Because what I have here is something which maps a left auxiliary particle and a right auxiliary particle to a physical system. And this is the same thing. It maps a left and a right auxiliary particle to something physical, which is now bigger. And you see I can do an inductive step, right? What I did is I kind of took this input system here and I attached another system, right? And here I can do the same. I can attach a system there and continue. So if I understand one step and it turns out to have sufficiently nice structure, I can inductively see what I get if I keep on doing that. So what do I get in one such step? Well, what I have at, so let's just write down what we have at the right. So we have P one times P two applied to omega sitting between sides one and two, which is what? Well, lots of summation indices for one thing. Then we have these P's. So we have A one, I one, alpha beta, I one, alpha and beta. So what do we have? That's I one, I two. We have an alpha, beta, gamma and delta, okay? Okay, so we tensor this with A two, I two, gamma, delta. This is just a big summation index now over alpha, beta, gamma, delta, I one and I two. So this thing here is P one times P two, right? Now I have to put omega. There's a one over square root of D and then there is a K, K. That's pretty lengthy, but if you want to understand what's happening, we have to see what belongs to what. So where's something non-trivial happening? Well, the non-trivial part takes place here. We have an entangled state and act with a map on it, right? So we have these two guys, that's the entangled state. Let me label my sides. So this is side A, B, C and D here, okay? So we know where we are. So this is A, this is B. This is C, this is D. And similarly here, this is B and this is C. So what we see is that this K belongs to this beta here, right? So what I get is I get the overlap of the, well, basis vector beta with the basis vector K, which will give me a delta, well, that's a bit of an unfortunate notation. So this is a conic at delta, right, unlike the other deltas. So this tells me beta and K will have to be equal. And on the C system, gamma and K meet, so this tells me that gamma and K have to be equal. And well, if I do the sum over K, it tells me that the sum drops out, the delta drops out, and together with the sum over K, it tells me that beta has to be equal to gamma, right? Because beta has to be equal to K, K has to be equal to gamma, but it can take all possible values with the same weight. So up to this one over square root of D, all what this maximally entangled state tells me is that beta is equal to gamma. Or if you look at this picture, there's a maximally entangled state, well, which is exactly of this form. So its effect is that it exactly forces this spin and this spin to have the same value, right? So it exactly enforces that what I have here and I have here has the same value. So from that, we can, so what does this give us? So we have the sum here, we have A1I1, alpha beta. The beta index here is gone, right? We only have this because this was connected with a K, only this one remains. Then here we had a gamma originally, but this gamma is now equal to beta, so we put beta. And now that's not perfect. Actually, we would like to express it the same way as the original P, we want an inductive procedure. We want to write it like that, something mapping input indices to output indices. So we'll just take this guy and put it here and let me single out a sum over beta, okay? So I just collected these two A's here and I collected my ket and bra vectors. So what we see now is this map is very similar. It takes the auxiliary virtual degrees of freedom alpha and delta now, like the ones at the boundary, in that picture and maps it now to two physical spins. And what's the coefficient? The coefficient is given exactly by that expression here. So we can leave it like that, but we can also note that if we think of this guy as a matrix, so if we think of this as a matrix, with alpha and beta as a matrix indices, what is happening here is that we're carrying out a matrix multiplication, right? Because we're summing the right index of the left matrix with the left index of the right matrix. So what this really is, this is the product of these two matrices and then evaluated the matrix element alpha delta. So then what we see from that, if we just write it in this form, is that this total composition of this map P1, P2 is the omega, is really almost of the same form as before, just that instead of having a single matrix A, like we had here, this matrix A got replaced by the product of two of these A's for the two consecutive sides. So now you can immediately see that we can derive an iterative recipe from that, right? So we can iterate this construction. And what happens is that if then we have something like P1 times P2 times P3, applied to omega one, two times omega two, three, what we would get, for instance, would be something like that we have to multiply three of these guys, A1, I1, A2, I2, A3, I3, evaluated at some, let me call this beta again, element alpha beta. And now we have a product of three matrices and so on, right? So we can just keep doing that. The only question is what happens at the boundary? And what happens at the boundary? Well, if we have periodic boundary conditions, so if we keep doing that, we always have this alpha beta here. Now in the very last step, what will happen when we have periodic boundary conditions? So we do this a few times and we always have this alpha and beta here. Now in the last step when we close the boundaries, it means that we connect this alpha index with this beta index here, right? So we're not adding any new P, we're just adding a maximally entangled state acting on these two indices and again it will force them to be the same. So we have this matrix here and we need alpha and beta to be the same in sum, so that's a trace of that matrix. So what we then get is that in the last step, and that's what gives us the actual state, psi. So that's the expression we ultimately get, right? So we keep multiplying these matrices in the last step, the last entangled state amounts to taking the trace of these guys. So that's really the full expression for such a state on periodic boundary conditions where for each side, there is a set of D by D matrices. So for each side and each value of the spinata side, there exists a D by D matrix specifying basically how the state has to be constructed. And that's why it's called a matrix product state because well, the coefficients, the C, I, I, I, N can be obtained as matrix product states, as matrix products. So maybe just a brief comment on the question of, well, what happens here for open boundary conditions? Well, in some sense we can again do what I said earlier, there's no need to keep all these dimensions fixed, right? So this could be a D1 times D2 dimensional matrix and D2 times D3 dimensional matrix and so on, just in a way where you know you can multiply them. Obviously the left dimension of one has to be equal to the right dimension of the preceding one, but otherwise you can vary the dimensions. So for instance, you can think that open boundary conditions is just a special case where here, what you do is you choose the dimension equal to one, the last one, right? So that's a specific way of thinking what would happen for open boundaries. So what this would mean is that the left dimension of the leftmost A would be equal to one and the right one of the rightmost one, which basically would just mean that on the very left and the very right, you don't put matrices, you put vectors and that's why there's no need for a trace. So for open boundary conditions, we would have that A1, I1, N, A, N, I, N, are, N, A, N, I, N, are, what? Column or row vectors. Okay. So before I conclude, let me just very briefly say, well, we wanted kind of a succinct and efficient description of states with an area law. Is this description efficient? Well, how many parameters do we have in the simple case where all matrices have the same size? Well, we have N sides, we have D physical states and to each side and to each physical state, we need to specify a D by D matrix. So we have D square parameters. So in total, we have N times small d times capital D parameters. So in particular, it means the number of parameters grows only linearly in the system size as opposed to exponentially for a general vector. And of course, the relevant question is how big do we have to choose D to get a good description of our state? And one intuition is that D should be related to the entanglement in the state. So if there's not too much entanglement, hopefully we can choose D at a moderate size, meaning we only have a moderate number of parameters. Of course, this has to be made rigorous because there's no strict one-to-one correspondence between this D, if you want to describe a specific wave function and the number of parameters. But it indeed turns out there's kind of a nice relation which allows for an efficient description. And that's where we'll start continuing in the next lecture. Thanks.