So, first of all I thank the organizer for giving me the opportunity to talk. Not in Triestry, because that's quite normal. But to this audience that's quite different from the audience which I am talking about this thing. And indeed it is been asked me to give an overview of what is known, general, about entangohen entropy, entangohen in many body systems, from the subtitle that I put, is from Field Theory from Condes Matter. z bistvenim prijevama, kaj počutim, ki v tudi obriga na zelo, ne znovu, da sam sem tukaj. Zelo nedaj ta k 마ča, ne znovu je. To je forno k mentalna začeliš, prav Queen prej ne bo, ki nije vse prijevama, da sporo laje napravo, ki počutim pa, da je, da jefasto. Na vse jest do vrne, tudi, da bomo izgledati, da imamo spetanje, včasno, v Vincenco Alba, Maritio Fagotti in Derek Tone. Čak je to? Čak je to? Včasno, da bomo izgledati, da imamo načine, ki so nekaj, da in tudi, in tango in entrobi, zelo vsega vsega vsega. Vsega vsega vsega vsega vsega vsega vsega, zelo vsega vsega vsega, ki lahko se predetilo s kosmologijami, pa nekako se vidijo kosmologije, zato si je zelo v izgledu, da je mi je prišlišil.  Zelo je zelo obrobilo V 220 in 2015, in postupa v 2016 so dobro,hire that is about 500. I will try to motivate here, why has all this is interesting in comets matter in all this fields which motivated also the a very big workshop at KTP two years ago, and even as you know are ten million grans of Simon Foundation like the one for woodstrap we had yesterday, and I will start the story really from the zero point, ki najbolj zelo, je, da, se ki počke kot zelo, naredimo počke, da nosimo všetko, izgleda počke, da ne všetko ne več, tudi počke počke, kej Fermeon, bozon in bi pristop, da je tako zelo, zelo, da bozon na metalj, pri Fermeon, v superfluidu, počke bozon. Ne všetko všeč, je všeč superfluid, in ki je kako vzelo. Svega vzela je, da je bilo vzelo. Kako je učinila, kako pošla vzelo, da je bilo vzelo. Je to vzelo. Nesi je bilo vzelo, kako je vzelo, da je bilo vzelo. Svega vzela je, da je bilo vzelo. Spodnije, da je bilo našli. Da je bilo vzelo. so many different phases and so many different things in the world, among which we can... I made a list of things here, which goes from typical statistical physics object, like multinsulate or topological state, or high energy things, like confined phases in quantum chromodynamics, of which we had a lot in this talk. And the reason why I think this talk, even not talking about the normalization group at all, in this conference, is because all these very strange phases of matter of the interacting system have been tackled with the normalization group very successfully. The normalization group nowadays is one of the best tools to understand at least part of this thing. And what I will try to tell during this talk is that entanglement entropy can tell a lot again about all the different phases of matter in a way that I will enter in the next few slides. So, what is the problem in describing this interacting system? One cool thing just... OK, you have the Hamiltonian. We are not talking about using strong interaction or weak interaction. We just... Basically, with QED we could describe everything that we were interested in that slide. Actually, much less than QED. We just, for example, would like to know what happened to a system of spin that interacts. And still, we are not able to do this. One very easy approach would be just put on the computer, make your simulation and try to get results. But this is not possible, because the problem is too complex.  That means a chain where on each side there is a spin. If I want to describe a state of the system, I should write in the basis of the spin, which is, let's say, it's one-half spin, so it's just plus or minus one, the value of the spin. Plus or minus one-half, OK, I'll just plus or minus. But the basis, if I have n spin, obviously I need to specify two n coefficients in order to specify a state in the Hilbert space. And two to the n coefficient is a big number, when n start being already of the order of magnitude of those n's. Now, there is the most powerful computer that can make a sub-diagonization, which means calculate the value of this huge matrix for something like 30, 35 spins, not more than that, on a computer, not on this small computer here, on a big cluster, on the largest computer on Earth, we can at most manage something like 35 spin and 35 spin are very far from thermodynamic limits. One dimension 35 could be also already something, but you understand that in three dimension 35 it means nothing. It's three times three times three makes 27. It's too little to be stored on a computer. OK, so we cannot tackle this problem numerically. And it's clear that if we want to gain something in this game, what we could try to search is to find a criterion that said physical states apart from the other. This is a new thing that emerged in the last ten years, probably was not clear before, that not all this two to the n coefficient states are really important to describe a physical state. They describe states of the Hilbert space that will never be interested in. Physical state, in a sense that I will discuss later, could be different from the other. OK, and we want to understand what makes physical state different from the other. And I will try to push forward the idea that entanglement is, at least, it's a criterion, maybe it's not the only one, but it's a criterion that puts physical state different from all the other states of Hilbert space in which we will never be interested in. Let's see first of all what is this entanglement entropy. OK, now that I gave you an introduction, I'm still introducing you, just defining quantity, but for the people that never saw the quantity before, it's better to give proper introduction. So let's consider a quantum state, where it is, which means I mean a pure state psi. OK, from this pure state, I can construct, obviously, density matrix as the projector on this pure state. And I consider the Hilbert space where the system leaves, and I bipartite it, which means I divide in two parts. OK, as the tensor product of HA times HB, usually in quantum information language people say that there are two guys, one called Alice, one called Bob, which one has access to some degrees of freedom, the other one has access to other degrees of freedom, and they can make measurement only on one part of the system, but this is not at all important. OK, and in a general state, as we already know, just from basic quantum mechanics, the state shared between Alice and Bob is entangled, which means that the measurement can depend on what the other one does. This is encoded in this midde composition, which tells us that for each state there exist two bases, one in A, one in B, that are not just by psi here. OK, such that the state can be written as a superposition of these bases, with only one set of coefficient, a set of coefficient here C, and that depends only on one index, and not on two, like would be for an arbitrary basis. OK, the statement is not trivial, but it's quite easy theorem of linear algebra, probably most of you have seen in the first theorem of linear algebra, but OK, if not just, it's a theorem of 19th century, so just believe it. And, obviously, it's much easier to see that this coefficient can be chosen by rotating a phase to be positive, and there some must be one, because the initial state we were talking about is pure, just for normalization of the state. Now, what is the entanglement? If the state is a product state, it's not entangled. If a state is a product state, it means that there is only one coefficient that is equal to one and all the other are zero. This is a non entangled state. A state is entangled and being different from one. OK, so there is, for example, there could be a bell pairs with two states having one over square root of two. This is something that we have studied, but more coefficients in the submitted composition appear, more the state is entangled. OK, but we would like, being physicists, we would like to have a measurement that say not just yes or no, we would like this is more entangled than that. Intuition would say, for example, clearly this state is completely unentangled, but if all the coefficients are non zero and they are only equal, this would be a maximum entangled state with some intuition. So more the state is spread among all the parties and more the state is entangled. It's an intuitive picture, but we would like to have one number that measures this. Now, in quantum information theory, many years have been and are still working out measurement of entanglement. OK, so object that can measure the entanglement. There is nowadays a quite precise definition of a measurement of entanglement which is called entanglement monotone because obviously you cannot measure the entanglement in meter or in seconds. You cannot have an absolute measurement of entanglement. An entanglement monotone is an object that says if given an illber space and a big partition one state is more entangled than another. This in this sense is monotone. This can be the statement can be made rigorous, but what they say is that entanglement is a quantity cannot reuse under local operation in classical communication which translates in rigorous terms to quantity that is able to say if one state is more entangled than another. So it's a comparison measure. In this sense is a monotone. And there are many entanglement monotone in the literature. One that is very good and that I will consider probably is the only one that I will consider in this talk probably depending on where I will arrive is the entanglement entropy. Entanglement entropy is a quite natural object. It's defined in the following way. Let's take the reduced density matrix of the part A of the system. So by taking the trace over the B degrees of freedom this means is the reduced density matrix that Alice can access. So this is whatever Alice can do. And excluding whatever Bob can do. This is probably something that you have seen doing in the first course of statistical mechanics when you say in a subsystem and a bat take the trace over the bat. We are doing the same thing, but now I am not saying anywhere that the bat is much larger than the system and I am not saying anywhere. I have no other assumption. I trace out something that is of freedom. I define the entanglement entropy as the fornojman entropy corresponding to this reduced density matrix. Since you can write explicit expression of this entanglement entropy in the smith basis that we wrote up here and you can see just by linear algebra, probably you see immediately that in this basis the entanglement entropy is the expression minus cn squared, long cn squared. So cn squared are like probability of being in a state and obviously the fornojman entropy is given by this expression. But ok there are very few very important properties. This is the expression in smith basis that is important, but from the definition it is clear that entanglement entropy is basis independent. And this is something that as a physicist we like a lot from the quantum information perspective from the information perspective as a basis dependent something that they don't care, but many entanglement measurements in fact are on the basis, but as a physicist we like something that you don't need to go in one basis to calculate, but it is the same whatever basis I go and this we like. Ok, and there is another very important property that you can see going to the smith basis that depending on your cn squared you have that either you take the trace of A and make the over B and make the fornojman entropy or you take the trace over A defining some row B and making the fornojman entropy of B you will get exactly the same result because this object is completely symmetrical in A and B in the smith composition basis. So you have that the entanglement entropy of A is equal to the entanglement entropy of B independently of their volume. These two objects have very different volume, but they have the same value. So this entropy is not like the classical entropy we are talking about, that is an extensive object this cannot be extensive. And this was already at the basis of some totes in the 90s. In fact, based on the fact that it should be the same one can reason in a very bad way, it's not a good reasoning but the only thing that where is the point that A and B are sharing is just the area between them. So the idea that came in mind to people that start now to be correct but that should be taken really with a grain of salt is that the entanglement entropy would be proportional to the area that separates A and B. This is true grand state of a local Hamiltonian. Nowadays this was pointed out by Shredkin in 1993, probably even before by Bambelli et al. in the context of black hole entropy. And it's proved nowadays to some it's rigorously proved in one dimension it's proved in physical but not rigorously that it's it's true even in higher dimension that entanglement entropy satisfies an area low if the Hamiltonian is local and it's gapped. These are the conditions. In fact first this idea was proposed in 1993 and a year after to find a counter example that's what we all like to do in this conjecture. Also in our friend Wilczek calculated exactly what happened to the entanglement entropy in a one plus one dimension conformal field theory that will be a good part of my talk here showing that the entanglement entropy is equal to C over three logarithm of value. Who are this quantity entanglement here? In a one dimensional system you can be part of the system choosing as A one interval of a given length and B the rest. It's just a number, it's two. It's the number of boundary points between the system. If the area low will be satisfied entanglement entropy should not depend on L for large enough L. And this is in fact true for gapped system as it has been proved by many people and then proved rigorously by asking back in 2007. Instead for conformal field theory which is not gapped or in a more field theoretical language which is massless, so for a massive theory you have an area low. For a massless theory you don't have an area low and in conformal field theory you have these log L divergences with subsystem size and the prefectoria that you see C is the central charge of the theory that people in the odds I want just to comment without entering in any details that nowadays in this formula is the best, most effective way to our disposal of all physicists to measure the central charge. If you have a theory and you don't know the central charge just calculate this quantity which is very easy there will be no numerical problem in the sense that I want to discuss here but if you are someone that are interested in this theory don't try other methods that were popular in the past like skilling of the gap I just measure the entanglement entropy you will get the result. Why this is important now? I want just to make this to close the story of the introduction and then entering in something more technical like showing from how that formula C over 3 log L can be get. The importance is that if I imagine now this full Hilbert space with 2 to the end state ok and I pick up randomly a state whatever this means, this sentence I would expect on the basis of physical intuition that I will get a state that satisfies a volume low not an area low Now this statement that I am telling and that each of you if think properly should say yes it is correct it has been proved rigorously by a mathematical physicist obviously what you have to define so define a measure in the Hilbert space what does that mean taken a state rigorously obviously with some measurement this works very well but this you can make you can make the statement that I am telling rigorously ok and what it is that you have this huge 2 to the end state that is full of states that are volume low and only have small corner in this space satisfy the area low so if I am interested in knowing what is the ground state of a physical system which is what we are interested in knowing for example to solve the problem of ITC superconductivity we want to know if the ground state of the 2 dimensional above model can have this superconductor or not we are not interested in all the spectrum of the superconductor ok that will be the at least the first starting point to make this thing if I am interested in this I should not scan the full Hilbert space with my numerical code 2 to the end state I can just reduce to a very tiny fraction ok now if we can find a way to numerically reduce for example the search in this tiny fraction of state we are in we are in good shape to solve the problem and in fact one meaning of the entanglement entropy from information perspective is exactly that the entanglement entropy gives the amount of classical information I need to store to specify the state on a classical machine like this laptop ok if a state has very high entanglement entropy I need a lot of memory in my computer if a state has very low entanglement entropy like if it is a product state I just give to to give two elements of the basis ok so this is quite intuitive but also this information perspective can be done can be made more serious than the word I am telling ok so you understand why all the circle closes that being the entanglement entropy related to the amount of information I need to store classically a quantum system obviously if the entanglement entropy is low for a physical state that I am interested in I can try to make an algorithm that is low state this is not just a dream nowadays there are these algorithms based on tetson network which is a very new and powerful set new it is not so new nowadays but probably to many people it is not known yet numerical based on the entanglement contents of system ok that carry very strange name like MPS which means math explore state entanglement error my position project entanglement state so it is just a lot of acronyms here in fact so will such as say that we are in front of an alphabet super proposals ok but independently of what this name says these are states which naturally encode these encode the area low in one dimension it is the only one that I will try to explain so these object encode the divergence of the entanglement entropy from the structure you see 3 is like the normalization 3 these object encode the peps the area low in 2D so it is good to make two dimensional system because of their shape and ok they are quite good people are using actually the one dimensional we are using back from 92 without knowing it in fact these mathical states have been understood only later in 2000 something that this famous density matrix organization group that is a numerical argument that many of you could have heard about works because the entanglement is low and because can be written in terms of magic process state what is a magic process state? I want just to be quickly this maybe most of you are not interested but it is important to know what is going on in the world ok you write the famous coefficient that were 2 to the n I am writing randomly I write this trace of some matrices here ok one for each one for each side this one to n and one for plus and minus depending on the basis ok I didn't make anything by writing this if the matrix is big enough I am ok indeed let's assuming that the system of variance all these matrix are the same and I should just put a to the n and let's do this so I have to specify one matrix to specify the state but for a genetic state if I want to describe these matrix should be 2 to the n times 2 to the n that means I didn't gain anything ok but let's say instead of taking a 2 to the n matrix I take a matrix of dimension chi ok that's what people do chi is a number and in my matrix of chi times chi you understand that in greasing chi more and more entango can be stored in this state actually by a simple counting of states you see that the maximum entango in entropy you can store in nps like this is logarithm of chi ok so obviously if this chi is 2 to the n the maximum entango in entropy that you can store is n and this is the maximum entropy possible so you are ok you can surely define anything but the point is who makes numerical method fix chi to be some number 3, 10, 10, 20, 100 then takes this state as an ansatz like a variational ansatz minimize the energy in this multiparameter space using the same, et cetera this is more complicated which best describe the state this is what is a matrix for the state and the famous and known dmr density matrix normalization group that you could have heard is just a practical way to find the variational meaning of this problem this has been just written down in all mathematical details the two things are completely equivalent so there is a famous review by willis worldwork that is called density matrix the normalization group in the times of matrix of matrix protostate and you can find all the proper derivation of what I am writing here but now let's go to the physical state in one dimension the area is a number as we have seen before so the entango in entropy must be constant so a given matrix protostate with a finite chi can describe it so if I arrive to chi I try with chi equals 3, 4, 5, 10 at some point I will see that the energy I am calculating does not change anymore so I converge and I am at right chi to stop my simulation if the state in state is critical we have seen that it grows like log L the entango in entropy the chi should grow algebraically and having putting on a computer an algebraic growth as people in information theory say that polynomial growth of the complexity is very easy if you ever run something you know that you can do what you cannot do is an exponential but power law you can deal with with the better and better computer you can do with my stupid laptop you can make simulation up to extreme precise simulation up to system sizes of thousands without problem which will be impossible as I was telling in the most powerful computer in the world in higher dimension what happened the error law says that entango in entropy grows n to the d minus 1 so the chi that you should put is exponentially big still this is not good in fact the matrix protostate does not work but still it is a reduction of the problem before we had exponential of n to the d now we have exponential of n to d minus 1 we went down we reduced the complexity of the problem exponentially it's already something but again as I was telling if you go to this state here you can reduce even more the complexity of the problem and that's what are doing people in 2d in 3d there are still not concrete and working algorithm ok, the expert on these are working and probably in a few years I will say there will be there will be some also in this direction and by the way there is another simon foundation program exactly to finance this thing for fermions so it seems that all the money of some foundation are giving to one of the topics covered by this conference so that's good up to now it has been an introduction just to give a motivation why to do these things I hope I motivate well enough why to do things but at some point we should start doing things not only saying why and ok, I will give my own perspective on how to do things which is how we can calculate this entanglement entropy and find these nice results that are at the basis of what I have been telling plus many other developments that some of them I will try to explain some I will not have time to explain as field theorist here we are all field theorist so I am really easy to go on with what we should do is trying to define this entanglement entropy integral and that's what we did back many years ago in 2004 and ok being sure that you are all field theorist I think I can go relatively fast on this so what is the density matrix in pat integral formalism it's just the pat integral on a slab let's say at fine temperature beta it's the pat integral on a slab with a given equation action ok, the slab is with beta inverse temperature and the element of the density matrix is given just by the boundary that on the boundary of this club that define the element of the matrix that here I am calling phi 1 and phi 2 ok, so the pat integral it just means the pat integral over the clinical range with boundary condition that field is equal to phi 2 tau equal to imaginary time equal to 0 and is equal to phi 1 at imaginary time equal to beta ok, what is the effect of the trace the effect of the trace is trace means take the argument and sum over there in a continuum theory this amount to take the cylinder fold take the slab, fold to form a cylinder of circumference beta and sum over this diagonal element you may have heard you all have heard that the partition function of a system is just pat integral on a cylinder at fine temperature most of the time this is shown in filthy lecture by Matsubara frequencies are crazy difficult things this is a nice way to make things wringles but it is much easier just to go in a pat integral and make explaining what is a trace for some reason this thing is not found too much on the textbook but in some textbook is there but not in all of them it is much more intuitive than making Matsubara and it helps in understanding many other things what is the reduced density matrix in in this pat integral formula what I should do, I should take this slab I should take the trace only in part of the system which means while I am closing to this slab to form a cylinder I should sum on some part of the system the B part but I should not sum on the part A of the system because I am not tracing over there so I take this slab here I take this slab I close to form a cylinder but I leave some open cuts on the part A of the system you have the part A in this plot for some reasons I put three different intervals so it means this is B where I close to form a cylinder and this I left open this is a reduced density matrix this is the element of the reduced density matrix ok now from this density matrix I want to make the entanglement so take minus trace of rho a log rho a taking a log of an operator is some difficult object difficult operation that all the people here working with this order system should know very well that it is difficult to make the log of an operator but always in the from this order system we know that there is a way out through this to make this average of the log8 which is using a replica trick minus trace of rho a to the a minus trace of rho a log rho a is formate the same as taking the minus the limit for end that goes to one of the real thing with respect to end of trace of rho a to the end this is just a trivial algebraic expression just write on the again values of rho a and check that this object is correct ok but now what is the simplicity that when the end is integer and only when end is integer trace of rho a to the end is a simple object is a simple object because is just taking end of the cylinder that I plot before sew them together in a cyclical fashion because I have to make the trace by column matrix operation I mean making a cyclical operation ok and if I do this you will realize just by drawing that this is nothing but the partition function on an n-sheeted reman surface because I take end of this cylinder I couple them together let's take the limit that beta goes to infinity cylinder become a plane so I have end of this cylinder each sheet here is the reduced density matrix then I should identify the diagonal element the element of one matrix with the element of the next one because ok, just let's be clear the trace for example of rho a cubed is just rho i j, rho j k rho k i ok, so I should identify j with k which means that the first here should be equal to the the like rho element of this sheet should be equivalent to the column element of this one rho 1 of this should be equal to the column of this one and basically of the trace the first one should be equal to the last one ok, that's what I'm drawing here so this is trace of rho a to the n which it's a partition function and as you know calculating partition function in free theories is easy calculating operator is difficult but it's easier than calculating horrible object like this and that's why things will become easier and become even easier in a conformal field theory when a Riemann surface is a natural object a Riemann surface can be obtained by conformal mapping from the plane that's the fact I will use and this is what help us to make things even easier in conformal field theory where you can use conformal mapping to make everything you like ok, on passing and this is important that not only this replica trick gives the entanglement entropy as the finalist for Riemann entropy but just even for any integer this object trace of rho a to the n define in this way 1 divided 1 minus n log item of trace of rho a to the n are called rainy entropy of order n and these are very interesting objects for two different reasons the first reason is that they are entanglement monoton also so they are very good measurement of entanglement from the information perspective so if you are more from an information perspective you are just satisfied with n equal to 3, 4, you don't want to make this limit n equal to 1 after you are satisfied with one of them from a physical point of view from more field theory and statistical physical point of view you could prefer still the entanglement entropy ok, I just mention on passing why and the reason is that for example, the von Neumann entropy satisfies sub-subdivitivity while this object here does not satisfy sub-divitivity that's not very nice and we will see later why also we prefer the entanglement entropy but the rainy entropy is a very important thing that it became measurable just last year so it's something that can be measured in experiment which is very surprising for how old field went ok this being said, let's see this trick I have been doing the entanglement entropy defined as the replica limit of the partition function on an unsheated surface is independent of dimension theory lattice or no lattice this is just a general statement it's based just on sum stuff together as now I draw things in one dimension because I can draw it imagine you have either dimensional object connected in higher dimensional way and this you can do up to this point is valid for any system and it's used by many people for any system in most of the cases it's the only way to access the entanglement entropy then what I will do after I will start to specify to something that I know better but keep in mind up to this point this is valid for anything and there is no assumption on nothing what I want to show you is in one slide how you can get this famous silver trilogel and it's very easy actually if you know a bit of CFT what you have to do is to take this entanglement surface where is the guy this entanglement surface you try to map to the plane but this is very easy in complex analysis you can imagine that you just take this reman surface with a branch cut between u and v first of all you map the branch cut between minus infinity and zero and this you made with a simple mobius of bilinear transformation I don't know how you call this transformation here which map the w plane in this zeta plane like w minus u divide w minus v if you don't remember this from your complex analysis just notice that when w is equal to u you go to infinity and what map is take the branch cut and you put in the negative real axis then you have this surface here what does mean this surface that I start going around this branch point and every time that I cut the branch cut I go on the other sheet of the reman surface up to n times and after n times I'm back to the original point I cannot believe that there is one person in this room that didn't see this object and this is just the surface in which is defined in complex analysis the root of order n so you must have seen it if you know the square root and this object is uniformized on the complex plane just by the root of order 1 over n so I take just zeta through 1 over n and I get into a complex plane a complex plane in which each corner of opening angle 2 pi over n was one of the sheet on the original surface each of this sheet it's mapped in a corner of opening angle 2 pi over n and sheet make 2 pi ok, when you have this transformation what you have to do to get the trace of rhoe to the n is just perform this transformation for the stress energy tensor where one can use the famous formula of Bella Vipolakova examinological to relate the stress energy tensor to the plane to the stress energy tensor in this new surface one make this algebra that is extremely easy and I don't show just make first, second and third derivative of this transformation this stress energy tensor gives you the reaction to scale transformation as Slava was explaining yesterday of the system so which means this is nothing but derivative with respect to l of trace of rhoe to the n this is the partition function ok, then you can integrate this equation and extremely just elementary algebra you get the trace of rhoe to the n is equal to some constant that you cannot determine times u minus v to this power minus c over 6 and minus 1 over n c is the central charge now you put u minus v equal to l and you get that the entanglement by the replica limit is just c over 3 log n ok and this is a very easy way to get calculation if there is anyone in the audience that no conformal field theory just understood the logic of my calculation without showing details because it's extremely easy for the people that don't know conformal field theory probably even showing the details that's why I didn't write that one important thing to mention is that trace of rhoe to the n is equivalent by this construction to the two-point function of some operators which in the literature called branch point is filled and this is extremely useful for making many interesting things that I will not have the time to do nowadays this is nice but for many years I think in a very not fair way people have been saying ok but what you are doing, you not me your community is not measurable it's just uninteresting because it's not measurable I disagree on this statement but it's very interesting one of them I always say is the way function we cannot measure the way function but it's not a reason why not to calculate it and I think on the entanglement entropy there is a similar stories but it will be too long to go into all the details but what turned out last year in 2015 there was this interesting paper by the group of Marcus Grein at Harvard which measured the entanglement entropy in a cold-draffum experiment entanglement entropy as the fundamental entropy but he measured the rainy entropy for n equal to 2 and how he did that he construct the replica he made two copies of his cold-draffum system he coupled in a way to mimic what I was explaining obviously in real time it cannot make imaginary time but ok if we made this replica construction back in 2004 then some people worked out what is the equivalent in real time it's a bit of thinking with the operator but it's doable so you just make two copies identical of the same system coupled them in a proper way it can do for a relatively small number of stuff that's what he's done here and then from that measure the free energy in this system that gets the rainy entropy that he can do and he gets these results for example for this system which is a Bose Abarth model matter really much what it is Bose Abarth model obviously this was the first experiment he made the system of six pins no, here it's more than six actually this one is one, two, three, four five, six, seven, eight spins eight atoms, sorry eight sides and it be partied in several different ways ok, that's what he can do but they can do you can worry but eight is a small number and ok the important thing is a proof of concept this experiment is doable and things are scalable, oh only things are scalable so are interesting so what he did he made this experiment and what he found nice is that this system was known since ages to have a multinsulating phase and a superfluid the entanglement entropy you see that is constant here because it's increasing subsystem size because it's up, then when you pass the transition you go into a fluid phase which is a conformal theory and you see that it grows here with system size and if you put this continuous line at the theoretical numerical prediction for this small system and the points with the error bars are the experimental data for several division for several things that he did everything agrees between theory and experiment and so the entanglement entropy is measurable that's extremely interesting so it's not anymore something for us the rotation to play but it's something that can be measured and in a not too difficult way ok, since there is not much time I prefer to skip to this more difficult problem that was, what happened when you have more this joint interval it's nice to see that in this case we have Riemann surfaces with non-genous surface and this makes the calculation much more complicated you end up in objects like these that are very nice especially if some of you has to do comes or knows string theories of the early days conformal field theory on higher genus string on surface towards the everyday meals of these people and ok, what I can tell is that with this new all these, all the results of the early days of conformal field theory found an application that actually is measured, so I'm going in the wrong direction found an application that is even checkable on on your computer and nowadays probably also on the experiment ok I won't just to advertise quickly that there is a booklet of reviews that we wrote down in 2009, it's not up to date so many things up in these 7 years that it's impossible that they were all here but still there are the main theoretical tools to tackle the problem were really present at the time so if you are interested not in the growth of the result that this is probably outdated but if you are interested in how to make a replica trick and any continuation and all these kind of things you can find a series of reviews here where you can find many things and I want to mention that there have been many many further developments of these like it became a standard theory a standard way to detect criticality in general it's a way to detect topological order ok, even through the entanglement spectrum which became a very popular tool but I don't want to enter in this and there are C-theorem analogos of entanglement entropy that's a very popular thing in the energy community there is this F-theorem you can have heard in many places there is a geographic prescription for the entanglement entropy another field ok, what I wanted to discuss is two developments this one I will not have time so I will just explain to you what entanglement entropy help us to understand the non-equilibrium processes and this is very important because we are out of time what I am doing is to consider one of this many-body system out of equilibrium now, not anymore in the ground state of the system where there is an aerial law I increase the I take a system in a given state that I call psi-0 so prepare a system in a many-body state it's not an against state let this evolve according to an amytoin a different one psi-t e to the minus e ht psi-0 the question that we wonder is does it exist a stationary state in which sense exist a stationary state first of all psi-t it's obviously a pure state so we cannot have termitization but still we won't termitization we expect termitization, there should be a way to get termitization and the way out to this paradox is again through the reduced density matrix as so natural entanglement emerges because while psi-t is a time-dependent pure state the reduced density matrix rhoe of t equal to trace of b of rho t is not anymore a pure object is a mixed state and I will say that a stationary state exist if taking an infinite before the infinite volume limit for the full system and only after the infinite time limit there is a limit of this reduced density matrix that I call rhoe infinity if this limit exist we define a stationary state to exist and this can be proved in some cases to be the cases now this rhoe infinity is equal to reduced density matrix that you get from a thermal state then I will say that the system thermalize clear the logic? so it's not the full system that thermalize any subsystem thermalize so for all practical purposes the system thermalize but I don't enter in the paradox that the system is pure and cannot be thermal so any final subsystem of an infinite system is thermal what thermalization means but explaining why all this will be another story talk I want to continue to talk about entanglement entropy I make a very short long story very short telling that entanglement entropy just if you make the calculation on Riemann surface you play you make many of these things you get the entanglement entropy gross linearly up to time equal to l divided by 2 and then it became asymptotic ok so what you get is that first gross linearly and then subsystem there is this horizon effect which is ok I don't want to discuss this other thing which can be understood in terms of quasi particle picture which is very nice in the sense that the initial state emit pairs of quasi particle which move in opposite direction with the velocity vk quasi particle emitted from the same point entanglement entangled while particle emitted from different points don't know about each other there is a light cone only point separation l become entangled and even correlated when left and right movers originate from the same point reach them if there are not particles reaching this two point this two point cannot be entangled either correlated and if particle move all at the same speed entanglement and correlation are frozen afterward so that's why in conformal field theory you go back in this way you have this plateau straight because the particle move all at the same speed in a real system you have slow positive particle still arrive and don't make abrupt ok I'm done almost so so clear this argument because it's very important hopefully yes now if you understand this argument you can say first of all is it true that correlation are frozen for times smaller than that this has been measured in experiment numerical simulation that show this that was the case numerics we don't care too much nowadays we won't experiment ok there was this famous experiment by the group of manual block in Munich in 2012 where it showed that the light cone spreading of correlation was indeed correct as it's here but what does it mean for the entanglement I want to go to the entanglement not on the correlation if I assume this quasi particle picture etcetera I would say that the entanglement is proportional of the interval of length L proportional to number of pairs of particles that emitted from arbitrary points one reach A and one reach B ok in formula the entanglement is equal to the integral of all the particles emitted with some probability Fp that then from many point X in all the system minus infinity to infinity that from X travel at speed Vp one with positive and one with negative speed and one is in X prime and one in X second where X prime is in A and one is in B ok that's what I'm telling I'm putting in formula in a very easy formula what I've said even by word this integral is an integral of delta function you don't need to be very you don't need even mathematical to integrate it and get this very simple result ok which tells you that the entanglement for short time which means for T smaller the velocity if it exist the entanglement just grows linearly with time because this quantity is exactly zero then for larger time both of the integral contribute if the velocity is not all equal ok but for very large time this piece becomes zero and the start becomes extensive ok so this formula in a very easy way encode all everything I have been saying this has been checked explicitly in calculation I don't want to show I want to go to the interpretation of this interpretation is that the extensive value that we get at infinite time of for the entanglement entropy ok I have been starting by saying entanglement entropy is not thermodynamic entropy something else completely different but in a non-equilibrium process after we make the time evolution ok a time equal to infinity we get an entanglement entropy it is extensive it is extensive in the subsystem of an infinite system so we have no paradox s a equal s b I am talking about a finite system in an infinite subsystem so the other part will not be true because obviously it is infinite ok so it is the thermodynamic entropy of the mixed state just because in a thermodynamic system the entropy must be extensive by definition of thermodynamic the fundamental entropy of this trace of rho infinity is just extensive so it is the same wherever I put it this is what extensively means and should be the same of the full mixed state from which I started so when I make this non-equilibrium evolution the entanglement entropy become the the thermodynamic entropy and this is understood in more complicated situations that I want to discuss but not only in more complicated situations have been measuring experiments again this is another experiment published just not even one month ago in science by the group of Markus Greiner in Harvard where he repeated the experiment that we were talking before of coupling the replicas but in a lot of equilibrium what he is telling he makes this very intuitive picture and if this is the full system while time evolves there is entanglement created by many pairs of particles and this thing is what gives the mixedness to the subsystem that this is the picture but don't think to picture we repeat this experiment you make the replicas as you see in a very different way you make measurement and you got this you got the first the entropy increase and more or less saturates why does not perfectly saturates because obviously he has a finite system as a science system of like of 16 so you see revival effects you cannot have perfect saturation but you see that already with 16 you see this linear increase of the entropy and followed by saturation and this is very it's quite amazing that this is measured in experiments showing how thermalization entanglement so how we get thermodynamics out of entanglement this can be used even as a tool but I don't want to discuss this I will just to conclude the conclusion are just some the core messages entanglement is a very useful concept one important things and I gave only the example of the central charge because I skipped on many more complicated things but there are more complicated things and codes for example all universal properties of one dimensional many body system something that to people making RG should be very interesting ok and from the universal point of view this is also very interesting conceptually because all these is in the vacuum I only need to know the ground state of the theory I don't need to know the spectrum I don't need to calculate other things in the ground state of the theory in the universal information about the theory if you are about many body organizations this is a completely different story because there is nothing universal we are talking about universality and you just need to know the vacuum which is not vacuum at all from the conceptual point of view is plenty of information and all information is there as I was telling it's a tool to design better performing numeric algorithms this for many practical purpose will be more than in after going on it provides a very natural mechanism for thermalization which was not taught before all these ideas people are thinking about thermalization since the time of Von Neumach in 1929 he started by saying what is the thermalization of a quantum system and I should say that this mechanism despite of almost 90 years now of working on the field was not taught before and there is many other things as I was telling you that I would like to say but already now I am out of time so I just stop here


and traders and residents of Agura Buribe in Ikorodu Local Government area of Laker State, a second government intervention after a downpour three days ago ravaged the entire community, leading to a shutdown of businesses. Some traders and residents at the entire road in the area from wall posts down to inner parts of the communities in Agura Buribe had become impassable for motorists. The victims were lamenting their situations that the rains had left many businesses in Rhin and several shops shut. They are that all the residents and businesses who called no longer cope with the disruption being caused by floods in the area were seeking accommodation elsewhere. Welcome to Agura Buribe, we are here today to talk about the I'm sure that's the only way. I'm alright with everyone. I'm alright with everyone.


What's up, everyone? It's your girl, Kenzie here from Layton Sports Cards, and I'm ripping a personal for Vincent Z. We've got a 2022 Panini Donores Optic Choice Baseball Locks. Alrighty, Vincent. Best of luck. Here we go. Oh, did someone ask? Alright, if someone asked a question. I'm not seeing Sean's. Yep, we offer free worldwide shipping. International shipping. You're just subject to any taxes, tariffs, or import fees that your country may charge. But free shipping worldwide, yes. Thanks, Steve. Still don't see that comment if they asked. We have a hollow of Kyle Hendricks. Diamond King's Hollow. Corbin Burns. And we've got our snakeskin. We've got 40 Rays by Dalbruhan. 51 of 99. Green Connor Wong. Red Sox. Man, Red Sox have been coming out this morning. Nine of 99 there. We've got our candy cane. Boba Shet. There you go. Blue Jays. 33 out of 46. Boba Shet. And another candy cane here of Trevor Rogers. Marlins. 5 out of 46. Autos. Oh, wait. No, that's not an auto. Snuck one behind there. Nicoladolo on the Rated Rookie for the Reds. 8 out of 99. Thought we had an extra auto for a second. Alright, auto time. We've got Alejo Lopez Rated Rookie on the Blue for the Reds. Reds Blue. 38 out of 99. Then we've got Purple Rated Rookie of Tyler McGill for the Mets. Oh my gosh. What is going on this morning? Yeah, that's the next one. 77 out of 88. What is going on this morning? Vincent! How about a green? Gosh, this is banana. 67 out of 88. Wow. There you go, Vincent. We'll mag this Franco for you. Gosh, what is going on this morning? Love it. Congrats again, Vincent. Thanks so much for hopping in. We'll get these right out to you.


the great Mansa Musa. First, Mansa is not his name. Mansa is his title. It means king. King Musa ruled over the Mali Empire and the great city of Timbuktu for 25 years during the 14th century. Mansa Musa, a black man of West Africa, to this date is recognized as the richest man to ever live. His net worth in 24 karat gold has been valued at over 400 billion bucks. He's worth more than Elon Musk, Bill Gates, Michael Jordan, and LeBron James put together. He moved an entourage of over 60,000 men, 800 camels on a 4,000 mile journey traveling through West Africa on foot crossing through the burning sands of the hot Sahara desert resting in Cairo Egypt while on his way to the holy city of Mecca to fulfill one of the five pillars of his faith, Mansa Musa of Mali, the generous black king of Africa.


What's shaking, booktube? My name's Cam. Welcome back to another video. Today's video isn't actually about me. Devastating. I know. So I'm gonna keep it brief, but I just wanted to kind of reflect a bit on when I started out on booktube and authortube. When I started out, I was lucky enough to make a few friends and get a few shoutouts that kind of boosted me. In fact, I imagine a lot of you watching may have found me through other people. I was lucky enough for that to happen, and now I'm lucky enough to have a platform of like over 11,000 subscribers, which is still just crazy to me.  But rather than me just telling you why I think they're great, I think I'm gonna let them use my platform here to introduce themselves. I'm not telling you to go subscribe, but at least hear them out, maybe check out their channels, and I think that much will say a lot more than I ever could. Take it away, guys. Talk about bookish videos. I am going to be doing a little bit of stuff on the side where I talk about some Asian dramas I watch and Star Trek and things like that. Most of the bookish stuff I talk about has to do with the types of genres I like to read, which is anything historical fiction, nonfiction, biographies. I also love fantasy, YA, science fiction, stuff along those lines. I do talk about Bahamian culture, language, and history, and those videos will be on my channel as well. If you like, check it out. Bye. Hi, everyone. My name is Ben from Ben Reads Books, and I'm a sci-fi and fantasy writer who also has a booktube channel. Normally what you'll see on the channel is reviews from sci-fi and fantasy with a little bit of a literary fiction thrown in. The reason why is because I want to talk about all the books that have helped me and influenced me as a writer and has really, really developed my craft along the way. Within the channel, you'll be able to find reviews from different genres, writing tips, and me documenting my journey as I try to build the ultimate goal of owning a publishing house. And I want you to see every single step of the way, whether it be me taking editorial classes, talking to different people in the industry, I want you to see it all. So, this is my channel, and I'm Ben from Ben Reads Books, and I hope you have a fantastic day. And if you're interested, take a look. I hope to see you soon. Bye for now. Stick around, and something wonderful might happen. Or something awful. Hi, I'm Olivia here from Olivia's Catastrophe. I love to talk about books and travel on my channel. I talk about pretty much every kind of genre. I love to read all sorts of genres. I love to read all age audiences going from middle grade to young adult to adult. So you'll definitely see a conundrum of things. Usually I like to travel as much as I can and do reading vlogs while I travel. I've started a new series on my channel, which is called Read It Backwards, where I read some of my favorite series backwards and tell you about my experience doing so. And we just love to talk books and make sure that we are having a lot of fun doing so over there. So I hope to see you there. Hey everyone, I'm Zoe from Zoe's All Booked, which used to be called Your Booking Your Zoe. My favorite genres are adult fantasy and adult romance, but really I'll read anything that's not horror or nonfiction.  Nonfiction puts me right to sleep. On my channel we like to have a lot of fun, a lot of shenanigans. I have a cat, a dog, and a two-month-old that pop up in my videos often. Recently I brought back what I like to call dramatic readings, which are exactly what they sound like. I dramatically read the synopsis of a book and make a complete fool out of myself for the enjoyment of the wider world. There's the cat. I like to keep it real on my channel. I just filmed a video in a onesie, so if you're into that and you're into a lot of swearing, come check out my channel and say hi to Draco. Bubsy, can you say hi? He hates me, it's fine. I don't have a cat hair all in my mouth now. The novel I hear from her is Eva Sanders. I am a book blogger and host of an author interviewing Instagram live series, as well as a freelance cinematic style book trailer creator for authors all around the world. I post my writing webinars on my YouTube channel as well as my author interviews, the book trailers I've created, and other writing tips. I also am going to be documenting my self-publishing journey as a debut self-publishing author of a young adult thriller named The Beckwith Brothers. The Beckwith Brothers is a young adult thriller about teen angst, forbidden romance, secret societies, and murder all intertwining in three mysterious brothers return home to their small town, Halloween night 1995. So be sure to subscribe, add the novelette, see these videos, and more. Thank you all so much.  I like, I write middle grade, so that's why I read middle grade. That's why you will see middle grade. Anyways, I like fantasy, mystery, thriller, rom-com, so it's a big variety. I chat it up, I make cool little short films, fun little videos, so it's a big variety. Come onto my channel. It's fun. So, hello all. My name is Aphrodite Lee, and I'm an author. While even now, saying it out loud is still unreal to me, but it is true. I wrote a book. Wait, wait, I wrote two books. No. I wrote dozens of books, but I published two books. There you go. And so I can't wait to share like the rest of them with you all. My main concentration is New Adult Paranormal Fantasy. That's how I roll. Now, of course, if you are familiar with the two books that I've already published, you know I also delve into other territories. But my main jam is fantasy. That's just, I'm a fantasy girl. On my channel, I mainly share my process, and I try to have fun while doing it. I also do a lot of silly stuff because for some reason I'm highly addicted to tags. Every time I see a new tag, I'm like on it. I need therapy. I do. I realize this. The reason I started my channel though, because I wanted to like track my process and then also share my process at the same time. And I hoped that anything that I did would inspire other people in some kind of way, no matter what. Just looking at my madness and seeing like, hey, well, I can do this if she did it kind of thing. One of my deepest goals has always been to be the person that I need it. So I hope that that's what comes across the most on my channel. Also, writing is lonely. Writing is hard. And I wanted to find people who were like me, who were doing the same thing so that I could have like some buddies. So I started it for selfish and unselfish reasons. I love being a part of altitude. I love it. It literally changed my life. It changed my life. So if you want to learn more about me, come to my channel and hang out for a little while. I would love to meet you. Until then, remember this, you hold the power as long as you are driven. Hey, everyone, my name is Aspacia, but you might know me better as Asparagus is reading. My channel is mostly about books, but you might find a few videos I'll talk about other things such as bullet journaling, college, plants, video games like Sims and Halo, or things about my dog or really just anything that I'm interested at the moment. I am currently uploading every Friday at 10 a.m. Eastern Central Time. I do occasionally upload on other days such as Monday and Wednesday if I have any vlogs or like other videos I need to get out at a certain time. I definitely think that you all should check out some of my videos because there's been some wild stuff happening, some changes going on. There's been some laughs, some jokes, some new things that I'm planning for the future. I think it'll be a lot of fun for me and also anyone that is watching and participating. So if you're interested in that, go ahead and check it out.  I know my channel is pretty small. I just recently hit foreign subscribers, but I do hope that if I continue with this, you know, like new ideas, staying motivated, uploading weekly, having fun, doing all different types of things, communicating with all of you guys. I definitely think there is an opportunity for growth and to have more fun and just to get to other places with it. So if you guys have any ideas or anything at all or just want to hang out, please feel free to do so. And yeah, thank you guys for watching this clip and supporting all these wonderful YouTubers. We all definitely deserve it. And I will see you guys hopefully in another video. Bye. Hi, Cam. Thank you so much for having me. Thank you to everyone who's watching this video and giving me a little bit of your time. I appreciate you. My name is Tamara Woods and I talk about books and writing on my channel, which is named Tamara Woods and very creative. One of the things that I really like to do with my channel is to figure out ways to connect with the community. Each week I host a live stream right in on my channel every Thursday, 2pm Eastern, and we write stuff and we chat and we have a good time. On a monthly basis, I do writer's workshop where myself and a group of friends, we talk about a book that the Goodreads group picks. It's a nonfiction writing book. And if you were looking forward to going to some writing events and they've been canceled because the beverage bug has destroyed us in 2020, I have a free offer for you. My friends, Cara Brown, Cache Warren, Dana Gallant and myself created Evergreen Writing Oasis. We wanted to create something that's free, something that was really great for the community. We have so many people who are going to be involved. There's going to be panels and workshops. And again, it's free. So I'm not sure when this is going to air, but the event is the weekend of July 17th. Again, Cam, thank you so much for having me here. I really appreciate you taking the time to reach out to all of us. And I hope that there's some people who are watching this who will stop by and say hi over on my channel. Bye.


QuickBooks Desktop 2023 statement of cash flows. Let's do it within 2-its. QuickBooks Desktop 2023. Support accounting instruction by clicking the link below, giving you a free month membership to all of the content on our website, broken out by category, further broken out by course. Each course then organized in a logical, reasonable fashion, making it much more easy to find what you need than can be done on a YouTube page. We also include added resources, such as Excel practice problems, PDF files, and more like QuickBooks backup files when applicable. So once again, click the link below for a free month membership to our website and all the content on it. Here we are in QuickBooks Desktop, sample rock castle construction practice file provided by QuickBooks going through the set of process we do every time maximize in the homepage to the gray area. The view dropdown, noting we have the height icon bar and open windows list checked off, open windows on the left hand side open, then go into the reports dropdown, company and financial, opening up the P and L profit and loss range and the change in 010124 to 123124 January to December. Customize it, fonts change for 12, okay, yes, okay. Then we're gonna go to the reports dropdown again, company and financial this time, the big balance sheet. We're gonna change the dates this time with the dropdown looking at this fiscal year, which is 123124, customize that report, fonts and the numbers to change it to 12, okay, yes, okay. There's the set of process we do every time. Looking at the major financial statement reports that being the balance sheet and the income statement. This time we're gonna be looking at the statement of cash flows, which is another major financial statement report. And you might have been asking as we've been going through our practice problems, why do you just open up the balance sheet in the income statement or profit and loss and you neglect the other main financial statement report of the statement of cash flows. That's because the balance sheet and the income statement are generally the reports that we think of constructing first on whatever basis we are using, generally the accrual basis. But remember, you may still be on a kind of like a cash basis depending on the type of industry you are in and how you're entering data into the system. Let's recap that because it will be important when we look at the statement of cash flows and why we have the statement of cash flows, noting that if we just look at the revenue cycle or customer cycle, if you had a system where you're waiting till something clears the bank, for example, using bank feeds before you record it as income, you are in essence on a cash-based type of system. If you're just entering the create sales receipts, say you have a register, possibly in a food truck type of situation, you're still in essence on a cash-based system because you're recording the revenue at the same point in time that you're getting paid. If however you're in the kind of business that you need to bill someone or invoice them for work done in the past, expecting payment in the future, then you're on an accrual-based system when you're using, in essence, this invoice. No matter what system you are using, we're gonna use these forms to construct the balance sheet and the income statement as our main two reports. Now the statement of cash flows then is gonna usually take what we have already constructed on the balance sheet and income statement and redesign it so that it's on a cash flow type of basis. So the easiest way to kind of think of the statement of cash flows is to compare it to our performance report, which is going to be the profit and loss report, which again is in essence on an accrual basis if we're using accrual components, such as invoices, when we're entering the data into the system, and then try to think about, well, what if we were to convert the income statement, the performance report to a cash-based system? That's in essence kind of what we're doing with the statement of cash flows, attempting to get the best of both worlds, the better comparability that we get with an accrual-based system, allowing us to be comparing month to month, year to year, without the kind of distortions that go into place when you have simply a cash-based system, when it's not appropriate.  So the comparability gets kind of messed up. If we use the accrual concept, we would put it on the books as an asset and then depreciate it over when we use that item, and that would make the comparability on a performance standpoint better. However, we also wanna see the cash flow because the cash flow is gonna be important as well. That's why we have the statement of cash flows. So from a construction standpoint, if you were to build the financial statements, for example, if you were to do this in Excel, and we do have courses in Excel to kind of demonstrate this concept, you would first do the financial transactions and use that to create the general ledger, use that to create a trial balance. The trial balance would then be used to create the balance sheet and the profit and loss. And then typically, we're gonna use the balance sheet and profit and loss to construct the statement of cash flows, basically just reorganizing the data on a cash flow basis. Okay, so we're gonna go to the reports dropdown. Statement of cash flows can be found in the company and financial. All the way on the bottom down here, the statement of cash flows, the third major financial statement report. Now note that all other reports, I wouldn't really call financial statement reports. The main financial statement reports are the balance sheet, the income statement, or profit and loss. And then the statement of cash flows, which is still kind of like a supporting report in that it's gonna be adjusting the data. And then all the other reports that we have are typically gonna be expanding on some line item, usually on just the balance sheet and the income statement, or the profit and loss report. So let's go ahead and customize this report to make it a little bit larger on the fonts and numbers, changing the font size up to 12, okay, yes, and okay. So the statement of cash flows will typically have three parts to it. Let's go ahead and minimize everything first and try to build it up as we go, as we saw with the other two reports. So I'm gonna collapse everything first. And so it usually has these three components to it. It's gonna have the operating activities, the investing activities, the financing activities. That's gonna give us the net cash increase or the change in the cash. Notice here, it's actually a decrease because it's a negative. So sometimes these terms, you could call it change for a generic term so that it would be right, whether it be an increase or a decrease, but in any case, and then we got the cash at the beginning of the period to get the cash at the end of the period. So the end of the statement of cash flows, the 67, 8, 19, 29 should match what's on the balance sheet. If I go back to the balance sheet, notice it's a 65 here. You might say, well, what's wrong? That's because remember that undeposited funds is really a cash account and they put it in the other area, I believe. So if I was to say 65379.29 plus the undeposited funds 2440, then we've got 67, 8, 19. Statement of cash flows is the 67, 8, 19, 29. So in that way, you can kind of think of it as the statement of cash flows is expanding, giving you more detail on the cash account, but it's really giving you more detail than that. It's really trying to give you the information from an operating investing and financing standpoint on a cash flow basis, as opposed to basically on an accrual basis. So the first component or section of the statement of cash flows is usually the longest section and the most complex one to understand, that's the operating activities. Now the operating activities you can in essence think of as kind of like the income statement, but you're gonna convert the income statement from a cash flow basis, I'm sorry, from an accrual basis to a cash flow basis would be the general idea. But even that is a little too simplified because if we were to do that, you would think that you would go to the profit and loss and every time we had income up top, we would convert that income to, instead of recording it when on an accrual basis, we would record it on a cash flow basis. In other words, if I go to the homepage, instead of recording income when I have an invoice, if I'm using invoices, I would record income when I've received the payment because that's when we got the cash. So that would be the difference in a timing difference. There's a timing difference between an accrual basis and a cash basis. So you would think I would go line by line, income recorded on a cash basis as opposed to an accrual basis, expenses record them on a cash basis as opposed to an accrual basis on the expense side of things. If I go to the homepage, if I'm using the bill item, this bill would record the expense, but not when we paid it at that point in time because we entered it into accounts payable. So it would then record the expense instead of here, it would record it here when we actually paid for the expense. Again, you might be on a cash basis because you might just be paying with bank feeds and entered into the system that way and not using an accrual component, but just know that's when the difference, that's when you're gonna have a timing difference between a cash flow and a accrual basis. So then if I go to the statement of cash flows, that's what you would think would happen, but we don't typically do it that way. I'm back on the profit and loss. We could do it that way and that's called a direct method of doing it, but oftentimes we're gonna do it with an indirect method. That's the method that QuickBooks is going to be using here. And there's a couple of reasons for that. One is that that gives us a reconciliation between the net income on a cash flow basis and the net income that is on an accrual basis. So we're gonna start with the end result and reconcile or back into the cash flow, basically net income number, which would be the operating activities on the statement of cash flows. The other reason is because for reporting purposes, they usually require the indirect method, even if you use a direct method, so which those two reasons are kind of tied together, the reason that they prefer this indirect method of starting with net income and then tying it out to recording the difference, showing the change from the net income on an accrual basis on the income statement or profit and loss, to that on the statement of cash flows, the net cash provided by operating activities, in essence, net income on a cash flow basis kind of, is because it gives you that reconciliation. It shows you the difference, whereas if you were to reconstruct it, you don't have that exact tying out from one to the other, so it's a nice added component. It's more theoretically easy to see if you were to do a direct method, but usually you're gonna see an indirect method and you'll have to use an indirect method for reporting purposes if you have to report statement of cash flows. Note, like a small business, oftentimes doesn't need to report a statement of cash flows because you don't need it for taxes, generally. Usually you're gonna do your taxes with an income statement, but if you get a larger company and possibly if you want a loan, they might ask for a statement of cash flows if you're looking for a loan, and corporations as they get larger are gonna need to have reporting requirements. If you can have an audit, you're gonna need a statement of cash flows, and of course for internal use, if you can understand the statement of cash flows, it gives you the best of both worlds because it gives you the comparability of the income statement on a performance measure as well as the cash flow to help make sure your cash flow is doing well, cash flow being important because cash is the lifeblood of the company. The cash is involved in every cycle of the transaction, so the cash flow is still an important component. Okay, so on the operating activities, we start with the net income, that comes from the income statement, and then we've got the adjustments to net income. So this can be a little bit confusing on how this whole thing is basically constructed. If you can understand how to construct the statement of cash flows, then really that's a great exercise for understanding the double entry accounting process. We do have a course on constructing the statement of cash flows. Also note that the statement of cash flows in QuickBooks will make some assumptions to kind of construct the statement of cash flows, and you might have certain transactions, such as disposing equipment or something like that that had depreciation on it where you would need to do some more adjustments to it to really get the statement of cash flows dialed in exactly correctly, but it gives a pretty good baseline of it. So the typical way you're gonna construct the statement of cash flows would be that you would look at the balance sheet and create a worksheet generally from the prior period, 1231, 23, for example, and then the current period. Let's do a comparative balance sheet and just take a look at that if I was to say, let's take the 010124 and look at the prior period, which will be last year and look at the dollar change. So I'm gonna say, okay. And so now we've got 2024, the current year, and then the prior year, and then we can take the difference between each of these items. Now you might be saying, why would I take the difference on the balance sheet when I'm trying to construct a statement of cash flows for the operating section, which you just said was kind of like the income statement. And the reason for that is because if I go back to the balance sheet, remember the balance sheet is the double entry accounting system, assets equal liabilities plus equity, or assets minus liabilities equals equity. The equity down here represents kind of the book value of the company as of a point in time. The income statement is the distance, how we got from one point to the other. So one way we can analyze that distance is we can say, well, why don't I take the difference from where we stood last year as of a point in time equity section minus this year, that difference is the 119, 154, 64, which mainly is gonna be net income typically, although there could be investments from the owner and there could be draws from the owner. But, and then, and so I can see the difference between the two as the change. And then I can try to look at my accrual accounts up top and say, well, let's look at all of them. If I take the cash account, that's gonna be the bottom line change. So that's what we're kind of reconciling to. In other words, if I go to the statement of cash flows, you've got the cash here, which is gonna be meaning you've got the activities, the flows from operating, financing and investing. And then you've got the net increase for cash, which is this negative 3068. So if I go to the balance sheet, that would be then, let's take this change here, the 12, 743.65 minus this 15, right there for undependent positive 15, 812.08. So there's the 3068. So this change, this difference in the cash flow is kind of what we're reconciling to. But we don't wanna leave it just there. That's why they add to it the beginning balance, which is gonna be this 52635.64 plus the 18252.08. And that gives us our 70 statement of cash flows, 788772, so that we can arrive at the cash flow at the end of the period, because that's the thing that's easiest to tie in to the actual balance sheet would be the general idea. If I go back to the balance sheet, then I can take the differences in all the other accounts. Now the accounts receivable is an accrual account, meaning if you were on a cash basis, you wouldn't have accounts receivable. If you didn't invoice clients, you would have no increase in accounts receivable. So this then represents, this change represents the timing difference between when we record income on a cash basis and when we record income on an accrual basis. And so we're gonna kind of use that to go to the statement of cash flows and show that change right here, which is gonna be a decrease. So if I start with net income here, I'm gonna decrease the accounts receivable because if I go to the balance sheet, you can see this accounts receivable wasn't increased. It was higher in 2024 than 2023. That means that we had income that we reported that we had not yet received the cash for. Therefore, if I start with net income that includes that income and subtract it out, then I can kind of back that out and get closer to where the net income should be on a cash basis. And we do that for all the differences on the balance sheet. So we can do that same kind of thought process for inventory, employment, prepaid, and so on. And that's where you get these changes, these differences are trying to back out the accrual components so that we're gonna be on a cash flow basis. And if done properly, that should give us the net cash provided by operating activities, which is in essence net income on a cash flow basis. So then we also have some cash flow stuff that isn't basically kind of like recorded on the income statement. So in the investing activities, the main items that we might have here would be say we purchased something and it wasn't an expense, like we purchased property, plant, and equipment. So this name investing can be a little bit deceiving because if you hear investing activities, you might just think stocks and bonds that you're investing in, CDs, cash. But investing from the corporate or a broader term of investing that we're using here is that we the company are investing stuff, capital cash into something so that we can use it into the future to generate revenue, meaning we're investing in assets. We're investing in long-term assets. Typically that's gonna be property, plant, and equipment. So stuff related to property, plant, and equipment, which when we buy it, even if paying cash, we don't expense it. We put it on the books as an asset. Therefore it's not part of like the operating activity up top, although you could have like some depreciation that could be up there. But in any case it's gonna be down here in the investing activities. When we buy and sell furniture and equipment, it's gonna be activities down here, which will have a cash flow impact. And then we have the financing activity. Also we could have other investing items that would be normal natural investing activities that you would think of stocks and bonds in investing too. Then we have the financing activities. Those are us financing our operations. So if we have assets on the books, we finance those assets that we have on the books either through loans, liabilities, or through equity. Us generating the money, investments from the owner or the equity that has been accrued or generated in the business. Those are the kind of things that would be down here. If we took out a loan, that would be cash flow that would be possibly increasing from the loan we took out. If we're paying off the loan, that would be a decrease in the cash flow. If we the owner put money in, then that would be a cash flow coming into the company from the owner. If we took the money out as a draw or a dividend in the form, if we're in the form of a corporation, then that would be cash flow going out from the finance activity down here. So then those are the three cash flow activities. And if everything is done properly, we should be able to add those three cash flow components up to get to the change in the cash flow, which should match as we saw the difference between cash and the prior period in this period. And then if we add that to the prior period cash, so now we got the prior period cash, the starting point, and then the change to the cash, we should get to the cash flow at the end of the period. And that would be the general idea of the statement of cash flows. So notice that QuickBooks can put together a pretty good statement of cash flows here that at least is starting out and reconciling, although there could be some still confusions when you get into more complex components, such as a lot of times it has to do with these depreciable assets and loans. So you can imagine situations where you took out a loan to purchase a vehicle, and that can be a little bit confusing for QuickBooks to properly allocate on the statement of cash flows when you dispose of assets that had some depreciation related to it. That could be a little bit confusing for QuickBooks to properly allocate as well. So you might need to do some adjustments for reporting purposes, but this gives you the general idea. And if you want to learn more about the statement of cash flows, we have got a course on it. It's really a good process to put together. If you could put together a statement of cash flows from a balance sheet, I mean, if you could reconstruct this, if you were to take this balance sheet right here, take this worksheet and try to reconstruct the statement of cash flows in Excel, be a huge exercise that would probably give you a lot in terms of knowing the double entry accounting system better. Now just one more thing to note here is if you go to the file tab up top or edit, and we go to the preferences, then we've got the report and graph preferences down below. And if I go into the company preferences, you've got the classification preferences here for the statement of cash flows. So if I go into that, then you can see how these changes, the differences in the balance sheet are being applied to operating, investing or financing activities. And if you need to make any adjustments to that, then you can go here to make those adjustments. And you can kind of think, like if you think about which category it should be in, notice it's checking off all the balance sheet accounts, and you try to say, okay, which category should these items be? And you can think of what are the normal transactions that record accounts receivable? Do they have an impact on the income statement? If they do, accounts receivable going up with an invoice, the other side being revenue, usually it's gonna be an operating activity. If they don't, like property, plants and equipment, when I buy property, plants and equipment, then I usually, even if I pay cash for it, I decrease cash balance sheet account, other side goes to property, plants and equipment, no impact on the income statement. Therefore, it's not gonna be operating. It's gonna be, in this case, investing activity. And then down here, if you think about, of course, accounts payable, you have a similar kind of thing. And that the other side of accounts payable is gonna be an income statement account of an expense oftentimes. Therefore, it's gonna be an operating activity on those. And then the loans, when we take out a loan, cash goes up. The other side is gonna be going to a loan, right? When we take a loan, it's no income statement impact until there's interest involved. Therefore, not typically operating, something else, in this case, investing activities.