Juan Maldacena - Entanglement Entropy 1

Juan Maldacena - Entanglement Entropy 1

SUBTITLE'S INFO:

Language: English

Type: Robot

Number of phrases: 1435

Number of words: 10188

Number of symbols: 44529

DOWNLOAD SUBTITLES:

DOWNLOAD AUDIO AND VIDEO:

SUBTITLES:

Subtitles generated by robot
00:04
for today and for this week yeah we have plenty of time because our fish could not make it because of the visa so okay you feel like you need to go over time we have for example one slot after your switch is free so we are going to okay same for tomorrow no no just the
00:33
blackboard okay I'm ready to start yeah so it's a pleasure to be here so I will be taking off from where my van ran so we'll have left and then I'm kidding I'm
01:25
joking I will the numbers were because my leg we swapped and initially he will stop going to talk the first week and I was going to third week but now we have the other order so I will start with more basic things so of course you all know information is very important and you know that for example Jan Shannon's thinking of classical information was very useful for understanding thermodynamics in in a new way and of course monta mechanics now will have to
01:58
do with bantam information and all physical theories in some sense are processing information in some way you give some information about the experiment you are trying to do and then you get some information out so they are about information now one could go over the top and say the physics information that might be true but we don't know suddenly our physical theories have a little more input on how precisely we process this information and but something that has been very productive and useful is has been to
02:29
think about the constraints that information places on physical theories of course the concept concept of entropy is a very clear example of that but thinking more specifically about quantum information and how quantum information is stored in relativistic quantum field theory and also in quantum gravity has been it's very important and understanding it in detail has led to some progress and so my talks will be mainly about this topic so how quantum
03:00
information we store in the theories we know about in the theories of relativity quantum field theory and first I will talk just about the case of relativistic quantum field theory and then we'll talk a bit more about quantum gravity so examples of of results come very well concrete results that have been obtained using this thinking of how quantum information is encoded in relativistic field theories are for example the C and
03:29
F C and F theorems then going to the case of gravity so thinking about clearly about the black hole information problem requires you to understand if you're going to solve the quantum information black hole quantum information problem you need to understand some details of how quantum information is stored then it may be by thinking about those Thetas you can even rule out some scenarios ok and quantum
04:04
information if we're in the constraints of information might have something to do with the motions of gravity and it suddenly information constraints the organization of the state so it's important so states in quantum field theory are organized according to the normalization group flow and thinking about this in the quantum mechanical in a quantum mechanical way involves understanding how quantum information is stored in the state and this has been
04:33
fruitful to to try to understand okay so well physics from information well I said that you can go over the top and say that physics is all about information I will not do that so but on their hand if you there seems to be some indication that thinking about gravity for example in terms of the Lagrangian as we usually think of or thinking in terms of giving a formula so that Lagrangian gives you a formula for the dynamics of the theory
05:05
and you can alternatively give a formula for the entangled entropy it's a concept I'll discuss later but the area formula for entanglement entropy seems to give rise to the same equations that the Einstein lagrangian gives rise to so I'll this might be more in mine runs norms lectures maybe else a few things about that and that's not a complete equivalence yet but something that might develop into a complete equivalence in the future now regarding this issue of
05:38
whether hole physics is information I encourage you to read wheelers article on it from bit if you've never read it so but I don't have anything new to say but if you want something over the top you can read that he has this picture of a big u this is the universe it's very small in the beginning and big here and we are here looking at the universe we created by the things we look so the
06:10
beginning of the universe is because we ask questions we have questions about the universe and by the questions we ask we create what we say that's for me may be over the top but maybe that's what we'll feature will develop suddenly winner was right about winner was right about many things and so maybe he's right about this too um okay so but the first so what will the we'll discuss here will be mostly at least in the very beginning will be mostly about
06:42
entropy about computing various kinds of entropy and properties of entropy of quantum entropy and mainly of sub regions so mainly entropy of sub regions in quantum field theory should I write bigger or everyone can see good and also the concept of so-called relative entropy so that just given intro and
07:18
then how this can be used in quantum field theory quantum field theory and gravity so that will be the topic of these first few lectures and depending on how we go we might see some other things so first I'll talk about some basic concepts and relations so basic concepts and properties of relative entropy and so on and entropy quantum entropy and a reference or a book you might want to read on this this one by
07:53
all oh yeah sorry oh yeah and that's because all the it's a book so it's called the quantum entropy quantum entropy and it's use and contains many of the things and if you want some something online you can find a review by bedroll which is that's this number it's also a nice review okay so now in
08:29
quantum mechanics the basic the basic concept is the density matrix so normally in all the spokes when they talk about the state they typically think about this density matrix and in the particular case that the state is pure and that country then we have the draw is equal to subside so normally in quantum mechanics we think of science the state vector but you might this works read that the state is really
09:02
specified by the density matrix and that's the more general description of the state ok so then once you get the density matrix there is a notion of entropy which is the four nine four Norman entropy which is minus trace of Rho log Rho and this for normal entropy
09:32
is a measure of how much you don't know about the quantum state of the system I mean of course in quantum mechanics you cannot know everything so even if you had the state with its complete characterization like this side you cannot predict the outcome of some experiment right you still have some randomness but this for normal entropy measures the extra randomness that you have beyond the one you have in quantum mechanics okay so you can if you had a pure state then this entropy defined
10:03
this way would be zero and then you have the term in your state as much as you can determine it by the roots of monta mechanics okay so and of course this formula when Rho is a diagonal matrix reduces to the usual Shannon entropy so s which is sum over i p I log of P I and
10:35
I'm sure you've seen both of these formulas in your courses on statistical mechanics of course in this case for this particular state s is equal to zero okay so now another concept that will be useful is the concept of relative entropy so here in this case we are
11:12
given two two states or two density matrices so we have Rho and Sigma okay so we could have the system described by density matrix Rho and the system described by density matrix Sigma and this concept called the relative entropy depends on both density matrices and is by definition so this just the definition is trace of Rho log Rho minus
11:45
Rho log Sigma okay so these are matrices so this Rho and Sigma do not commute so this is defined in this particular way but of course you can calculate the logarithm of the density matrix then multiplied by Rho and actually compute this this quantity and now it might be that you get infinity so there is a possibility to get infinity here if for example Sigma has a zero eigen value at
12:18
the position where Rho has a nonzero eigen value right for example imagine they are diagonal even when they are diagonal you can have the possibility that you have an infinity okay and well in that case we define it as infinite so no no problem okay so that's a definition and in order to gain some intuition about the definition one can think of the particular case where Sigma is equal I'm not going to write down the normalization constant of course this this density matrices are normalized so
12:50
that yeah perhaps I should have I said already here that we always have the trace of Rho is equal to z21 okay yeah yeah I should also have emphasized that this is not symmetric and their interchange of Rho and Sigma there's only one Sigma that's a good question and so the next remark will try to make that a little more natural so
13:21
imagine the particular case where Sigma is proportional to e to the minus beta H so it would be the Sigma that corresponds to a thermal system right so you imagine you have a thermal system and so in this particular example and this part for this particular case then s s of Rho and Sigma just becomes equal to the difference in free energy the
13:52
difference in free energy between the state characterized by Rho and the state characterized by Sigma and you can you can check this so in other words so you'll have so you'll have so the free energy is a minus D times s so there is a term that involves the expectation value of the energy and that's this first term right so this first term is the so here when Sigma is that particular expression then this
14:22
logarithm of Sigma becomes just H right minus H this minus becomes a plus and that's the expectation value of the energy right so that's the energy term so let me call it the expectation value of the energy and then this term is just minus the entropy right up to well they're factors of a times R into one but so there is a better here so this is beta e minus s so that's in the state
14:53
characterized by Rho and then you can subtract the same things in the state characterized by Sigma right and those terms really give you essentially zero once you or malaises Sigma properly okay so we if we normalize this Sigma properly then the would have been an extra term here that would not have been the Hamiltonian exactly but there will be an extra shift and that shift is precisely canceled by evaluating a minus D / take T time cells with this particular density matrix is that clear
15:26
no should I do this more explicitly or maybe you can do it as an exercise - should I do it or not No so I can do it as an exercise now from so the point is that now this expression really is the expression for the difference in free energy that you would have for a system whose equilibrium state would be given by Sigma right so if you had such a hamiltoe if you are able to choose
16:01
whatever Hamiltonian you want to pick you can choose a Hamiltonian such that the Hamiltonian is the logarithm of this Sigma and then this expression would be just that difference in free energy for the Hamiltonian if you'd chosen and that's so that's the case now from this discussion you suggest that well what should be the role that minimizes this relative entropy what would be the role
16:33
that minimizes this difference in free energy yeah that's right so the iquili-brium state is the one that minimizes the free energy and so you expect that if you take any other state the free energy should be bigger and therefore the the role that would mean my States would be equal exactly equal to Sigma and and so on so in fact
17:05
you can actually you can prove that for any density matrices so s of Rho and Sigma it's a bigger equal than zero for - you can recall that sometimes it can be infinite but that's fine that's consistent with this inequality it's always bigger equal than 0 and is equal to 0 when Rho is equal to Sigma and it's not so you can somewhat easily show it
17:37
for diagonal density matrices and if you think about it a little more you can also show it for non diagonal density matrices now ok so a property that is sometimes used related to this is the first law so so you know the first law of thermodynamics that says that the change in entropy is equal to the change
18:11
of the change in energy right up to a factor of beta and you can think of that as just demanding that the free energy is minimal so when we just make a small displacement we can only increase the free energy therefore any small change the first order should not change the free energy and to higher order it should only increase the free energy right so if we think of this as a function of Rho so in the space of all possible rows of course that's a very multi-dimensional space so we have Sigma here and so here we'll have a minimum
18:41
and then it should increase and that implies that we make a small variation so we change row if Rho is equal to Sigma plus some Delta Rho then we will have that the Delta s so the difference in entropy yep so we will have that what we will have is that this due to the
19:12
fact that this minimal first order variation of this should be zero and this is the entropy minus the energy in where the Hamiltonian is essentially given by the log of Sigma and this is sometimes written as Delta s is equal to beta well people sometimes put here factors two pi ignore the factors of two pi if you don't like them times K times Delta K where K is just the Hamiltonian so usually in this
19:45
context people define matrix K which we should have simply the logarithm of Sigma just this is just by definition the logarithm of Sigma and it's called the modular Hamiltonian I didn't invent the name I don't know why so it's basically the Hamiltonian that would correspond to view in Sigma as a thermal density matrix and by convention we choose the temperature to be 2 pi so
20:25
ignore the factors 2 pi the unlike number but this is this is sometimes called the entanglement first law or should we call the this relative entropy first law it becomes an entanglement first law when we get Rho and Sigma 4 think from thinking about entanglement and will come that later but this is a relationship that it's being used it's been useful and perhaps I should also
20:56
rewrite these relations of the relation that says that this is positive can also be rewritten as saying that again 2 pi Delta of the let's say expectation value of this K is bigger or equal than Delta of s where this Delta is an arbitrary difference not an infinitesimal difference but an arbitrary difference this is just the same as this equation for the free energy the difference in free energy of just so we said that this was bigger equal than 0
21:27
and I've all I've done is to reshuffle the terms in such a way that we get this relationship so these are completely general relationships and ok so this relative entropy should be thought of as some kind of distance between between Rho and Sigma so it's sometimes people call it a distance is not very good to think of it as a distance because it's not symmetric so distances are normally symmetric but if you read
22:00
the reviews they call it a distance probably not a good name so maybe I shouldn't even say that so it's a measure measure of the distinguished ability or the distinguished ability of between Rho and Sigma so imagining that imagine that you are looking at a system
22:32
some random system and you are trying to guess what the the let's do this classically first so let's imagine we have a classical system which consists of you know there are some variables for example a coin that I can toss you can be either heads or roads and I'm tried trying to figure out what the probabilities are so what the method for determining the probability is to toss the coin n times right and then say that
23:03
the probability of the coin having the value I so value I means up or down in this case there are only two values right will be equal to the number of times so number of times that I appears appears divided by n right so it's just the frequency of the eyes result in a specific throw of the coins that we threw the coin n times okay so that's one thing you could do and that define
23:33
some P I and in this way we get some P I but imagine that the true distribution of the coin was not the pie that we got in this particular instance when we did the experiment with n times but it was actually Qi okay so the idea is that the so you would like to ask well how is so you of course if if we would like to understand how I'm probable the result P
24:09
I different from Q I would be okay and you can calculate how I'm probable that result would be you can think about this and it turns out that the probability of getting a very particular qi upon throw-in the coin n times is going to be proportional to precisely this relative entropy of p and q okay so q is the true distribution p is something that could
24:41
have come out of doing it ten times and this sorry I forgot the factor of n here okay so if you throw in n times then the probability of getting confused between P and Q will start decreasing right as you throw it more and more times you will be less likely to be confused now it's a symmetric and okay so let's try
25:15
to understand this form a little more and why this are symmetric and so imagine that P is equal to a fair coin so this is usually and q is equal to so this has probably that means that the probability is 1/2 1/2 okay for the up or down and then Q is an unfair coin so Q is a completely unfair coin so
25:51
always heads so Q is equal to 1 0 okay that's too small okay well okay so first one is an unfair this part last part was too small yeah but I should get trying to it looks big to me okay so so suppose we have these two cases right so then now in this case you can go back to the
26:30
formula and what what do you what do you see that what's what's s for this case what's relative entropy infinite air here yeah in this case it is infinity so let's try to understand this infinity so this is saying that I will never ever ever confuse this P if the true distribution is Q okay I will never ever get this P okay so what what does it mean to get this speed means that you threw the coin n times and half
27:08
the time was heads and half the time was it was tails right but if the true distribution is always heads it's impossible to get half the time the wrong value tails okay so that explains why it's infinite and why it's reasonable that we get this infinity now we can do the other example that's so we could do P is an ant the unfair right
27:37
heads and Q is even jus is fair coin okay so what do we get for relative entropy in this case well this is a little more complicated well if you do it you can do it as an exercise and you will get log - okay and we get here so in this case in this particular case this probability will be 1 to the 2 to the n okay so now let's
28:22
check that this is reasonable right so let's say we have a fair coin right so we have a fair coin that has same probabilities have been heads or tails and you threw it n times and it turns out that the end times showed up heads right so what's the probability of that that's 2 to the minus n and that's indeed what this formula is saying so you might get to the wrong conclusions about the the probability distribution with this very small probability okay
28:54
very good so that's that's just this definition I would set this in detail just so that we understand the definition and also because this is usually set in the wrong way the wrong order not usually maybe some people selling the wrong word okay so now so enough with the relative entropy now subsystems okay so imagine
29:33
that let me see well so first let's talk about entropy of subsystems so that's so in this case we a simple situation for this is to imagine that H is some Hilbert space where we take it to be a direct product of two Hilbert spaces for
30:07
subsystems a and B and this will be the structure of the Hilbert space if you imagine that you have to specially let's say separated systems like a bunch of spins here physical spins here and another bunch of physical spins on their side so here the full Hilbert space is direct product of the two Hilbert spaces so it's really machinist to decouple systems or that are initially separated in space or separated because they don't interact or whatever reason they are separated and then we can think of the
30:39
Hilbert space in this way we can separate the Hilbert space in this way we can take the original density matrix that is matrix in the original Hilbert space and we can calculate reduced density matrix Rho way which comes from taking the trace over the sub part of the Hilbert space H a of the original density matrix Rho okay it's is it clear
31:12
how we are defining this should I have you seen this definition before or should I go through it a little more in detail who wants me to go through in detail to this definition it's important you it's crystal clear to you what we are doing when we do this yes or no who wants more details no one wants to say one more deep it's okay okay good so how do we do this so we choose a basics VI of the hilbert space here and let's say
31:45
W J so these are some states in the two Hilbert spaces the states of this Hilbert space are labeled we can choose a basis which is e AI x WJ right where I runs from one to the dimension of a J runs from one to the dimension of P dimension of this is the sum of the dimensions now the product of the dimensions right dimension of this is the product of the dimensions of a times B and so the density matrix then
32:17
normally the density matrix has two indices alpha and beta but in this case each index alpha is really these two indices right so we have Rho of I and J and I prime and J prime okay so that's the total density matrix we can after we write it in this basis you will have this index structure this is a density matrix in subsystem a so this should
32:46
have which in this is true that have I or J or alpha who thinks the indices of this density matrix should be J and I oh yeah the problem is I I good but they we're paying attention I really needed to give more details because I wrote the wrong formula okay so indeed we have in this is I and J sorry I an i prime and this is the sum
33:34
over J of Rho I i prime IJ i prime J okay maybe I wrote it too small so it's I I i prime those are the indices of the hilbert space a and we shall salmon over the big one is that visible to everyone okay now we will continue will continue discussing the centerpiece of subsystems but i would like to now make a point
34:20
regarding relative entropy so particularly inequality that is very important and somewhat difficult to prove so I I will not prove it which is that if you start from two density matrices Rho and Sigma so now we'll combine the two concepts so we had two factors in the hilbert space and we'll go back to relative entropy we have two
34:48
density matrices and and now from these two density matrices we can construct through tracing out be the density matrices Rho a and and Sigma a okay so these were the density matrix is that we have if we look at the whole system right and now we have the density matrices of just a subsystem such as this is subsystem a
35:20
and we could calculate the corresponding relative entropy s ro a sigma a smaller or equal sorry and we can compare this to the relative entropy of Rho and Sigma an important thing is the inequality between these two now recall just to remember to remember which direction the inequality goes relative entropy is a measure of how different the two states are right now if you only look at a
35:52
subsystem right let's say at the part of the state part of the degrees of freedom do you think they are going to be more or less distinguishable less yes so everyone will think there will be less and indeed that's what happens and you can prove this inequality mathematically so giving more credence to the idea that this is a good measure of the distinguished ability of Rho and Sigma ok so this is this is a this is an important inequality that despite non-trivial
36:26
to prove in for non-community in matrices and and there are other others that derive from it so we'll discuss the others later but I guess that intuitively should be true it's a clear okay so now we go back to the discussion of the entropies of subsystems and let's
37:02
discuss so imagine that row is a pure state right so rock would be pure but if we okay so let's consider the following situation so imagine that you have a bunch of spins so this is subsystem a and subsystem B are here to separate it let's say some spins here some spins over there and we have some raw in the total Hilbert space which is pure so the entropy is zero so then can we if we compute the entropy
37:40
of subsystem a could that be nonzero oh sorry first I need to define what the entropy of the system is so the entropy of subsystem a is just the form an entropy of the density matrix of associated two subsystem a right so we have we start with the food density matrix and this is defined even if Rho wasn't pure so in general so this is a general definition whether Rho is pure or not so you can define the entropy
38:13
associated to subsystem a as the trace of this reduced density matrix defined in this way okay similarly we could define the entropy of subsystem B which is a similar construction but where we now trace out a subsystem a and we keep subsystem B and sometimes the total entropy of the density matrix Rho sometimes we are going to write it as a union B okay so
38:48
that we put together the spins in subsystem a and subsystem me okay we define this as a union B usually also in relativistic quantum field theory usually we'll going to take a to be a separation of space and then it becomes really the union of the region's geometrically but while internal is the union of the two and this is just the usual entropy that we define before for the density matrix R or original density matrix Rho
39:22
so now there are a bunch of inequalities that whole okay so one is this sub additive 'ti property which says that well first of all first that let's do an example and then we'll discuss
39:55
inequality so if if Rho is pure then the entropy si might still be nonzero so you could have a situation where this entropy is nonzero even though Rho is pure right when what's an example of a situation like this BR yes so if you have two spins in an EPR pair then the this entropy will be log two for each each of the spins but the total entropy would be zero and then
40:32
well so that's that was an example and in this cases where Rho is pure and we have a non zero entropy we assign this entropy to entanglement so the center we can only arise from entanglement entanglement and it is sometimes called entanglement entropy so when the original robe was pure this si of a subsystem is sometimes called entanglement entropy but of course if Rho initially was not pure to start with well it could have been this si might
41:05
arise from classical correlations they don't have to arise through entanglement now since this is sometimes called entanglement entropy people then sometimes call everything entropies of subsystems always entanglement or special in the physics recent high energy Theory literature so you should be aware that it's not exactly what sometimes advertised okay so there is the form and entropy of sub-regions defining this way when Rho is pure then we can think of
41:43
that as entanglement entropy okay now suppose that you have these two systems amb if Rho is pure what do you expect to be the relationship between SA and SD yeah so in that case that will be equal and in general there is a relationship that says that s a union B this would be zero for a pure state but it might be nonzero for a non pure state this P are
42:18
equal then as a minus SP B or equal than the difference right so this inequality says that if this was pure and this is zero then indeed si is equal to SB but the generalization of that intuition to the non pure case is this inequality that you can come true and also there is an inequality which says that si plus SP
42:48
is bigger or equal than this and one way to intuitively think about these inequalities so you can think of a and B I don't think this intuition translates into mathematical proof but so you can have things that are entangled between a and B and things which are entangled between the outside world right and if you think about the centerpieces as counting the number of lines that you cut then you get this type of inequalities okay now we can
43:53
now as we saw in this diagram so we so si might be nonzero because the there is entanglement with some external system right and but it could also be because there is some correlation between a and B right and there's a good some measure which is called mutual information actual information which is a measure of the correlation between a and B so it's called denoted by a and B this is this
44:27
now one is symmetric between a and B and it's a plus SP minus s a union B and by the inequality we wrote there that's always bigger or equal than zero and this is basically the is the correlation it could be quantum correlation or classical correlation so it could be that you have a mixed state but so you have coins that you throw and when you
44:58
throw the cons are up in one side they are down in their side completely classical but that will give rise to a mutual information of this type okay so one cool thing about this mutual information is that it bounds correlators so it's related to correlators in the following way so imagine that we have again these two subsystems and we have an operator o a in system subsystem a and an operator o
45:37
be in subsystem B then we can form the so that these operators will have some expectation values in each of the of course since it they are they act on subsystem a whenever I write expectation values we are imagining we are doing trace of Rho oh right and since this acts only on subsystem a we could also write it as trace of Rho a oh wait okay it's too small again yeah
46:10
it's get smaller when you get further yes this angle effect okay I think we are discovering a new low physics very good so now I'm going to consider the connected correlation functions so Oh a will be connected which are simply the correlations they're really true correlations between a and B where we subtract the expectation value of a and the expectation value a and B okay so that's
46:49
the connected correlator and the nice inequality is that this correlator can essentially be no bigger than the mutual information so so you take a be connected you take the square of this
47:21
but you need to divide of course you can always rescale the operator by some constant so we need to risk I'll risk a doubt so we'll define a norm of an operator to be the maximum eigen value so that's the norm of the operator similarly for OB so all these are things that are well defined in systems of finite numbers it's the freedom of course if you have a field operator in quantum field theory it's not but we'll discuss that case when we come to it so
47:56
this is less or equal than the mutual information between a and B okay and again all these things are relatively easy to prove once you find the right trick and those reviews the discussed okay so this is some these are some of the basics of entanglement and want to mentor P and so on and there are many other things that one could say that they will not say one could discuss Bell inequalities and how to make sure that
48:28
they are are not obeyed when we have when we divide the system into more than two parts then the characterization of entanglement is a little more subtle and I'm not going to discuss it and another thing I'm not going to discuss is that some of this can be phrased not so much in terms of dividing the hilbert space but in terms of the operator algebra and thinking about algebras and sub algebra and so on and assigning entropies to this algebras and again I want to discuss it from that
48:59
point of view but those are other topics I'm not discussing okay so now we'll so all these were purely preliminaries now we'll get to applying many of these ideas to quantum field theories and to apply them to quantum field theories there are many subtleties and the first one is that well the hilbert space is
49:30
infinite dimensional in quantum field theory and and so on and we'll try to deal with disabilities and the spice despite some of the subtleties there are non-trivial statements that you can make so some of these most of these inequalities are I mean are proven for finite systems or subsystems of finite systems but in many cases you can also prove them for sub systems of infinite systems of infinite dimensional hilbert spaces it's a little more subtle but it
50:04
has also been done for many of these entropy inequalities our yes yeah so separately a separable separable hilbert spaces but infinite dimensional so they'd been proven but we will the approach will take is that will and this has been taken it's been useful is to think about the continuum field theory by first doing a lattice regularization so that if you choose in many cases that will reduce the finite dimensional
50:41
hilbert space and then you apply these formulas with finite dimensional hilbert spaces sometimes you also need to discretize the target space and so on because even a harmonic oscillator has an infinite dimensional hilbert space sometimes you need to cut off with some energy cut off and so on to get to finite dimensional hilbert space okay so so now we are going to start talking about entanglement or entanglement in quantum field theory and the main the
51:13
main point is that in quantum field theory you can very good question yeah I thought I was putting this one one feature of this this entropy discussions and entanglement entropy is that which is I feel it's an uncomfortable feature is that entropy is not something
51:45
directly measurable and the the point is the following that anything that is directly measurable is some operator oh okay and then the results of oh or the expectation value for are computed for example by computing the expectation value with some density matrix but the entropies and the properties of any operator this kind is that it's linear in the density matrix but entropy is nonlinear in the in the density matrix
52:17
so if so it looks a bit like an expectation value of something but it's really something nonlinear so it's really not expectation value and so in order to determine entropy would have to somehow determine the density matrix and then compute this log now of course in thermodynamics we use entropy so you could say well I mean experimentalist measure entropy with no problem there is no problem but how do they measure entropy they measure entropy using the
52:49
first law of thermodynamics so they start from the system the system they have they lower the temperature they see how much heat comes out at each temperature they divide the heat divided by the temperature they sum this up they go to zero temperature and that's how to determine the entropy of a system not least in principle so that's that's how you do it in in principle and here you could imagine for a big system doing something like this but that's something that you cannot do somehow for a system
53:20
with a finite number of freedom you have to repeat the system many many times and do this anyway so that's an uncomfortable feature of this entanglement entropy but many nice results have been obtained by thinking about entanglement entropy so it's been so far mostly a theoretical tool because it's cooled from the theoretical point of view it's as we said difficult to measure or well if you take this very in principle point of view it's impossible to measure but nevertheless it's been
53:51
useful theoretically now what what should you think in a situation like this maybe this is the wrong thing to focus on but if you if you dismiss it completely you are throwing away all the nice the radical results you derive from it maybe there is some other quantity closely related that is more physical I feel in some sense that relative entropy is a little more physical notice the but close connection to the first loaf and tankmen's and so on and yeah so maybe
54:23
there is something better maybe there is some way to measure it is people have proposed indirect ways of measuring them but they are not very general no no no what I did is sorry I did I defined the so one defines the relative entropy and one notice is that if you do a small variation of row then because it's always bigger equal than zero and zero for Rho equal to Sigma then for a first order variation it's always stationary it's always zero so it's not the definition is something that follows
54:59
from the definition and the properties of the I mean follows trivially you can immediately check check check it please just exercise check that I mean you take that definition you take Rho equal to still written there somewhere no arrested okay just check it from the explicit definition and if you try to check it you will see it doesn't work unless you demand that trace of Delta is
55:30
zero because you know you need to keep the trace of Rho equal to 1 or Sigma quantum on anyway so okay so that's the point of it this been mostly theory so in quantum field theory we can have so one property an important property of quantum field theory is that we cannot localize particles so if we have let's say something like a single particle
56:01
state can we say that the single particle is localized within this very small region or not and well of course approximately we can write in in practice we can certainly localize this box is here and not somewhere else but if we go to very short distances then this is not possible we go to distance is smaller than the Compton well the wavelength of the particle because we try to measure the particle we create more particles and so on and so we cannot really localize particles we cannot say a particle is localized to a very small region of space but what we
56:34
can localize in quantum field theory are operators so operators our local so we can think of operators acting on the localize degree of freedom very very tiny and we have the intuition that the underlying degrees of freedom that are the fundamental dynamical variables of the quantum field theory are local ER is like having a local spin at this at each space-time point at each space spatial point okay and so we can
57:11
we have these operators which are local and we can in principle localize these operators to finite regions finite solution now this is a separation in space so this is a region in space in space now of course we also have time so we should think of let's say space and then we also have time so we we were
57:41
saying that we localized something we can think on your fob of operators which act somehow on this region of space the operator might be non-local in this region so you can have a Wilson loop for example a Wilson line that involves operators at different positions or it could be the definition you could measure an operator which is the product of a spin operator here times the spin operator here that's perfectly allowed right what's not allowed is an operator which is a measure of the spin outside
58:12
so you have all these operators it's a this is a sub algebra operator algebra so you can multiply operators and so on and furthermore we know that all the if you have some other local operator here the locality and causality in quantum field theory tell us that the properties of this operator are only determined by what Scott was going on in the past like on right so all operators which are in this coastal
58:44
wedge are determined by the data on the surface so we should think of this operator algebra not but we could think of it as a region in space that's perfectly fine but we could also think of all the operators in space-time which are localized within this so-called cosine wedge so cosine wedges you take the light sheets that come from the boundary of the region you take them backwards and forwards and you consider all the points in the interior so okay
59:20
so if you had if you had a field here and you had a cold I mean you have a second order equation the values relativistic second order equation the values of the field here would be determined by the values in this region very good so I said that states are not localized now the vacuum is of course the vacuum is just a pure state so if we
59:53
took the density matrix of the whole of the vacuum just that would have zero entropy but if we consider two sub regions so just the fact that correlation functions so let's say we have two sub regions a and B so we have a point X here and a point X Prime here you know that the expectation values of the field operator are nonzero so these are non zero so this implies that
01:00:23
through the bound that we were discussing over there the bound on the expectation values of operators there should be a non-trivial mutual information between this two and in particular there should be non-trivial entanglement entropy associated to each sub region and because this this expectation values can be very large when you take the points very close to each other you realize that this entanglement entropy will have to be infinite okay
01:00:53
now strictly speaking to apply those bounds me strictly speaking the norm of local of an operator of a field operator like this is infinite for two reasons one is very localized at the point and the second is that the actual amplitudes of this field go from plus to minus infinity so in order to say this a little better you would have to integrate the field operator over some region some small region R and then put that integral in the exponent right so now we get something which is
01:01:24
bounded eigenvalues now if you want to apply that argument but in any case the final conclusion will be that we expect this entropies to be infinite ok now so they are infinite so something that this infinite is not so well defined another thing is that really the the Hilbert space cannot in the continuum field theory cannot be this described separated into the Hilbert space inside
01:01:57
the region and the Hilbert space in the region outside or a complement ok so this is not strictly valid in the continuum limit so in the continuum limit we cannot do this and however is something we can do and the typical approach is to do it in first to a lattice regularization and then and then
01:02:29
everything is fine ok so then you can really divide the Hilbert spaces into the two parts and then we'll remove the so the the the strategy is first to lattice regularization then calculate this entropy si for example and then identify the regulary say the divergence is that arise when the lattice regulator goes to zero and understand them well enough so that you can remove them
01:03:00
right so these calculations of entropy are somewhat similar to calculations of any other observable we can most observables we do in continuum panting field theory when you try to compute them naively you get infinity and you have to be a little more sophisticated to compute something that this finite okay and so understanding the origin of the divergences is part of understanding how to get something finite out of these things and there isn't the recent progress has been in trying to understand those finite pieces okay so how much when I am I supposed to
01:03:34
end okay yeah okay all right good so yeah we'll discuss and yeah so the relative entropy is already finite and mutual information is already finite but in order to see why it's finite I'll need to let me know that's the point that was going to try to explain here so to understand why they are finite we need to understand the divergence is a little bit and today right now I'm going to just state what the divergence is look like and then
01:04:14
we'll probably discuss them in more detail in some more specific cases just understand why that is the case so the result so there is some epsilon will be the ultraviolent cut off would be the lattice spacing when we regularize the theory by putting it on a lattice and so here we are considering a region of in space so let's say could be a third color could be any shaped region in
01:04:45
space and we are going to calculate the entanglement entropy associated to or the phone Neumann entropy associated to this region and the first result is that the divergences go like the area of the region area of the boundary of the region and say in a second to the dimension of space-time minus two so this area is the area of the boundary of a so the boundary of a is this surface
01:05:15
here right so in space where two dimensional as in the blackboard so we have two dimensions of space one and two and in addition we have the time direction that I'm not drawing so in this case Capital D would be three and this would be one over epsilon and this area would be the length of this curve here okay is that clear so imagine you are in one plus one dimensions right so there's only one direction of space so that I drew here
01:05:49
and then we have an inter region is this interval what do you expect this formula to say what's the boundary of a are these two points right so we have two points and when this is - this becomes a logarithm of epsilon so in the particular case of the equal to two we have that s of a will have some divergences the lead in the brush senses will be the log of epsilon times some coefficient some number some times the
01:06:20
number of points number of points in the Boundary number of points the number of points would be two in this case but if we had written a where this disconnected region of two intervals right we would have a factor of four right but the idea is that these coefficients especially in this two dimensional case is a universal constant we'll discuss later in the higher dimensional cases the coefficients depend more strongly on the cutoff because if you rescale the cutoff of course this number gets rescaled here
01:06:52
since the coefficient is of a log it has a more environment okay and we'll we'll calculate it later but the important important point is that the divergence does not depend on the state that we have so if we had the back if we have the vacuum we have these diversions with some number if we keep the same regularization procedure and we consider a state which is not the vacuum then we'll get exactly the same number in the same diversions the finite pieces would be different but the divergences are going to be the same so this is yeah
01:07:26
yeah so here we are using a relaxed so all these things are true in any relativistic quantum field theory okay yeah you can have anything that is a quantum field theory so this does not apply to gravity for example so gravity will discuss later but if you have a theory that has gauge fields matter fields and so what gets is a little more tricky but it also has this property yes a local pants oh yeah okay so when I say
01:08:04
relativistic quantum field theory what I mean is a theory that is what we call local no not just causal but local that he has a local stress tensor its local in the sense that he has his localized degrees of freedom that exist at every point and so on so all the the typical theories we consider in physics are of this kind so the standard model is of this kind and the theories that arise in condensed matter physics are of this kind well the ones that are relativistic are of this kind and those are the theories I'm discussing here yes ah
01:08:37
sorry yes yeah I'm discussing here is for any state so I'm discussing the divergences and this is true for any state yeah yeah yeah so this is supposed not to depend on the Lagrangian is supposed to be something completely general now I'm in condensed matter physics there is a lot of discussion of area law and so on and when system can have an area law but they are they're interested in the finite pieces so they have a more
01:09:13
I mean that there is a you be cut off which is the atomic scale and then there will be some finite pace and the system could have an aerial or could not have an aerial or and it depends on more on the system okay so very good let's see what was I going to say yeah so we have this diversion terms and in general there is a whole series of diversion terms with less and less powers of Epsilon
01:09:43
so in two dimensions it just stops here and then it's finite that's for D equal to two for D equal to three we also get only this term and then something finite but for example for D equal to four we also get a term proportional to a time proportional to curvatures called em curvatures it could be the curvature of the space where this is define or equal also be the extrinsic curvature of the surface and so on that is down by some
01:10:16
some powers of epsilon to the D minus four so for D equal to four this is a logarithmic term okay and then we go to five dimensions this would be a one over epsilon and those would be the only two diversion terms and then we'll have to go on and on and there's been some characterization of this curvature term so they could be here various curvature structures involving the various contractions of the extrinsic curvatures and curvature of the ambient space but I won't discuss
01:10:46
them in detail but all these divergences are essentially state independent it's what I'm saying it's not 100% correct but let's say the first order correct like many things we have the first so there are sometimes some some subtleties in this direction for the one yeah sure yeah so they even so this yeah so here we can understand this even nested say if we think in terms of the extrinsic curvature right so you might
01:11:25
wonder why don't we get the term which is linear in the string see curvature the string see curvature K would be proportional let's say if you have a circle here Direction length L would be further 1 over L but if we consider the vacuum of the theory then the SA and SB should be the same right but the extrinsic curvature as viewed from B has the opposite sign so term which is odd in the extrinsic curvature cannot appear okay so that's why you don't get here the term which is linear in the extrinsic
01:11:57
curvature we would have only one power of epsilon and yeah so so that that gets rid of the odd terms which could only come from the extrinsic curvatures and then riemann curvatures and so on have to two derivatives so dimensional analysis together with this argument says that you have this even expansion but of course if the theory has some other mass scales and so on they could also appear in here and you could have a more complicated structure of
01:12:27
divergences this is the simplest situation where you let's say you have a scale invariant theory so when you have other mass scales they could also instead of a power of epsilon you could get the power of the masses and so on so have something a little more complicated but the basic idea is that whatever you have will involve in this case curvatures in the most completely correct statement is that it will involve local operators evaluated on
01:12:58
this boundary so that's the most general statement so when I said that there are some subtleties I mean in some cases you need to have subletting diversions that involve expectation by these local operators on the boundary they are not important for what we are going to say so worry about those good now the fact that these divergences are staking pendant implies that the well that will imply that the relative entropy is well-defined
01:13:28
okay because the relative entropy recall that raised the formula but could be viewed as the difference in entropy between the difference in free energy and difference in free energy is different in energy and entropy right at least for the entropy pieces is clear that we we have something finite because we are subtracting two infinite quantities and actually the relative entropy is really finite finite it's the thing that is best defined of in
01:13:59
continuing Pantone field theory it's relative entropy the other thing that is finite is mutual information so we have two regions we have the finishing of mutual information here for a and B and if we have this is region a and region B is a disconnected region separation that is separated from a then you find that here in SI you have a diversions which
01:14:31
is proportional to the area of a right and similarly for SP and in s of a union B will again get the area of a plus the area of B so that will cancel those diversion terms will cancel in this difference okay so that's something that is finite so something better define okay well yeah also stability in terms also because yeah so the reason that they are they cancel is that especially in this
01:15:06
case of mutual information they cancel even if we have well if you have curvatures it's always the interval of some local thing here and well you will have the sum of the two terms right so these terms are not only yeah maybe you should have emphasized that they are not only well in state independent and it is the area but this this area is just some local integral over the boundary so there is the boundary Shanae and we are just integrating square root of G over
01:15:35
that boundary so it's the integral of a local quantity along this boundary and here again we're integrating the curvature along the boundaries the fact that it is this integral make sure that they cancel so there is some integral here integral here the councilor okay so I could finish here it's a good place to stop or I could go for 10 more minutes good which some more

DOWNLOAD SUBTITLES: