### 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

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