 izgledaj se o Tentanguenu in Funtumfield teori. As the title says, I will talk about Tentanguen in Funtumfield teori, I understand that the background here is very vast, so many of you are not familiar with some of these concepts, yet in three lecture of two hour, I cannot make an introduction of Funtumfield teori, so I will just try to organize some lecture in such a way that will be let's say at least decent for both expert and non-expert in the field so that everyone can learn something. Clearly I'm happy to eventually re-normalize the lecture if I have some feedback that it's too easy or too difficult. The overall structure will be that today will be quite easy, just I will explain what is entanglement in many body system, why it's important, et cetera. Next lecture will get a bit more difficult when we start using bit of Funtumfield teori, but I will just give you some ingredients that you need without saying where they come from, some of you may know where these ingredients come from, some of you just have to believe me when I take as axiom of our calculation. In the last lecture probably will be a bit harder for the non-experts, but eventually depending on your feedback we can try to get it somewhere. So what are we going to talk about? The main subject is entanglement in Funtumfield teori and if you want, end in many body system. As you all know entanglement is a characteristic feature of quantum mechanics, you started probably at undergraduate and usually it's a property that is connected when it started at undergraduate student with the behavior of few particles, like you say that one spin is entangled with another spin and you know if you make a measurement somewhere you get an outcome that can influence the measure everywhere else. Still, what then it has to do with many body system. And to understand these, let's try to understand in general what is the structure of entanglement in a given quantum state. So we imagine to have a, let's start with a pure state and I will mainly focus on pure state during this lecture because time is what it is. A pure state psi and this state psi belongs to some Milbert space H that I assume can be factorized as the tensor product of two different space, h8 times hb. And the overall idea is that you have a Milbert space and there is an observer that traditionally is called Alice that can make measurements only in the part A of the system while there is another observer traditionally called Bob which can have access only to the complement of A. So overall this is the total space and there is, and Milbert space can be written as a tensor product of h8 and so there should be in this way. Now there is a standard here a millenial algebra which is called Schmid decomposition. Many of you have started already a thunder cloud but I will just repeat here. Schmid decomposition is guaranteed that psi can be written as sum over alpha lambda alpha wA alpha tensor product wB alpha where w alpha A is a basis of Milbert space HA while w alpha B is a basis. How many of you know this theorem? Ok, so there is not much to all of you know. Just for the very few that didn't hear before keep in mind that the basis here the tenter in the Schmid decomposition depends on the state itself. The Schmid decomposition ensures that for any state it exists a basis such that the state can be written in this way. But the state depends on, the basis depends on the state. So it's not that there is one general basis where this can be written that will be clearly wrong. Just, there's our mind but you seem to be very familiar with this concept already. Yes, so especially for you that are there. I will, thank you. I hope I don't have to rewrite what. Thank you for... Now, you see from this expression now that already entanglement start popping up because if there is only one lambda alpha equal to one Ok, I forgot to say lambda alpha should be such that some of lambda alpha square is equal to one because of normalize... See if you want the state to be normalize. If psi is... But this is just trivial. If there is only one of this coefficients equal to one the state psi is a tensor product this means that there is no entanglement between A and B. Contrarily, if many lambda alpha are different from zero the state psi cannot be written as a tensor product. If many lambda alpha are different from zero the state psi cannot be written as a tensor product. And then we will say that psi is entanglement between... Ok, so what we conclude from this very brief discussion about the Smith decomposition is that the presence of these non-again values being non-zero sigma the presence of entanglement. There is first a very important observation which I wanted you to get in your mind now for both the rest of this lecture and hopefully forever the entang... You will read papers that are working on the surface and go the entanglement in the state is that and that and maybe you have written paper already with the statement which in some sense is correct but it's also wrong at the same time because the entanglement is a property yes of the state but also of the B partition that you make on the state. So you take a state psi and you consider a given B partition of the inverse space. So in the state and in the partition you have some entanglement encoded in this lambda alpha and now we will try to make even something better out of it but I can very well find another B partition of the same state where there is no entanglement or I can find one B partition where the entanglement is big and one where the entanglement is small. So the entanglement is a property of the state but also of the B partition. Ok, so don't... It's trivial, I know from the introduction but then many people forget this and just start discussing that state is entangled and many other stories. If you have two spins, obviously there is only one B partition but you cannot divide in different way but if you start having ten of them just you see very easily that there are many possible B partitions. Ok, so this is a very important observation and I hope that you will not forget. Then we can continue and trying to get something more quantitative because just from this mid decomposition we arrive to the point that we are able to distinguish whether a state is entangled or not but we are physicists, we like to measure things and we don't want just yes or no. We would like to know, characterize if I have one state and one partition like that then another state and another partition as is written. Similarly to know if there is more entanglement in one case or in the other. These are the kind of questions we are wondering. So we want to go through a measure of entanglement. What we are? We are going for a measure but I didn't ask you before maybe I should have asked how many of you know how to measure entanglement compared to mid decomposition the number really went down so this introduction is really needed. Ok, so to do so let's start by introducing the let's introduce, reduce density matrix this is the most important concept for defining the entanglement entropy we shall define this rho a equal to the trace over hb of the density matrix rho and the density matrix rho for a pure state you need to be just the projector on the state itself rho a equal to trace over hb of rho rho rho is projector on psi psi psi and you don't read down here this is too low the density matrix for the entire system is just the projector on the state and let's write everything here also rho a this is the most important formula is the trace over b of rho and usually this is shorted as trace over b of rho not trace over hb this is short for the same thing ok, now if you take this mid decomposition and you plug into that formula ok, it's a very simple exercise you can try that since these objects are orthogonal you get that trace over rho a is sum over alpha lambda alpha square w is the sum of the projector on this basis and this is the weighted weight the square of the Schmitt again values ok, so before getting to this definition of the entanglement I want to observe another thing that I could analogously define the reduced density matrix of the part b of the system as the trace over a over rho ok and if I write in the Schmitt basis this guide is the sum over alpha alpha square wb alpha wb alpha and the fact that the form of the two reduced density matrix is the same show once again that entanglement is a reciprocal property between a and b and not a property just of the state still you see that already this formula has something that often gets overlooked h a and h b can be extremely different pieces I can imagine to have a state I know done by let's imagine psi is a state of 300 spins of qubit call as you want depending on which language you like but I can be partied this thing in many different way in particular I can make 150 and 150 and then the two wilber space will have the same dimension but I can even be partied the system in such a way that h a corresponds to one spin for example and h b 299 in this case h a's and as dimension q while h b as dimension q to the 299 with no surprise 299 is much larger than q everyone should know still the reduced density matrix can be written exactly in the same way in this case let's see if you are getting something with entanglement how many most be different from zero then what does it mean that you meet the composition I have two lambda alpha different from zero so I have let's say a is the one of dimension 2 so b are sum over alpha goes from one to two of these two guys and two states are obviously are basis for one qubit what about the other alpha I mean the other alpha is bigger than alpha for the b part there are two again values that are non zero and then the other 2 to the 298 identity but I say that meet the composition that is a theorem improved by this mathematicians meeting in 1850 say that it is a basis and the space of dimension 2 to 299 cannot be written a base of 2 where are the other 2 to 298 states it is a basis it is a theorem it is written even on Wikipedia if you don't trust me it is the source of all the culture of nowadays Wikipedia so if it is in there it must be true obviously it is a tricky question it means that in the remaining part of the Hilbert space you can choose whatever basis they will have just taken value zero and then basis is okay this is always the case it sounds trivial it's adding a lot of zero eigenvalues and the eigenvector in the kernel can be important for something don't forget that they exist this is something that often brings to proving wrong theorem so don't forget that they exist something else so in general for example this alpha that you never find specified you can write alpha runs from one to the minimum between the dimension of HA and dimension of HB we say this once alpha goes from alpha to the minimum of the still dimension and we will not say anymore but keep in mind that for the larger system but also for the other there can be zero again values everywhere let's go back that we were searching for a measure of entanglement and we see we have said that from this definition from the Smith decomposition we see that the entanglement is encoded in this coefficient lambda alpha and the intuition tell us that the most coefficient more coefficient lambda alpha are different from zero more entanglement there is more coefficient obviously lambda alpha itself doesn't matter too much should be the absolute value of the object that matters because you can always rotate this guy and keeping in mind that sum over lambda alpha square is equal to one as we already said we can interpret we know that lambda alpha square is just the probability that the subsystem A is in the state given by this basis so a measure of how much the state can be of how much we don't know where in which basis is the state in the subsystem can be the entropy minus sum over lambda alpha square log lambda alpha square ok, this is a probability so you can always define this entropy which is noting that minus trace of rho A log rho A which is also since lambda alpha correspond both to rho A and rho B minus trace of rho B let's put parenthesis and we avoid all dubs this is the definition of phoneme an entropy ok, you may know it so this quantity trace of rho A log rho A or equivalently trace of rho B rho B is usually called entangament entropy ok which is the the measure of entropy we were searching for the measure of entanglement we were searching for now I don't want much to insist on why mathematically that object is a good measure of entanglement because this will bring us in a detour quantity information theory that neither is my expertise nor probably is what you want to know just to first of all I want to tell you what is the official story ok and then people interested in it can go to original literature or to Wikipedia again as you must know that quantity information theory has very horrible Wikipedia pages while physics has very horrible Wikipedia pages so this is probably fault of the Ph.D students because Wikipedia pages in the end of Ph.D students then in quantum information theory one say that the entanglement entropy ok the entanglement entropy that we define there is an entanglement monotone ok, so the entanglement monotone let's write it big but in equivalent way of saying that is a good measure of entanglement and is a good measure of entanglement because definition of entanglement monotone is such that does not increase that's why monotone under LOCC and LOCC means LOCC LOCC operation and classical communication ok, so this is the definition I don't want to enter too much in this story and just give you a an idea of how the thing work ok if I want to measure if I want to have a measure of entanglement ok, and the people in quantum information got this idea of entanglement monotone ok and more or less to give a very rough explanation is that you cannot really say if the entanglement is 3, 4, 5 of one state and one bipartiton what you can give is an order between the entanglement of different states and the partition ok it's not like obviously I have this table that measure probably 2 meters 2 meters and half ok and I have this chalk that is much shorter for the entanglement you cannot say that this guy is really absolutely one tenth of that but you can say that definitely this chalk is shorter than the table ok, so there is a monotonic there are some objects that are called entanglement monotone which tells you if one state in a given bipartiton is more entangled than another so it's a property of a state in a given bipartiton you understand the logic that I don't want to question at this point I will tell you a bit better again at the same level of what I'm telling now but ok, the first point is I don't have an absolute measure of entanglement ok, I can have several measures ok, all of that are monotone in the sense that they provide an order between states ok, so if I have that if I have so I have an invert space with a given bipartiton ok, so I should start with this this is how things are well defined don't compare in quantum information state with different bipartiton or part of the game ok, this is important for quantum matter but not in the other game so I have this bipartiton then I have two states, psi and phi belonging to H I can make what I can tell is whether the entanglement of psi is greater or equal than the entanglement of phi and this I can use this entanglement monotone if I found in one monotone that this relation is true this should be true in for all the entanglement monotone ok, sometimes with someone can be equal but there is this idea of ordering but then in one case one can be the double in one case one can be one third times more the scale is not defined at all so this is the idea of monotone it's an autonic relation between the entanglement of the state in a given partition yes, sure, sure, but this is not ok, I am telling you why for normal entanglement and then we discuss what else you can do with that I am trying to convince you at least then this ordering of state according to the people that developed this theory ok, that are many people in quantum information ok, is that any entanglement monotone should not increase under LOCC local operation in classical communication and what local operation in classical communication means it's the name itself that tells you you had this state with the B partition A and B local operation are those object that acts only in A or in B ok it's clear that if I have one state and I make some local operation in A or in B I cannot increase the entanglement I should make something between A and B to increase it a local operation can be a unitary restricted way in that case you expect the entanglement does not move at all but you can make a measurement for example in A and in this case you can bring down the entanglement to zero clear? can make a measurement of all the observable in A the state will be projected in one of the smith basis the entanglement will be automatic at zero in fact it didn't increase but when it's obvious that only in A or only in B I cannot increase the entanglement ok that's the it's obvious it's the definition from the mathematical point of view but the physical intuition tell us that this is what we want clear to everyone classical communication on the other end is as an operation is a bit more complicated I don't want to enter in what it is the famous part when you discuss about EPR paradox you know that there are these two states entanglement you make a measure here and you discover that immediately on the moon the state is in your spin is in another state how this is compatible with causality because to know that something happened on the moon you need some classical communication that tells you what happened in the moon that was the explanation of the EPR paradox the acceptable explanation according to quantum mechanics and that part of the that famous phone call that is telling you what happened on the moon is the part of the classical communication ok so you need if you A is a spin on the art and B is a spin on the moon is true that as soon as you make the measurement on the art on the moon something went differently but in order for the people on the moon to know that you made the measurement you have to tell that and this takes some classical communication to do it ok that's you knew this explanation rise the end or you knew the explanation not as many you know who knows the EPR paradox please all rise the end ok so you know the EPR paradox and the solution I just told you is that what Einstein didn't like the solution is that in order for the other observer that didn't make the measurement to know that the state must be in one state and it's not yet in super position is that you need to communicate the other person that you make a measurement because until you make the measurement he doesn't know ok even when you make the measurement the other observer doesn't know you have to communicate this communication which is not within classical mechanics mechanical communication must be done, take some time because cannot go faster in the speed of light and this is the explanation why there is no paradox in the EPR protocol ok this kind of operation for mechanical for classical communication and also under this classical communication the entanglement we expect that should not increase it because we are doing something classical so we cannot change the entanglement ok the reason why it's obvious but what is more complicated is to give in operation in operational sense to discuss communication but I will not tell telling you as a very reasonable story ok is the story reasonable to you yes ok this is there are way-ups also of this but it takes it's not it's local ok you can make I understand that quantum mechanics when we are discussing here et cetera is not relativistic but ok you can make relativistic quantum mechanics which is quantum field theory and ok you can make everything fitting in that and still the entanglement is there ok this will take a longer way the only thing ok I don't want to discuss EPR protocols I want just to discuss my main point is that the entanglement entropy is entanglement monoclonal this can be proved ok I will not even show the proof ok so we have this concept of ordering between entanglement entropy respect this property that's my main message and I wanted this as clear then if you want to discuss aside or between you or the other implication very welcome but ok we will not we will not go anywhere if we start discussing this too much ok is this clear for the rest of the lecture and also for and also to make contact with what is nowadays done in the literature only the entanglement entropy defined as ok so there we have trace of rhoe log rhoe is a good measure of entanglement the the rainy entropy that probably are less known concepts rainy entropies who knows rainy entropies here they became very painful remember like 5 years ago I was making lecture about the entanglement entropies and no one new rainy entropies switch completely the rainy entropies are defined just s a of order n s1 divided by 1 minus n log of trace of rhoe to dn ok modification of that formula for if you take the limit for n that goes to 1 of this object obviously you find the entanglement entropy that we defined before but it's a one parameter family of entropy and all of them are good entanglement model so all of them are good measure of entanglement still there are physical reasons to prefer the entanglement entropy that I'm not sure we will discuss during this lecture depending on how times go and what we what we do any of you has some idea why obviously not the people working in the field because they should know but for the other people any of you have some idea why entanglement entropy means for Neumann entropies better than rainy there is no reason from the point of view of entanglement since they are all good entanglement models so there should be some other reason no computation is the other way around we introduce rainy because we are able to make instead entanglement entanglement entropy is more complicated both to measure and to measure in experiment especially we are still not able to measure in experiment entanglement entropy by the way only the rainy version ok let's extend the question to the expert one of the expert tell me why not Marcello obviously one of the junior expert probably people are scared to say why ok one of the main property of the entanglement entropy that the rainy entropy does not is the so called strong subaditivity for Neumann entropy satisfy strong additivity while strong subaditivity it means that if I have different be partition like I have A1 A2 ok, so this is A1 there is another piece A2 and there is the rest SA1 plus SA2 it's larger or equal then SA1 plus this is not true for rainy and you see that why this object is important from condensed matter or field theory point of view there was a question but first let me tell why this is important because this relation allows you even to get some more the ideas when you value the be partition not only the state entanglement monotone it's a property ok within a given be partition it ordered the states given a be partition so I cannot compare according to quantum information and what it gives us the entanglement entropy in of a state in one be partition we do one in another be partition this is we do, we will do it many time but it's not an allowed game within the definition of quantum information ok, we should go out of this of that and the idea that there is this subaditivity that relates entanglement of different be partition provides the tool to go through that ok, so that's why it's very important subaditivity and also it's a physical it's a very physical thing so it's important ok, there was some question around I had a story before subaditivity it's additive in the entanglement will be additive not subaditive, but this is something more ok it's part of the game that will be the equal sign for classical object so when this sorry when this this equality is saturated it means that you are in a classical state or in a thermal state as we call it that's a very good point this is saturated in thermodynamics but it's not all the we are talking about one single quantum state that we didn't even tell what it is with many body system or would be just a state that I pick up because I want to play with it there is no reason why should satisfy the state in thermodynamics ok, but obviously, since this is a general relation should be like also for thermodynamics state and in that case will be saturated this is the kind of question that please do it, because allow to make contacts of what I'm telling you what you already know and that's where you get better ok, so to go through this first 50 minutes again, what we say is that ok, we start from abbi partition of anilbert space we want to measure to understand the measure of the entanglement in this state, in this abbi partition and what I try to motivate you is that this object here, which is called the entanglement entropy which is extremely important that you know now and you don't forget anymore if you want to continue to be a converse matter physicist it's a good measure of entanglement and it's even not only a good measure of entanglement because even the rainy entropy is a good measure of entanglement but there is some other physical properties like strong sub-budditivity which will be very important to make connections between entanglement and thermodynamics that is something very important in this lecture but it's good that you know ok, so this is the sense of this first 50 minutes still probably you find you can still find this quantity a bit abstract so I want you to show some example of of what it is to make it more down on earth and not just a crazy definition of of someone I gave you this definition and if this definition makes sense it should first of all make sense in the easiest possible case which is a 2 spin state or a 2 qubit state ok so let's see what happen ok let's take a generic ok, not so generic, quite entangled state of 2 qubit ok, so I take a superposition of plus minus minus plus with coefficient cos alpha and sin alpha if these two coefficients are equal 1 over square root of 2 which means alpha equal pi over 4 this is the p r p r that you admitted already to know ok, so this is very simple state and we have we know that for alpha equal 0, pi over alpha et cetera, it's periodic so I don't go after pi over alpha for alpha this is an entanglement state it's a product state so I have zero entanglement it's clear while I would expect entanglement maximal at pi over 4 correct, no? when the two coefficients are equal it's the maximum uncertainty that we have you need to motivate more this thing, no? and I would expect even that between 0 and pi over 4 is monotons this entanglement, no? from pi over 4 to pi over 2 should be again monotons by decreasing function of alpha that's an expectation graph will correct so let's try to calculate the entanglement entropy first of all we should get the reduced density matrix and rho a is just trivial will be cos alpha square the projector on plus plus sin alpha square projector on the minus state you see how to might get it when you sum over the minus just cos 2 of cos alpha and instead in the other case you will get 2 sin alpha ok try to make the exercise if you don't see it getting this reduced density matrix from that state is an exercise with a 4 by 4 matrix if you don't know it by art because I'm sure that half of you know for the other half that never made this exercise just during the coffee break the lunch break make the calculation it's very easy if you get stuck to just ask of your person on the right or on the left that probably knows it but do it because they want to lose the 3 minutes doing it and it's important if you don't know that you do it given this definition it's obvious that lambda alpha square and then are just cos alpha square and sin alpha square and then the entanglement entropy it's just minus cos square alpha log cos square alpha minus you can plot this quantity maybe you can just ask matematica to plot it for you and if you do so you got a car that is done like that entanglement entropy is 0 in pi over 2 it's maximal at pi over 4 and the maximum value at pi over 4 is you see directly from here if you put one half here and here is one half log one half plus one half log one half which makes, with the minus sign here makes log 2 ok so the maximum entanglement entropy in this case is log 2 which by the way is the maximum entanglement that you can have between 2 square cube bit for these very specific states it's clear that the entanglement entropy makes what it should do question at this point what do you mean how you generate depends on whatever experiment these are also for example these are standard state in photon that's how bell inequality were proved for example but there are thousands of ways depends on which experiments you have in mind come on nowadays like for in cold atom, superconductor et cetera there is the possibility to manipulate and play with at least 50 but hundreds also of spins two is just easiest one for each system that's why I ask you what you mean for each system will be generated in a different way it's really not to talk about the fact that when some if a particle decay in two particle these usually are entangled in the k it's not much different from that you have to change basis et cetera but the overall story is the same more questions I expect I said no no the intuition tells me that this state is maximally entangled where the weight between the two parts are the same if they are the same after I make the measurement 1 I have 50-50 probability of getting something but if the weights are not the same like this weight is 70%, 0.7 after I make a measurement I will have 70% of getting something and 30% of something else so the maximal uncertainty is when the weights are equal and this I expect that it is connected with the maximal entanglement what do you mean contribution when I have now two spins I expect the entanglement between the two spins to be maximal when these coefficients are the same this is something fair enough to say what I am doing now is sanity check on the measurement that we introduced to justify that it is a good measurement there is nothing rigorous here the rigorous part is I have this definition entanglement modern I check that the definition satisfies my axiom and then this is this is rigorous from a physical point of view this is unsatisfactory you want to see that your definition match with what you think it is physical and for me this is physical and that is there is no way to prove this but the physical is that the entanglement must be maximal where these two things are equal this is my expectation I, people in quantum information constructed measurement that satisfies this requirement it is not the other way around it is not that this is maximal because that object is maximal that object has been constructed in such a way that is maximal here if I construct an object that is not I am wrong this is what do you mean obvious there is physics and there is math max is based on axiom that follows something that you have to measure what you have there is no way to define physics define physics is that when Gaileo let it fall the balls that is the same there is no way I construct an object that fits what I know it must be true and don't change the the order of the operation the physics come before the math then the math must be consistent I cannot make expectation and so I make a calculation 1 plus 1 ok, to fit my expectation should be 3 ok, if you make this the math your expectation must be consistent with the math that you define but there are some things that are part of the axiom of your theory and this you cannot they should just fit your expectation and that is what I am telling this object entango entango and monotone this is part of the math and the axiom is decided that a good measure of entanglement should be entanglement monotone but this is an axiom it is not that you can prove it this axiom is there because it fits very well what you want that this maths describe there is extremely good maths that describe nothing and we are not talking about that sometimes it is confusing but keep in mind it is the never forget it why the entropy measures the entanglement obviously when I say that the entropy measures the entanglement the value where the entropy is maximal is the maximal entanglement but who tells you that the entropy measures the entanglement it is not exen chicken exen chicken you do not know who comes first here you know very well who comes first if on the same state you compute the other reny entropy you can make the game and you will find plots that look like this for different alpha ok so this is for Neumann which means alpha equal 1 and this object a bit smaller for example is alpha equal 2 and so and so are the other more or less the same shape and all of them are monotone functional this alpha which is expected because there must be good measure of entanglement and sorry it's this guy here sometimes is referred to and sometimes is referred to alpha but alpha is also the angle so in this case cannot do it you are right so this is n equal 1 and this is n equal 2 but in the literature you will find that you symbol interchange in this case we have already an alpha so we cannot if you don't have any question alpha is the right moment to make a small break and how you don't usually is the break 5, 10 million, 5 or 10 you want 10, ok so we reconvene at 17 so try to enter at 10.15 so at 10.17 we start this is in also ok welcome back just an observation one of your fellow reminded me that what I wrote here is just sub-budditivity and not strong sub-budditivity it's true that everything what I said was correct I just gave the wrong name this is sub-budditivity not strong strong sub-budditivity is that you cannot even as a intersection a2 which indicates that I wrote here was was 0 but ok in general strong sub-budditivity refers to the fact that there is also no intersection while sub-budditivity is without effect they are both true for the entanglement entropy and none of them is true for the rainy entropy so not change was just a a miss wording did you think to any other question in the meantime or just be consistent obviously in best you log 2 makes 1 this is convenient but in quantum information because you don't have all this log 2 around and you have just 1 and the maximum entropy is 1 things are easier in fact in quantum information you use log 2 in contents matter the natural basis is much more natural for other reasons because ok when you will make contact also with thermodynamic entropy you have e to the minus beta h not 2 to the minus beta h and you know very well that you could have the final you can make thermodynamics with 2 to the minus beta h this is just a renormalization a different normalization of the temperature but when Boltzmann define the ensemble he define with a and so we continue using the natural entropy if he would have used 2 probably we will all use log 2 more question please don't be ashamed in the break I got several questions and some of them would have been useful also for the other 2 years the answer I don't remember but it's good if you make don't be ashamed to be trivial there is not in trivial if you are not understanding better to ask and not possible I prefer to make half an hour less of lecture but that whatever the other 5 hour half will be clear and not to make a lot of things and no one understood anything ok so I made this example on new spin since we are talking about many body physics I should provide at least one example of many body physics where things are easy in order to understand what is going on ok and easy way to think to the entanglement in many body physics let's let's think to a many body state of spin one hour always ok in which there are many EPR pairs ok what I mean is that ok you have this like imagine they are on a QD lattice now that matters where they are let's draw many of these spins then let's take another color and the state is such that there are EPR pairs between spins some close by some very far apart but only pairs ok this is a so in practice this state psi is the sum over pairs of the singlet state that we wrote there that guy there with alpha equal 1 over square root with alpha equal pi over 4 just singlets EPR pairs singlet plus minus minus plus I was going to write in the ij belonging to the pair this is the state do you understand what is this state if not ask again obviously all the pairs are connected I don't have to this is a very reasonable many body state and what is the entanglement let's choose the B partition here let me use another color so that it doesn't mix up with anything this is this is the B partition it means that the states that are inside are in A the ones that are outside are in B and to get the entanglement entropy of this B partition I don't tear so far away what you said all are pairs everyone is paired with someone else so pairs I didn't let anyone alone it's a nice word where everyone is not alone so what is the entanglement of this state in this B partition you have just to you see that let's change this one and let's put inside here let's change this to you see that some spin are paired with outside and some not the entanglement between A and B depends on how many spin are connected with the outside and since it's a product state you can just write the entanglement as the sum in this very specific case S A I am breaking 1, 2, 3, 4 singlets so the entanglement entropy will be in the green case here S A will be 4 times log 2 because log 2 is the entanglement of one pair but let me take another color I can as well take this B partition here orange in this case here I am not cutting anything I have 1, 2, 3 singlet at the time breaking so in this orange B partition S A is 3 log 2 so in general the entanglement entropy S A is equal to the number of singlet between A and B let me write here times log 2 you see in this very particular simple basis how the very proceed property between A and B is there singlet between A and B that matters it's not something only of A or something only of B entanglement is always a properties between the two ok yes I was going to ask now ok this is not some just a game to make an example but it's a very important state in physics it's the valence bond state everyone knows what is a valence bond state if you don't this is a valence bond state in valence bond state is extremely easy that's this is a new ok this is actually you can write the valence bond state also provides a basis and you can write any state as superposition etc etc and then there is more entanglement is more complicated to be calculated then ok then it's as difficult as in the general case if you have few terms ok you can try to do something more and more becomes complicated and there are people just trying to make this game which is not a game it's very serious stuff but ok it's a combinatoric it's a way of rewriting the entanglement as some combinatoric game yeah you can you can make many modifications of the game you can make that any single pair is less entangled you can give any pair as some alpha and then ok you just sum up in any pair structure you sum by the way this these two questions reminded me that I get during the break that I won't use answer where I want to share with you is that in all these lectures we will just talk about bepartite entanglement ok so I wrote there is a bepartition and I measured the entanglement between a bepartition etc obviously you may know or you may not know but it should be quite natural there is even entanglement that is not bepartite but it's more partite if you have three object, four object you can have something that is that is a property of the tree and is not a property of the various pairs ok this is called bepartite entanglement and it's not measured by what is done here and it's a very interesting line of research that I'm not mentioning ok so all the entanglement in these lectures and actually 99% of the entanglement in condense matter is bepartite entanglement more questions ok, it doesn't enter in the entanglement just give it zero it's a product an unpaired object does not enter it's a product when you make the reduced density matrix either is in A or in B but doesn't make entanglement spin one out from the lattice is very difficult to move around I have a lattice of spin out one out don't change the problem if you have some itinerant object obviously you are talking in the wrong basis to write in number model choose another basis to make the game for understanding what is going on it's better to think to particle that don't move distinguishable particle that don't move there is no when you go to itinerant object you have to change basis and things becomes clear you don't want a basis that depends on time ok, if not now we try to make content with what you know that is probably condense matter because as you have seen until now we play the game of quantum information that it is, I have a state I want to know something about the state but this is like ok, there are many states in the space and some of them you can never meet in your life so we would like to know the question is what happens what happens when dealing with many body states that nature that nature provides us because I can make all the nice math calculation but this could be related to states that will never be encountered in any of your experiment ok, at this point first of all we should yes what happens when dealing with state that nature provides us not a state just in a little bit space I mean ground states thermal states ok, that's where we want to go but first before this we should specify the B partition we want to deal with when you have many body states you have too many B partitions and we will limit I will write very big because so that you remember we will limit to special B partition in which the Hilbert space can be written as always h8 hb but this corresponds to physical space ok, so exactly like the lattice that I had before divided in two parts ok, so you understand what this means special B partition now in the drawing that I have been doing many time this one a and b these are not Hilbert spaces but a real space x and y this is real space and there are very good reasons to limit to real space entanglement the real space is the one Hamiltonian is local it's not one of the many possible states usually you define a local Hamiltonian a real space and that's the space we want to talk about we can obviously take other B partition even in many body system some of them can look interesting some of them can look a bit less but we will just limit to this one so, what happened the entanglement entropy SA in the ground state of a many body local Hamiltonian who knows the answer many no, come on I don't believe, who knows the answer that there is the area low et cetera et cetera, who knows the answer not so many I'm surprised what does mean it's minimal I'm thinking what happened in the ground state eh this is a question that is around since many time, since many years ok it was first proposed in the end of 80s, beginning of 90s especially by Schraderike that the the entire SA let's write it SA should satisfy satisfy the area low which means it's proportional to the area of the special region separating A and B this is a two dimensional object the area separating A and B is a perimeter and so should be proportional to the length of this object it does not scale with volume like standard entropy ok which will be objective because of this reason but it's proportional to the area this is it's an idea that is going on since the 90s that should now you are going ahead slowly hmm what it has been shown ok, there was this idea it's been going around for a long time ok, so this is let's say from 88 to 93 this is an expectation that was built already in 94 in a seminal paper all Snellarsen wheel check showed that one plus one dimensional conformal field theory the entanglement entropy is equal to C over 3 log L where this is A of length L and this is the remaining of the system and you see that the area between A and B in a one dimensional system is just the number of points between the two parts so the entanglement the area should be two so it should not scale with L instead here you see that entanglement entropy is growing with this L, so does not satisfy area low follow no area low I'm taking a one dimensional system align and I take an interval of length L which is special can you re-ask this later and later on in time and then during all the notice 2000 several years ok, people have shown indeed that area low is valid only gapped amiltonje there are several arguments to bring this up and this is rigorously proved even in 1D ok, so rigorously proved in 1D it's a theorem from Mastin in the 2007 ok, so the area low is satisfied for gapped system and obviously there are violations there can be violations for gapped system including the Fermi liquid mention by your fellow yes but for gapped system area low is satisfied for gapped system can be or can not there can be violations it's quiet in 1D all conformal field theory will have this form so bosonic fermionic don't matter you will have always violation of the area low but this you may know because in 1D there is no distinction between boson and fermion ok you know this now boson and fermion in 1D is just in our mind but you can bosonize fermion or you can fermionize boson they are completely equivalent the result is the same in higher dimension boson and fermion are different ok and in fact it's well known that fermi liquid and in general fermion have violation of the area low instead it's bosons constructed example of bosons still satisfy the area low even if they are gapped but it can be that someone will not but ok surely it's not a theorem that it's a gapped system then it depends if you have finite degeneration nothing nothing change if you have extensive ground state degeneration ok everything but this is this is a very specific case you know what does it mean if it's degenerate just there are two degenerate ground state 3 degenerate ground state nothing this doesn't change area low or not you have to make up one more super position eventually and this will at most increase the entanglement of log 2 log 3 and this does not scale with del clear if you have extensive ground state degeneration you enter into another kind of physics that is a by the way it's very peculiar and depending on it's not standard situation I think none of you know states with ground state degeneracy so with extensive ground state degeneracy so let's not even mention the possibility yes I will even because it will be part of the lecture I was explicitly asked to make a course in quantum field theory ok so ok I know that some of you don't have the background there but ok that's what I was asked to do conformal field theory is field theory which is also conformal invariance which means it's which means it's in particular scale invariance ok and in practice for everything that matter CFT is gapless field theory with dynamical critical exponent z equal to 1 which mean that the energy is a linear dispersional relation ok this is what is conformal dynamical critical exponent which is the guy that tells you the scaling between energy and momentum and in this case is what if you have e in general equal like could be some dvk k square this is not conformal field theory ok note is a dispersional relation the relation between energy and momentum when it's linear it's conformal field theory and as you may know most of one dimensional theory have linear dispersional relation what falls in this category among the expert talks this CFT includes exactly, Latvinger liquid things in 1D almost everything that is gapless Latvinger liquid CFT includes almost everything who knows what is a Latvinger liquid then the organizer should give me an approximately sketch of your expertise Latvinger liquid is the reason why Duncan Alden got the Nobel Prize last year you should know why what else to tell something but then the question ok so since I don't remember the observation I wanted to make we continue ok this this analogy with the area law that I'm writing here in the 90s actually it was mainly motivated by the trying to understand the black hole entropy you may know that this megastane working formula for the entanglement entropy which says that the entanglement entropy the entropy of a black hole satisfy an area law exactly like the entanglement entropy and ok there was the idea there is still the idea it depends on the community that the entropy of the black hole is nothing but the entanglement that there is between whatever is outside the horizon that is the only part that we can measure and contribute to the entanglement entropy and what is inside the horizon that is as you know not reachable and interstellar what I'm talking about and there is a deep connection between what I'm telling here and the megastane working entropy and this was the motivation why people were studying these things in the 90s where entanglement was not yet existing in the sense all entanglement monotone and all this idea were just they were they've been built much later and still people were studying this quantity to try to understand the black hole entropy now the idea of this lecture for what follows will be the following in during next lecture in all the two hours I will give you a receipt to calculate the entanglement entropy in a generic field theory and this should be accessible to almost all of you that knows what is a field theory for the other will be bit more complicated and even a very low knowledge of field theory and many body system will be enough to understand the first lecture so the next the second lecture that will be tomorrow the last lecture will be a way to understand where this formula comes from which is extremely important to try to explain what is what are these numbers and what to understand about that guy and this will require some knowledge of conformal field theory that obviously I know that most of you don't have what I will do make the game of the axiom I will give you some theorem you will believe this theorem someone has proved but you can take them as axiom of what I was going to do we make a derivation of that ideas starting from these theorems that you have no idea where they come from some of you at least some may have idea where they come from and if you like the story then you can go and study where these things come from ok and ok this is so the lecture of tomorrow should be really accessible to everyone more or less at the level of what I say today not much more than that ok and probably I should I don't know if I have time now I should first let's try just to use this few information that I have given you to motivate for example in some cases why the study of entangoinentropy in many body system became such a central field of research ok and you see that this will also help understanding the language in general we say that if a system is gapped the entangoinentropy satisfy the area law while in one dimension for example when it becomes critical or gapless it it does not satisfy the area law ok so if I have a imagine you have a model let's take a system undergoing a quantum phase transition for me keep it in quantum phase transition for example the very famous transverse field design model do you know this field model no who knows the model so you all know that h equal to 1 the model is a quantum phase transition are you characterizes phase transition traditionally ok 1 that answer something 1 random one what you did to say that this model is a quantum phase transition what you did to say that this model is a quantum phase transition magnetization ok most difficult guy you chose are you prove that the magnetization is I doubt that when you study the model you put down the magnetization to calculate the magnetization in the model you should calculate the tube and function kako je ta tačno občutnja, in je to več proč vzelo, je zelo več taj delov. In tudi vsak, vzelo mi in entangomen entropi tudi da se zelo počutnja, je to celo občutnja, tako, kako je vzelo, je, da se vzelo, entangomen entropi zelo počutnja, ker je na celo občutnjo, je to počutnja, Na kritikalij počke, to občutje, ki je tudi, zelo, da je 1 over 6, log L. Ok? To je kar, da je h, če je 1. To je zelo konformativa teori. Kaj si je plozirati entangelo in entropi, na drugi počke, da je h če je 2, da je to zelo, da je vse zelo, in da je to saturizala. To je h če je 2. V hrvi je zelo na 1,5, da je zelo na vsega, pa je vsega zato. In ki je je na vsega, na različke, ki je vsega na vsega, je subzistem, in in infinitivnji. If I have an infinitivnji, where are the boundary conditions? In vsega, in in infinitivnji, so to no bother with that. vsak čas je najbolj modifikačna vse moraš tudi glasbo. Šta sem odpronočila, da sem ne so v hisi, vsak ne, ne zelo je skupno. Zelo sem odpronočila, da sem odpronočila, da sem odpronočila. Vse sem odpronočila, da sem odpronočila. Tako, transverse field icing model. To je pa, da jim bomo nekaj. To je prototip. Zgledaj na transverse field icing model. Jutko je, da je to je vse. Zaznala je nekaj, da je to je pravda in kaj je vse. Proto, da je injeljstvena entropija na našelj tem, nekaj je nekaj, nekaj je entropija, šeljena, našeljena, And log2 comes from the ground state degeneracy, that is because of boundary condition, if you take some boundary condition that is even not that log2. But okay, this is a parenthesis that you should not really be interested in. The main point is that you see that the entanglement entropy is able to characterize very nicely and very easily whether the system is critical or not in this case. Okay, you have this unbound and growth only at the critical point, while the growth saturate and actually you can take the entanglement entropy, the saturation value of the entanglement entropy to define the correlation length. So you say that the entanglement entropy for L large is equal to 1,6 log psi and use this formula to define the correlation length of the system, which actually agree with the other one as it can be proved, but I will not have time to get there. Okay, but this is a very effective way of characterizing the entanglement entropy, much more effective than any other you may know. If you want to measure the entanglement entropy from the decay of the two point function, good luck. This is very easy in state to do, even for a model as simple as this one. I don't like, how many of you have ever used a density dmrg or variation of it? How many of you know that it exists? So I can use this word. If you use a dmrg algorithm, the entanglement entropy is a stupid byproduct of your algorithm. You don't have to make any calculation, but for a dmrg you don't make any further calculation to get the entanglement entropy. Your code, you just have to make one line, if you were not doing it before, to create the sum of the weighted eigenvalues of the reduced density matrix. You have just to ask to sum up sum of lambda i log lambda i minus, and this lambda i were already in your code. Getting the entanglement entropy is a zero, is the cheapest thing you can do, and, for example, for the critical system, it tells you automatically whether you are critical or not, which is really a lot of information. And in other circumstances, we'll tell you, for example, in QD, it allows you to understand the topological phase, where the theory is, etc., and many other things. Basically, whatever you are doing, the entanglement entropy can be a very good hint. And now I remember the comment that I wanted to do, and I forgot, is that since we are studying just the ground state of the Hamiltonian, how the system can know that the Hamiltonian has a gap. You are just in the ground state. The first excited state is on top of the ground state. You just have your ground state, and still, by getting the entanglement entropy, you know if the model has a gap or not. That's quite peculiar, and I will give you the answer tomorrow, because I don't want to take more of your time. You have a very small break. So I will stop here, and we continue tomorrow from this point.