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HONG LIU: So first, do you
have any questions regarding

00:00:23.490 --> 00:00:26.630
this Hawking-Page transition
we talked about last time

00:00:26.630 --> 00:00:32.479
because we were running
out of time, so it

00:00:32.479 --> 00:00:35.556
was a little bit hurried.

00:00:35.556 --> 00:00:37.097
Do you have any
questions about that?

00:00:39.447 --> 00:00:41.530
AUDIENCE: So what's the
original thermal AdS stat?

00:00:44.230 --> 00:00:48.630
Did you put some random
[INAUDIBLE] onto that metric,

00:00:48.630 --> 00:00:50.614
and it will generate
some [INAUDIBLE]?

00:00:50.614 --> 00:00:51.280
HONG LIU: Sorry?

00:00:51.280 --> 00:00:52.280
Say it again?

00:00:52.280 --> 00:00:57.780
AUDIENCE: So what's the
original thermal AdS state?

00:00:57.780 --> 00:01:05.810
HONG LIU: The thermal
AdS state-- you couple

00:01:05.810 --> 00:01:09.030
AdS to a thermal bath.

00:01:09.030 --> 00:01:14.070
And then whatever
excitation will

00:01:14.070 --> 00:01:17.940
be generated by that thermal
bath will be generated.

00:01:17.940 --> 00:01:20.720
So mostly it's the graviton gas.

00:01:20.720 --> 00:01:26.760
It's the gas of the massless
particles inside AdS.

00:01:26.760 --> 00:01:28.810
So that essentially.

00:01:28.810 --> 00:01:31.620
The field theory procedure
to go to cleaning space

00:01:31.620 --> 00:01:35.060
and to make the
cleaning time periodic.

00:01:35.060 --> 00:01:39.444
And that physically should
be interpreted as you

00:01:39.444 --> 00:01:41.360
just a coupled decisions
to your thermal bath.

00:01:47.782 --> 00:01:51.021
AUDIENCE: I have a question
not really related to this.

00:01:51.021 --> 00:01:52.562
But I'm just wondering
why do we want

00:01:52.562 --> 00:01:56.170
to consider [INAUDIBLE]
on a sphere,

00:01:56.170 --> 00:01:59.260
because previously we
discussed about [INAUDIBLE] 4,

00:01:59.260 --> 00:02:01.460
which is reality.

00:02:01.460 --> 00:02:02.220
What is this?

00:02:12.780 --> 00:02:14.840
HONG LIU: So normally when
we look at the theory,

00:02:14.840 --> 00:02:17.110
you want to look at the
theory from as many angles as

00:02:17.110 --> 00:02:18.460
possible.

00:02:18.460 --> 00:02:22.550
So some of them may not be
able to directly realize it

00:02:22.550 --> 00:02:24.060
in the experiment.

00:02:24.060 --> 00:02:27.130
But still, it's useful from
a theoretical perspective

00:02:27.130 --> 00:02:29.160
because that gives you
additional insights.

00:02:29.160 --> 00:02:31.640
So we see that the
physics on the sphere

00:02:31.640 --> 00:02:33.020
is actually quite rich.

00:02:33.020 --> 00:02:37.200
So that actually gives you
some insight into the dynamics

00:02:37.200 --> 00:02:39.371
and also into the
duality itself.

00:02:39.371 --> 00:02:39.870
Yeah.

00:02:39.870 --> 00:02:42.070
AUDIENCE: So there's no
condensed matter system

00:02:42.070 --> 00:02:43.991
essentially like that.

00:02:43.991 --> 00:02:46.240
HONG LIU: Oh, if you're
talking about condensed matter

00:02:46.240 --> 00:02:50.170
applications, you may even
imagine in some systems,

00:02:50.170 --> 00:02:52.420
you may be able to
put it on a sphere.

00:02:52.420 --> 00:02:55.100
Certainly, you can put it
on a two-dimensional sphere.

00:02:55.100 --> 00:03:00.610
And I can imagine you can put
it-- you can manipulate say,

00:03:00.610 --> 00:03:03.380
and put some spins on the
two-dimensional sphere.

00:03:03.380 --> 00:03:05.280
Yeah, you might be
able to do that.

00:03:07.920 --> 00:03:11.260
Any other questions?

00:03:11.260 --> 00:03:12.250
Yes?

00:03:12.250 --> 00:03:13.982
AUDIENCE: [INAUDIBLE]
you said there

00:03:13.982 --> 00:03:20.480
are two kinds of
[INAUDIBLE] connected

00:03:20.480 --> 00:03:21.490
on the same [INAUDIBLE].

00:03:21.490 --> 00:03:23.127
HONG LIU: Yes.

00:03:23.127 --> 00:03:24.460
AUDIENCE: So it's like two CFTs?

00:03:24.460 --> 00:03:26.390
HONG LIU: No, it's a single CFT.

00:03:26.390 --> 00:03:30.282
It's just a different sector
of the CFT contributes.

00:03:30.282 --> 00:03:34.530
AUDIENCE: So it's like 0 quantum
critical point [INAUDIBLE]?

00:03:34.530 --> 00:03:36.990
HONG LIU: No, this is a
first-order phase transition.

00:03:36.990 --> 00:03:39.920
So it's not the
quantum critical point.

00:03:39.920 --> 00:03:42.760
So the picture is that
you have a temperature.

00:03:42.760 --> 00:03:44.330
So it's some TC.

00:03:44.330 --> 00:03:46.860
So below TC, you
have a phase which

00:03:46.860 --> 00:03:48.950
we call the thermal AdS phase.

00:03:48.950 --> 00:03:51.850
And in this you have a
big black hole phase.

00:03:51.850 --> 00:03:54.600
So translated from the
field theory side-- so this

00:03:54.600 --> 00:04:01.500
goes running to the states which
energy scales into the power 0

00:04:01.500 --> 00:04:03.020
contributes.

00:04:03.020 --> 00:04:08.300
And here, it's dominated by the
state of energy O (N square).

00:04:11.880 --> 00:04:14.160
Of course, in the
thermal ensemble,

00:04:14.160 --> 00:04:16.370
every state in
principle contributes.

00:04:16.370 --> 00:04:20.079
And in this phase, it's
dominated by the contribution

00:04:20.079 --> 00:04:22.650
of the no energy state.

00:04:22.650 --> 00:04:25.780
And here, it's dominated
by the high-energy state.

00:04:25.780 --> 00:04:28.820
And really the high-energy state
also has a much higher entropy.

00:04:28.820 --> 00:04:30.695
And then they dominate
your thermal ensemble.

00:04:34.710 --> 00:04:36.954
Does this answer your question?

00:04:36.954 --> 00:04:40.363
AUDIENCE: Yeah, this looks
like in any transition,

00:04:40.363 --> 00:04:46.210
you have [INAUDIBLE] carry
energy and also [INAUDIBLE].

00:04:46.210 --> 00:04:48.420
HONG LIU: Yeah, it is a
little bit similar to that.

00:04:48.420 --> 00:04:51.230
There's also an
entropy thing there.

00:04:51.230 --> 00:04:53.460
Yeah, it's a balance
between two things.

00:04:53.460 --> 00:04:55.370
But the details
is very different.

00:04:55.370 --> 00:04:56.970
The details are different.

00:04:56.970 --> 00:04:58.493
There's some
qualitative similarity.

00:05:03.770 --> 00:05:08.716
AUDIENCE: So I read on
the paper [INAUDIBLE].

00:05:08.716 --> 00:05:13.870
He said that the Hawking-Page
transition on the field theory

00:05:13.870 --> 00:05:18.045
side is due to some
kind of deconfinement

00:05:18.045 --> 00:05:21.600
and confinement of QCD.

00:05:21.600 --> 00:05:26.500
HONG LIU: Yeah, so that's a
heuristic way to say about it.

00:05:26.500 --> 00:05:30.890
So the key thing is that here,
it's not really a confinement

00:05:30.890 --> 00:05:33.780
or not confined because
in the [INAUDIBLE] series,

00:05:33.780 --> 00:05:35.700
there's no confinement.

00:05:35.700 --> 00:05:38.080
It's just scaled in
[? Warren's ?] theory.

00:05:38.080 --> 00:05:42.230
And so in the sense he says
this is confined-- so he

00:05:42.230 --> 00:05:45.720
called this a confined phase.

00:05:45.720 --> 00:05:50.540
And this is a deconfined phase--
it's based on the following.

00:05:50.540 --> 00:05:53.720
It just refers to
these two behaviors.

00:05:53.720 --> 00:05:55.720
It just refers to
these two behaviors.

00:05:55.720 --> 00:06:05.290
So in QCD, which will generating
confinement or deconfinement--

00:06:05.290 --> 00:06:08.660
so in QCD, we have
[INAUDIBLE] 3.

00:06:08.660 --> 00:06:11.390
But if you scale
into infinity, then

00:06:11.390 --> 00:06:15.080
you will find in QCD
in the confined phase,

00:06:15.080 --> 00:06:18.610
the free energy will
scale as N with N power 0.

00:06:18.610 --> 00:06:20.090
But in the deconfined
phase, which

00:06:20.090 --> 00:06:23.760
is scale-- obviously
the N square.

00:06:23.760 --> 00:06:26.370
So that aspect of
scaling is the same

00:06:26.370 --> 00:06:29.446
as a serious risk confinement.

00:06:32.980 --> 00:06:35.150
So that aspect is similar.

00:06:35.150 --> 00:06:39.820
So he essentially
refers to this aspect.

00:06:39.820 --> 00:06:44.520
But on the sphere, every
state has to be a singlet.

00:06:44.520 --> 00:06:46.770
So in some sense,
this notion there

00:06:46.770 --> 00:06:49.360
is no genuinely
notion of confinement

00:06:49.360 --> 00:06:54.364
as we say in the QCD
in the flat space case.

00:06:54.364 --> 00:06:55.780
If you read his
paper, he actually

00:06:55.780 --> 00:06:56.946
described this very clearly.

00:07:01.514 --> 00:07:03.680
Yeah, he just called this
a deconfinement transition

00:07:03.680 --> 00:07:04.960
the heuristic way.

00:07:04.960 --> 00:07:06.298
Yes?

00:07:06.298 --> 00:07:09.770
AUDIENCE: I have a question
about the sum by which we

00:07:09.770 --> 00:07:11.782
calculate partition function.

00:07:11.782 --> 00:07:14.872
The previous class on
Monday, you wrote it as sum

00:07:14.872 --> 00:07:15.842
over energies.

00:07:15.842 --> 00:07:19.660
The degeneracy of that
[INAUDIBLE] factor.

00:07:19.660 --> 00:07:21.280
And you noticed
that both of these

00:07:21.280 --> 00:07:24.940
grew at exponential to
something times N squared.

00:07:24.940 --> 00:07:28.096
One was positive,
and one was negative.

00:07:28.096 --> 00:07:30.794
I was wondering if it's
possible that sum diverges?

00:07:30.794 --> 00:07:32.460
HONG LIU: No, that
sum does not diverge.

00:07:32.460 --> 00:07:34.360
That sum is always
proportional to n square.

00:07:37.190 --> 00:07:39.690
Yeah, depending on what
you mean the sum diverges.

00:07:39.690 --> 00:07:45.320
So that's just from the
definition of your partition

00:07:45.320 --> 00:07:48.090
function.

00:07:48.090 --> 00:07:50.810
So it's just the sum
of all possible states.

00:07:50.810 --> 00:07:53.584
And then you can just
reach this-- yeah, when

00:07:53.584 --> 00:07:55.000
you have a large
number of states,

00:07:55.000 --> 00:08:00.700
you can just roughly write it as
a continuous integral and then

00:08:00.700 --> 00:08:02.470
with the density of states.

00:08:06.420 --> 00:08:09.860
And so the key is
that what happens

00:08:09.860 --> 00:08:12.250
to the density of states--
so you're asking maybe

00:08:12.250 --> 00:08:14.910
whether this
integral will diverge

00:08:14.910 --> 00:08:16.410
when it equals infinity.

00:08:16.410 --> 00:08:17.910
So when it equals
infinity, you will

00:08:17.910 --> 00:08:24.829
see that say supposing
E scale as N cubed.

00:08:24.829 --> 00:08:26.620
Then you will see that
the density of state

00:08:26.620 --> 00:08:30.300
actually does not
grow that fast.

00:08:30.300 --> 00:08:36.900
And actually, it's only
for the E of O(N squared),

00:08:36.900 --> 00:08:39.164
then they scale in the same way.

00:08:39.164 --> 00:08:40.289
They scale in the same way.

00:08:40.289 --> 00:08:42.320
So that's why you
have this balance.

00:08:42.320 --> 00:08:45.200
And if you are not in the
scaling with [INAUDIBLE].

00:08:45.200 --> 00:08:49.167
If you have N cubed, and
then this will dominate.

00:08:49.167 --> 00:08:50.583
So that factor
will be suppressed.

00:08:56.940 --> 00:08:57.773
Any other questions?

00:09:02.590 --> 00:09:03.450
OK, very good.

00:09:03.450 --> 00:09:05.440
So now let's go to entanglement.

00:09:40.320 --> 00:09:42.980
So first, let me say a few
words about entanglement entropy

00:09:42.980 --> 00:09:43.480
itself.

00:09:48.750 --> 00:09:51.620
So this is very
elementary stuff,

00:09:51.620 --> 00:09:58.560
even though maybe
not everybody knows.

00:09:58.560 --> 00:10:01.397
So let's consider a quantum
system, just a general quantum

00:10:01.397 --> 00:10:01.897
system.

00:10:06.434 --> 00:10:08.100
So let's separate the
degrees of freedom

00:10:08.100 --> 00:10:13.607
into two parts, say A
plus B. So AB together

00:10:13.607 --> 00:10:14.440
is the whole system.

00:10:14.440 --> 00:10:18.052
We just separated the
degrees of freedom.

00:10:18.052 --> 00:10:19.760
So essentially by
definition, the Hilbert

00:10:19.760 --> 00:10:28.030
space of the system-- then
we'll have a tensor structure.

00:10:28.030 --> 00:10:31.350
So the full Hilbert space
will be a tensor product

00:10:31.350 --> 00:10:33.990
over the Hilbert
space for the A part

00:10:33.990 --> 00:10:37.057
and tensored with the
Hilbert space in the B part.

00:10:37.057 --> 00:10:37.556
OK?

00:10:47.030 --> 00:10:49.220
And consider, of
course, in general, we

00:10:49.220 --> 00:10:51.250
have an infinite
dimensional system.

00:10:51.250 --> 00:10:54.150
But in the case if you have
say a [INAUDIBLE] Hilbert

00:10:54.150 --> 00:10:57.290
space-- so i f this M
dimension, this is N dimension,

00:10:57.290 --> 00:10:59.835
then the total Hilbert space
will be M times N dimension.

00:11:02.558 --> 00:11:13.520
And the typical wave function
will have the form-- will be

00:11:13.520 --> 00:11:19.880
some sum-- say you can write
it in some basis in A and times

00:11:19.880 --> 00:11:23.730
some wave function
writing some bases in B.

00:11:23.730 --> 00:11:27.020
And this actually means that
the Hilbert space is a tensor

00:11:27.020 --> 00:11:27.550
product.

00:11:27.550 --> 00:11:29.800
It's just your wave form can
typically have this form.

00:11:35.920 --> 00:11:47.570
So we say that AB-- so we say
that in the states psi that AB

00:11:47.570 --> 00:12:11.830
are entangled if psi cannot be
written as a single product--

00:12:11.830 --> 00:12:15.920
rather than the sum of product
and if you just have a single

00:12:15.920 --> 00:12:16.420
product.

00:12:22.480 --> 00:12:25.980
So for a simple state,
it might be easy to see.

00:12:25.980 --> 00:12:27.960
But if I write the very
complicated state which

00:12:27.960 --> 00:12:33.100
many-- in principle,
write the state

00:12:33.100 --> 00:12:36.070
in some bases which look
at the complicated sum.

00:12:36.070 --> 00:12:39.160
But they may be some other
bases and be a simple product.

00:12:39.160 --> 00:12:41.140
So in general, it's
actually hard to tell.

00:12:41.140 --> 00:12:43.640
In general, it's hard to tell,
even though it's easy to say.

00:12:43.640 --> 00:12:46.580
But in general,
it's hard to say.

00:12:46.580 --> 00:12:51.990
So the entangled
entropy-- so let me just

00:12:51.990 --> 00:12:54.870
call it EE just to save space.

00:12:54.870 --> 00:13:02.990
EE essentially
provides a measure

00:13:02.990 --> 00:13:23.210
to quantify the entanglement
between A and B--

00:13:23.210 --> 00:13:25.940
because even in the case
which you know this state is

00:13:25.940 --> 00:13:28.190
in entangled state,
you may still

00:13:28.190 --> 00:13:30.305
want to ask how much
they are entangled.

00:13:34.910 --> 00:13:38.055
Entanglement entropy provides
the way to quantify it.

00:13:41.600 --> 00:13:46.950
So the definitions
are very simple.

00:13:46.950 --> 00:13:50.280
So first we look at
this the density matrix

00:13:50.280 --> 00:13:53.360
for the total system.

00:13:53.360 --> 00:13:55.910
So if the system is
in the state psi,

00:13:55.910 --> 00:13:58.110
then the density matrix
for the total system

00:13:58.110 --> 00:14:02.115
would be just on the psi
conjugate-- psi itself.

00:14:04.650 --> 00:14:06.640
So in this basic matrix,
we just trace out

00:14:06.640 --> 00:14:13.640
all the degrees of freedom in
B. So since the Hilbert space

00:14:13.640 --> 00:14:17.090
have a tensor product structure,
you can always do this.

00:14:17.090 --> 00:14:19.470
Then what you get is you
get this the density matrix.

00:14:19.470 --> 00:14:23.440
And then here, you
get a density matrix

00:14:23.440 --> 00:14:28.037
which only depends on the
degree of freedom in A.

00:14:28.037 --> 00:14:30.120
And then we just calculate
the Von Neumann entropy

00:14:30.120 --> 00:14:33.025
corresponding to this row
A. So entangled entropy

00:14:33.025 --> 00:14:41.460
is just defined to be
[INAUDIBLE] entropy associates

00:14:41.460 --> 00:14:45.169
resistance in the matrix
rho A. So this rho A

00:14:45.169 --> 00:14:46.960
is often called the
reduced density matrix.

00:14:58.990 --> 00:15:01.490
And the Von Neumann entropy
associated with the reduced

00:15:01.490 --> 00:15:05.642
density matrix is defined
as the entanglement entropy.

00:15:05.642 --> 00:15:06.142
OK?

00:15:10.390 --> 00:15:13.010
So this provides a
very good the measure

00:15:13.010 --> 00:15:26.725
because say if SA is equal to
0, from our knowledge of the Von

00:15:26.725 --> 00:15:30.340
Neumann entropy,
you can immediately

00:15:30.340 --> 00:15:37.620
deduce where the SA is equal
to 0 if only if the rhos A,

00:15:37.620 --> 00:15:40.030
this density matrix
is a pure state.

00:15:45.590 --> 00:15:46.472
It's a pure state.

00:15:51.500 --> 00:15:56.482
And then from here, you can also
deduce rho A is a pure state.

00:15:56.482 --> 00:15:58.690
This reduced density matrix
comes from this procedure

00:15:58.690 --> 00:16:00.060
as a pure state.

00:16:00.060 --> 00:16:08.470
Only if the psi can be
written as a simple product.

00:16:22.900 --> 00:16:31.190
So this tells you as
when SA is non-zero,

00:16:31.190 --> 00:16:36.496
then you can be sure this
state must be non-entangled.

00:16:40.760 --> 00:16:44.740
So whenever SA is
equal to 0, you

00:16:44.740 --> 00:16:46.460
can be sure this
is non-entangled.

00:16:46.460 --> 00:16:49.880
When SA is not
equal to 0, you know

00:16:49.880 --> 00:16:56.800
for sure A and B must be
entangled in this state.

00:16:56.800 --> 00:16:58.540
So that's why this
is a good measure.

00:16:58.540 --> 00:16:59.800
OK?

00:16:59.800 --> 00:17:02.990
And then the value of
SA then will tell you

00:17:02.990 --> 00:17:04.736
how entangled the system is.

00:17:12.390 --> 00:17:15.200
And also, you may
immediately ask

00:17:15.200 --> 00:17:18.500
the question what happens
instead of defining the rho A?

00:17:18.500 --> 00:17:20.630
Of course, you can
also do the same thing.

00:17:20.630 --> 00:17:25.154
You trace out A to define
out rho B and then the

00:17:25.154 --> 00:17:27.750
define the entropy
for the rho B. OK?

00:17:27.750 --> 00:17:30.860
Define entropy for the
rho B. But you can easily

00:17:30.860 --> 00:17:42.970
show for any pure state
psi SA always equals to SB.

00:17:42.970 --> 00:17:45.530
So it doesn't matter
which are symmetric.

00:17:45.530 --> 00:17:49.840
So it doesn't matter which
one you're looking at.

00:17:53.040 --> 00:17:58.480
So this is very easy
to prove, essentially

00:17:58.480 --> 00:18:02.750
just following from something
called a Schmidt decomposition.

00:18:02.750 --> 00:18:06.520
And essentially right to this
state in terms of Schmidt

00:18:06.520 --> 00:18:09.560
decomposition between the
degrees of freedom rho A and B.

00:18:09.560 --> 00:18:12.710
And then you can show that
rho A and rho B essentially

00:18:12.710 --> 00:18:14.690
have the same eigenvalues.

00:18:14.690 --> 00:18:17.780
And if rho A and rho B have the
same eigenvalues, of course,

00:18:17.780 --> 00:18:20.440
then the entropy will be the
same, because the entropy only

00:18:20.440 --> 00:18:21.700
depends on the eigenvalues.

00:18:21.700 --> 00:18:22.200
OK?

00:18:24.710 --> 00:18:25.825
Any questions about this?

00:18:31.010 --> 00:18:32.350
Good.

00:18:32.350 --> 00:18:36.860
So here is what a mixed
state-- about a pure state,

00:18:36.860 --> 00:18:43.410
let me just say a side remark.

00:18:43.410 --> 00:18:56.510
So if AB, the total system is
in a mixed state-- so far always

00:18:56.510 --> 00:18:57.830
is in the pure state.

00:18:57.830 --> 00:18:59.840
But suppose it's
in the mixed state.

00:18:59.840 --> 00:19:03.960
Suppose the system itself is
described by a density matrix.

00:19:03.960 --> 00:19:08.130
So the total system
itself is spread.

00:19:08.130 --> 00:19:10.330
In general, SA is
not equal to SB.

00:19:12.970 --> 00:19:14.510
OK?

00:19:14.510 --> 00:19:19.440
Not only in general,
it's not equal to-- SA

00:19:19.440 --> 00:19:20.500
is not equal to SB.

00:19:23.960 --> 00:19:36.570
And in such a case, the
entanglement entropy

00:19:36.570 --> 00:19:55.750
also contains classical
statistical correlations

00:19:55.750 --> 00:20:08.010
of the mixed state--
of the mixed state--

00:20:08.010 --> 00:20:10.070
so in addition to
quantum correlations.

00:20:13.250 --> 00:20:17.950
It's almost trivial to see
because the people suppose

00:20:17.950 --> 00:20:19.180
here is not a pure state.

00:20:19.180 --> 00:20:23.690
So here you replace it
by the density matrix.

00:20:23.690 --> 00:20:27.500
And when you trace all the B
in this density matrix, then,

00:20:27.500 --> 00:20:31.280
of course, there's still
some original uncertainty

00:20:31.280 --> 00:20:34.200
in the previous density
matrix remaining in A.

00:20:34.200 --> 00:20:41.770
And then they will come into
this entropy-- so as defined,

00:20:41.770 --> 00:20:45.370
will depend on the
classical uncertainties--

00:20:45.370 --> 00:20:47.300
classical statistical
uncertainties

00:20:47.300 --> 00:20:51.520
of your original density matrix.

00:20:51.520 --> 00:20:55.440
So for a mixed state-- so
that's why for a mixed state,

00:20:55.440 --> 00:20:59.840
the internal entropy is not a
very good measure of quantum

00:20:59.840 --> 00:21:03.860
entanglements, because
it's contaminated

00:21:03.860 --> 00:21:07.240
by classical
statistical information.

00:21:10.350 --> 00:21:12.060
But for today, we
all need to talk

00:21:12.060 --> 00:21:14.820
about the pure-- mostly
talk about the pure.

00:21:14.820 --> 00:21:19.190
Yeah, we actually
talk about in general.

00:21:19.190 --> 00:21:23.100
But I want you to
keep this in mind.

00:21:23.100 --> 00:21:27.625
So any questions so far?

00:21:27.625 --> 00:21:29.750
So just to give you an a
little bit more intuition,

00:21:29.750 --> 00:21:34.949
let's look at this
very simple example

00:21:34.949 --> 00:21:36.240
to calculate entangled entropy.

00:21:36.240 --> 00:21:37.740
So let's consider
a two-spin system.

00:21:42.310 --> 00:21:45.840
So let's consider you
have two spins, OK?

00:21:45.840 --> 00:21:51.430
So this is my A and B. So this
has a two-dimensional Hilbert

00:21:51.430 --> 00:21:51.930
space.

00:21:51.930 --> 00:21:53.600
This has a two-dimensional
Hilbert space.

00:21:53.600 --> 00:21:55.849
Altogether, you have a
four-dimensional Hilbert space.

00:21:58.570 --> 00:22:02.520
So for example, so let
me consider such a state.

00:22:18.310 --> 00:22:21.780
So this looks like
a complicated state.

00:22:21.780 --> 00:22:25.190
But actually, this can be
written as a single product,

00:22:25.190 --> 00:22:41.670
because you it can
be written at OK,

00:22:41.670 --> 00:22:43.190
I hope this notation is for me.

00:22:43.190 --> 00:22:44.510
It's OK with you.

00:22:44.510 --> 00:22:50.300
You just A and B--
A spin and B spin.

00:22:50.300 --> 00:22:54.375
So even though in this
space, this is written as

00:22:54.375 --> 00:22:55.480
say a sum of state.

00:22:55.480 --> 00:22:57.230
It looks the entangleds.

00:22:57.230 --> 00:23:01.340
But in fact, it's not
because you can write it

00:23:01.340 --> 00:23:04.230
in terms of a simple product.

00:23:04.230 --> 00:23:05.970
So this is another
entangled state.

00:23:13.350 --> 00:23:16.100
Of course, for the simple
system, it's very easy to tell.

00:23:16.100 --> 00:23:18.430
But to give you a
complicated system

00:23:18.430 --> 00:23:20.630
with many, many,
degrees of freedom,

00:23:20.630 --> 00:23:22.595
then it becomes very hard.

00:23:22.595 --> 00:23:24.865
And then entangled
entropy becomes useful.

00:23:28.026 --> 00:23:30.400
So now let me give you an
example to calculated entangled

00:23:30.400 --> 00:23:31.150
entropy.

00:23:31.150 --> 00:23:36.591
So let's consider
a state like this.

00:23:45.970 --> 00:23:49.560
So let's see some parameter.

00:23:49.560 --> 00:23:53.920
So clearly this state is
entangled because you cannot

00:23:53.920 --> 00:23:57.280
write it as a simple product.

00:23:57.280 --> 00:23:59.680
So now, let's check it.

00:23:59.680 --> 00:24:01.390
Now, let's check it.

00:24:01.390 --> 00:24:08.630
So you can look at the full
density matrix of the system.

00:24:08.630 --> 00:24:13.850
You just look at this--
the bra and the ket itself.

00:24:13.850 --> 00:24:17.560
And then just do the product.

00:24:17.560 --> 00:24:34.300
And you get the quotient theta--
under the [INAUDIBLE] terms.

00:25:01.900 --> 00:25:07.990
So you just take the
product with itself.

00:25:07.990 --> 00:25:11.520
And then let's try
to find what's rho A.

00:25:11.520 --> 00:25:13.650
So you trace out
degrees of freedom of B.

00:25:13.650 --> 00:25:17.340
So this is our B,
the second spin.

00:25:17.340 --> 00:25:19.420
And now let's trace out
degrees of freedom B--

00:25:19.420 --> 00:25:26.690
trace out the second spin--
so rho A. Goes one into we

00:25:26.690 --> 00:25:29.580
trace out the B in here.

00:25:29.580 --> 00:25:32.480
So when you trace
out the B, you just

00:25:32.480 --> 00:25:35.030
take the end product
between these two.

00:25:35.030 --> 00:25:36.980
So these are the same.

00:25:36.980 --> 00:25:39.940
So this will remain.

00:25:39.940 --> 00:25:41.351
And so you have this.

00:25:44.510 --> 00:25:47.410
And similarly, this
one you have that.

00:25:47.410 --> 00:25:48.490
These two are the same.

00:25:48.490 --> 00:25:50.323
So when you take the
trace, this is nonzero.

00:25:58.030 --> 00:26:00.160
But this too will give
you 0, because this one is

00:26:00.160 --> 00:26:01.650
orthogonal to this one.

00:26:01.650 --> 00:26:03.677
And this spin is
orthogonal to that spin.

00:26:03.677 --> 00:26:04.635
So that's what you get.

00:26:07.020 --> 00:26:09.395
And then now you can usually
just write down the entropy.

00:26:30.890 --> 00:26:33.910
So this is the entangled
entropy for these two states--

00:26:33.910 --> 00:26:37.220
for this two spin system
as a general function

00:26:37.220 --> 00:26:40.560
of this parameter theta.

00:26:40.560 --> 00:26:43.882
So now let's plot these
as a function of S theta.

00:26:47.120 --> 00:26:49.320
So clearly, this is
a period function.

00:26:49.320 --> 00:26:51.177
We only need to go to pi over 2.

00:26:53.920 --> 00:26:57.187
And then you have
something like this.

00:26:57.187 --> 00:26:58.770
And you can easily
plot that function.

00:26:58.770 --> 00:27:00.230
You will see
something like this.

00:27:00.230 --> 00:27:02.580
So when theta is equal to 0.

00:27:02.580 --> 00:27:06.220
This is equal to 0
because this term is 0.

00:27:06.220 --> 00:27:08.510
This term is also 0 because
of the quotient theta

00:27:08.510 --> 00:27:10.770
square is equal to 1.

00:27:10.770 --> 00:27:13.710
When theta equals to
pi over 2, this term

00:27:13.710 --> 00:27:15.710
becomes 0 and this
term also becomes,

00:27:15.710 --> 00:27:18.670
because it's [INAUDIBLE] 0.

00:27:18.670 --> 00:27:24.300
But this maximum in the
pi over 2 or pi over 4.

00:27:24.300 --> 00:27:25.590
So I had a pi over 4.

00:27:46.910 --> 00:27:50.310
Of course, you can also go
to the pi equals to minus 4.

00:27:50.310 --> 00:27:53.409
OK, you can go do plus minus 4.

00:27:53.409 --> 00:27:54.700
I didn't go to the [INAUDIBLE].

00:27:54.700 --> 00:27:57.936
Yeah, anyway, so these are
called the maximum entangled

00:27:57.936 --> 00:27:58.435
states.

00:28:07.690 --> 00:28:10.299
This is a maximum entangled.

00:28:10.299 --> 00:28:12.673
So this is a state in which
you have the highest entropy.

00:28:19.860 --> 00:28:20.960
Any questions so far?

00:28:25.164 --> 00:28:26.613
AUDIENCE: You switched it.

00:28:26.613 --> 00:28:27.579
It should be up down.

00:28:27.579 --> 00:28:29.520
It's down.

00:28:29.520 --> 00:28:31.130
HONG LIU: Right, that's right.

00:28:34.490 --> 00:28:35.150
OK, good.

00:28:41.120 --> 00:28:44.710
So let me also say a few
things about the properties

00:28:44.710 --> 00:28:47.420
of the entanglement entropy.

00:28:47.420 --> 00:28:51.090
So there are many properties
you can derive from here.

00:28:51.090 --> 00:28:56.436
Let me only say a
few important ones.

00:29:08.850 --> 00:29:16.790
OK, so one property of
the entangled entropy

00:29:16.790 --> 00:29:18.670
is called subadditivity
condition.

00:29:26.250 --> 00:29:30.080
So if you can see that
the two systems A and B,

00:29:30.080 --> 00:29:33.550
then you can show that
S(AB), the entropy

00:29:33.550 --> 00:29:37.590
for the total system,
is smaller than the sum

00:29:37.590 --> 00:29:38.633
of the separate system.

00:29:41.390 --> 00:29:44.370
So when I write AB, I
mean the combined system.

00:29:44.370 --> 00:29:45.760
OK?

00:29:45.760 --> 00:29:49.338
And this is greater than
the difference between them.

00:29:54.750 --> 00:29:57.640
So this is so-called the
subadditive condition.

00:29:57.640 --> 00:30:02.560
And then intuitively,
you can understand.

00:30:02.560 --> 00:30:05.230
So if you have entropy
of A and entropy of B,

00:30:05.230 --> 00:30:09.630
you add them together, then it's
larger than the entropy of AB

00:30:09.630 --> 00:30:11.110
because there is
some redundancy.

00:30:11.110 --> 00:30:14.380
There might be some
redundancy here.

00:30:14.380 --> 00:30:17.280
Yeah, because when
you add AB together,

00:30:17.280 --> 00:30:19.740
there may be some common
correlation between them.

00:30:19.740 --> 00:30:22.000
And so this is
greater than that.

00:30:22.000 --> 00:30:24.321
OK, intuitively that's
what this inequality means.

00:30:28.920 --> 00:30:31.575
And also there are some property
for the strong subadditivity

00:30:31.575 --> 00:30:32.075
condition.

00:30:39.161 --> 00:30:40.661
AUDIENCE: I don't
quite understand--

00:30:43.540 --> 00:30:45.424
HONG LIU: Yes?

00:30:45.424 --> 00:30:48.360
AUDIENCE: What S of
A and S of B means?

00:30:48.360 --> 00:30:52.510
HONG LIU: So this is
the entropy for the A.

00:30:52.510 --> 00:30:55.290
This is the entropy for the B.

00:30:55.290 --> 00:30:57.660
AUDIENCE: Don't we
need this to partition

00:30:57.660 --> 00:31:00.270
system A into two subsystems?

00:31:00.270 --> 00:31:04.310
HONG LIU: No, you don't--
let me explain my notation.

00:31:04.310 --> 00:31:10.770
So S(A) means the
entropy equals 1

00:31:10.770 --> 00:31:14.030
if you integrate out
everything else except A.

00:31:14.030 --> 00:31:15.930
And S(B) means you
integrate everything

00:31:15.930 --> 00:31:19.000
else except B.
And S(AB) be means

00:31:19.000 --> 00:31:22.820
to integrate everything
else except A and B.

00:31:22.820 --> 00:31:25.760
And this is S(AB).

00:31:25.760 --> 00:31:29.694
AUDIENCE: But in our case, A
and B is all that there is.

00:31:29.694 --> 00:31:30.630
HONG LIU: No, no, no.

00:31:30.630 --> 00:31:33.582
Now I'm just doing generally.

00:31:33.582 --> 00:31:40.960
Once I have this definition,
so this can be-- even

00:31:40.960 --> 00:31:46.070
AB is a total-- yeah, so this
can apply both to the case

00:31:46.070 --> 00:31:49.070
I said earlier-- say
if you divide-- here I

00:31:49.070 --> 00:31:53.090
just-- so this AB does not
have to be the same as that AB.

00:31:53.090 --> 00:31:56.810
Here I'm just talking
about only two subsystems.

00:32:01.310 --> 00:32:04.290
AUDIENCE: So it's
for example, S(A)

00:32:04.290 --> 00:32:09.630
is like-- A and the supplement
as two parts of the system.

00:32:09.630 --> 00:32:11.310
And they could have
some entanglement.

00:32:11.310 --> 00:32:13.500
S(A) is just an
entanglement field.

00:32:13.500 --> 00:32:16.330
HONG LIU: A, S(A)
is the entanglement

00:32:16.330 --> 00:32:18.850
of A with the rest.

00:32:18.850 --> 00:32:24.940
And S(B) is the entanglement
of B between B and the rest.

00:32:24.940 --> 00:32:28.540
And the S(AB) is
entangled between the AB

00:32:28.540 --> 00:32:32.200
together with the rest.

00:32:32.200 --> 00:32:32.930
Yes.

00:32:32.930 --> 00:32:35.290
AUDIENCE: So to clarify one
thing, when you say S of A,

00:32:35.290 --> 00:32:36.430
we have this whole system.

00:32:36.430 --> 00:32:38.380
It means that trace
everything which is not A?

00:32:38.380 --> 00:32:39.936
HONG LIU: Yeah, that's right.

00:32:47.630 --> 00:32:51.090
Any more questions about this?

00:32:51.090 --> 00:32:53.580
So here when I write
to those expressions,

00:32:53.580 --> 00:32:57.902
I assume A and B don't
have interceptions, OK?

00:32:57.902 --> 00:32:59.652
I assume A and B don't
have interceptions.

00:33:04.000 --> 00:33:06.860
You can also have the strong
subbadditivity condition.

00:33:06.860 --> 00:33:09.595
So this is pretty easy to prove.

00:33:13.020 --> 00:33:17.010
If we have time, it takes
five minutes to prove.

00:33:17.010 --> 00:33:21.750
But the strong subadditivity
which I'm going to write down.

00:33:21.750 --> 00:33:26.320
So strong additivity is
involving three systems.

00:33:26.320 --> 00:33:28.790
Now, you have to add
the three systems.

00:33:28.790 --> 00:33:48.177
And then greater than ABC-- OK?

00:34:00.550 --> 00:34:04.050
So, again, intuitively, the
meaning of that inequality

00:34:04.050 --> 00:34:07.130
is clear.

00:34:07.130 --> 00:34:11.070
So the first
inequality just says

00:34:11.070 --> 00:34:12.720
that I have two systems here.

00:34:16.090 --> 00:34:20.524
And the sum of
these two systems--

00:34:20.524 --> 00:34:21.940
the entropy of
these two systems--

00:34:21.940 --> 00:34:27.320
is greater than the sum
between the combined

00:34:27.320 --> 00:34:30.610
system and the intersection
between the two systems,

00:34:30.610 --> 00:34:33.360
OK, because C in the
section between the two.

00:34:33.360 --> 00:34:35.640
And ABC is the whole
thing combined.

00:34:38.239 --> 00:34:42.679
And similarly here, he said
they have S(A) and S(B).

00:34:42.679 --> 00:34:45.600
I have two A and B. So
I trace out the system.

00:34:45.600 --> 00:34:49.960
Outside A, I get entropy for A.
I trace the system outside B,

00:34:49.960 --> 00:34:57.990
anyway for B. This
inequality says if now you

00:34:57.990 --> 00:35:04.740
attach A C to A and C to B, a
common system to both A and B.

00:35:04.740 --> 00:35:07.990
And then the resulting system
will be larger than the B form.

00:35:07.990 --> 00:35:11.130
So if you add something,
we increase the entropy.

00:35:11.130 --> 00:35:12.830
That, of course, is
intuitively clear,

00:35:12.830 --> 00:35:19.240
because if you have entropy
essentially parameterized

00:35:19.240 --> 00:35:21.190
the unknown part of the system.

00:35:21.190 --> 00:35:24.960
And if you add the third
system, and then this

00:35:24.960 --> 00:35:29.460
just increase your unknown and
then increase your entropy.

00:35:33.300 --> 00:35:37.590
So this strong
subadditivity condition

00:35:37.590 --> 00:35:39.980
is actually very hard to prove.

00:35:39.980 --> 00:35:41.265
It's very hard to prove.

00:35:43.910 --> 00:35:52.490
It can get rather mathematical
and requires some effort

00:35:52.490 --> 00:35:53.850
to do it.

00:35:53.850 --> 00:35:56.110
But this one is
pretty easy to do.

00:35:56.110 --> 00:35:56.610
Yes?

00:35:56.610 --> 00:35:59.885
AUDIENCE: Can [INAUDIBLE] about
the first of the two equations?

00:35:59.885 --> 00:36:02.420
For strong subadditivity,
what's the first equation?

00:36:02.420 --> 00:36:05.670
HONG LIU: Yeah,
so this means-- so

00:36:05.670 --> 00:36:08.490
look at this as a single system.

00:36:08.490 --> 00:36:10.260
This is a single system.

00:36:10.260 --> 00:36:12.950
That means that the sum of these
two-- the entropy of this two

00:36:12.950 --> 00:36:17.390
system is greater than
the sum of the combined

00:36:17.390 --> 00:36:19.815
system under the intersection.

00:36:28.960 --> 00:36:30.434
Any questions about this?

00:36:33.006 --> 00:36:33.506
Good.

00:36:41.040 --> 00:36:43.200
So this concludes the
very simple introduction

00:36:43.200 --> 00:36:44.680
to the entangled entropy.

00:36:44.680 --> 00:36:46.054
AUDIENCE: Does
the first one have

00:36:46.054 --> 00:36:50.664
any implications of topology?

00:36:50.664 --> 00:36:52.580
HONG LIU: Sorry, what
do you mean by topology?

00:36:52.580 --> 00:37:00.842
AUDIENCE: Because it's like the
intersection of the combined

00:37:00.842 --> 00:37:03.720
thing.

00:37:03.720 --> 00:37:08.400
HONG LIU: Yeah, so
as classical entropy,

00:37:08.400 --> 00:37:10.640
this is a very simple thing.

00:37:10.640 --> 00:37:12.540
You can easily convince
yourself if ABC

00:37:12.540 --> 00:37:16.620
are classical systems, and
just classical distributions,

00:37:16.620 --> 00:37:19.220
describe classical
probabilities,

00:37:19.220 --> 00:37:21.200
statistical distributions.

00:37:21.200 --> 00:37:24.590
And then with those things, you
can understand this inequality

00:37:24.590 --> 00:37:27.620
just by drawing this kind of
standard diagram corresponding

00:37:27.620 --> 00:37:28.659
to the different sets.

00:37:28.659 --> 00:37:30.950
The quantum mechanics proving
those things are actually

00:37:30.950 --> 00:37:33.500
not trivial are not trivial.

00:37:33.500 --> 00:37:35.430
Quantum mechanically,
they no longer come in

00:37:35.430 --> 00:37:37.380
a very intuitive way.

00:37:37.380 --> 00:37:38.634
Yes?

00:37:38.634 --> 00:37:40.540
AUDIENCE: I was noticing
something curious

00:37:40.540 --> 00:37:44.194
about the last line classically.

00:37:44.194 --> 00:37:51.120
If A and B are the same, I think
the equation will still hold.

00:37:51.120 --> 00:37:53.619
But quantumly--

00:37:53.619 --> 00:37:54.660
HONG LIU: It still holds.

00:37:54.660 --> 00:37:58.180
AUDIENCE: Doesn't it
not hold quantumly?

00:37:58.180 --> 00:38:01.190
HONG LIU: No, they are supposed
to hold quantum mechanically.

00:38:01.190 --> 00:38:05.504
AUDIENCE: But we are under the
assumption that A, B, and C--

00:38:05.504 --> 00:38:07.900
they don't overlap, right?

00:38:07.900 --> 00:38:10.790
I'm saying if A and B
are totally overlapping.

00:38:10.790 --> 00:38:14.990
HONG LIU: No, but
this inequality

00:38:14.990 --> 00:38:17.850
I'm writing in a way which
they are not intersecting.

00:38:17.850 --> 00:38:20.190
You can write them in a
way which they intersect.

00:38:20.190 --> 00:38:22.620
And I'm just
writing in this way.

00:38:22.620 --> 00:38:26.190
So the condition for
this particular form ABC

00:38:26.190 --> 00:38:29.222
is not supposed to intersect.

00:38:29.222 --> 00:38:33.870
AUDIENCE: Right, I'm saying
if we set A equals to B.

00:38:33.870 --> 00:38:37.740
HONG LIU: No, no, then you
cannot use this equation.

00:38:37.740 --> 00:38:39.215
Then you have to
write the equation

00:38:39.215 --> 00:38:42.560
in somewhat different way, which
apply to the intersection case.

00:38:42.560 --> 00:38:46.550
You can do them just choosing
to write in the way which

00:38:46.550 --> 00:38:47.936
did only intersect.

00:38:47.936 --> 00:38:50.580
AUDIENCE: OK, but the
thing I want to say

00:38:50.580 --> 00:38:53.130
is that we can still keep the
bottom equation classically,

00:38:53.130 --> 00:38:54.649
even if we took--

00:38:54.649 --> 00:38:56.440
HONG LIU: No, both are
applied classically.

00:38:56.440 --> 00:38:58.850
Both are trivial classically.

00:38:58.850 --> 00:39:00.930
You can understand very
easily classically.

00:39:00.930 --> 00:39:04.260
It's just quantum mechanically,
it's no longer trivial.

00:39:04.260 --> 00:39:06.410
Quantum mechanics is
no longer trivial.

00:39:06.410 --> 00:39:09.900
And there are various ways
you can write those equations.

00:39:09.900 --> 00:39:13.750
I'm just choosing to write the
way which ABC don't intersect.

00:39:13.750 --> 00:39:16.344
And you can also write
the way, just rename them

00:39:16.344 --> 00:39:18.099
so that they intersect.

00:39:18.099 --> 00:39:20.494
AUDIENCE: So [INAUDIBLE].

00:39:20.494 --> 00:39:24.280
So this is true for any
entanglement state we choose?

00:39:24.280 --> 00:39:26.667
HONG LIU: Any state,
yeah, density matrix.

00:39:26.667 --> 00:39:28.500
Whether it's a pure
state or density matrix,

00:39:28.500 --> 00:39:34.490
it doesn't matter--
a general statement.

00:39:34.490 --> 00:39:36.180
OK, good.

00:39:36.180 --> 00:39:39.550
So this concludes the
short introduction

00:39:39.550 --> 00:39:43.630
to the entangled entropy itself.

00:39:43.630 --> 00:39:47.450
And you may know that the
entanglement entropy actually

00:39:47.450 --> 00:39:48.810
plays a very important role.

00:39:48.810 --> 00:39:50.791
Entanglement and
entanglement entropy

00:39:50.791 --> 00:39:53.165
itself plays a very important
role in quantum information

00:39:53.165 --> 00:39:57.390
and quantum computing because
of quantum entanglement

00:39:57.390 --> 00:39:59.750
is the kind of
quantum correlations

00:39:59.750 --> 00:40:03.680
which you don't
have classically.

00:40:03.680 --> 00:40:07.464
So this EPR paradox and
the [INAUDIBLE] inequality,

00:40:07.464 --> 00:40:09.130
and the teleportation--
all those things

00:40:09.130 --> 00:40:12.895
that take advantage of.

00:40:13.870 --> 00:40:15.940
But that's not our
main point here.

00:40:15.940 --> 00:40:18.170
So what I'm going
to talk about next

00:40:18.170 --> 00:40:20.560
is actually entangled
entropy is also

00:40:20.560 --> 00:40:22.810
starting playing a
very important role

00:40:22.810 --> 00:40:27.640
in our understanding of
many-body physics and the space

00:40:27.640 --> 00:40:28.140
time.

00:40:43.900 --> 00:40:51.390
So now, let's talk about
the entanglement entropy

00:40:51.390 --> 00:40:55.660
in many-body systems in
quantum many-body systems.

00:41:02.360 --> 00:41:06.000
So I hope you're familiar with
this word many-body systems.

00:41:06.000 --> 00:41:08.810
It's just a system of a
large number of particles

00:41:08.810 --> 00:41:12.050
or a large number
of constituents.

00:41:12.050 --> 00:41:16.390
So quantum field theory issue
is a many-bodied system.

00:41:16.390 --> 00:41:19.910
Any quantum field theory
is a many-body system.

00:41:19.910 --> 00:41:23.740
But this also includes
many other lattice systems,

00:41:23.740 --> 00:41:26.380
which condensed
matter people use,

00:41:26.380 --> 00:41:30.847
which are not necessary
quantum fields-- can be written

00:41:30.847 --> 00:41:31.930
as a quantum field theory.

00:41:31.930 --> 00:41:34.520
OK?

00:41:34.520 --> 00:41:38.780
So now let's again consider
just a simplified case.

00:41:38.780 --> 00:41:46.390
Let's consider a system which
is composed of A and B--

00:41:46.390 --> 00:41:54.910
so A plus B. And then I will
say some trivial statements.

00:41:54.910 --> 00:42:00.485
So if H is HA plus HB.

00:42:00.485 --> 00:42:01.860
So now we talk
about Hamiltonian.

00:42:07.260 --> 00:42:10.140
So far, I'm not using
actually many-body systems.

00:42:10.140 --> 00:42:14.420
What I'm talking about are
pretty generic systems.

00:42:14.420 --> 00:42:16.910
So even the
Hamiltonian is actually

00:42:16.910 --> 00:42:18.800
just a direct sum
of the Hamiltonian

00:42:18.800 --> 00:42:21.670
for A and Hamiltonian for B.

00:42:21.670 --> 00:42:28.250
If they don't couple, then of
course, the [INAUDIBLE] state

00:42:28.250 --> 00:42:31.070
or, in particular, the
ground state is unentangled.

00:42:34.694 --> 00:42:37.500
So you just find the ground
state of each system.

00:42:37.500 --> 00:42:39.755
And then that's the ground
state of the total system.

00:42:42.870 --> 00:42:46.880
And also, in general, if you
start with the initial stage,

00:42:46.880 --> 00:42:58.300
which is unentangled, with
the initial state, which

00:42:58.300 --> 00:43:08.980
is unentangled, it will
remain unentangled.

00:43:14.810 --> 00:43:17.060
So if you started with
an unentangled state,

00:43:17.060 --> 00:43:19.770
you wove it using this
Hamiltonian, of course,

00:43:19.770 --> 00:43:22.090
it will just act on
the specific part.

00:43:22.090 --> 00:43:24.040
And then you will
remain unentangled.

00:43:24.040 --> 00:43:26.810
OK?

00:43:26.810 --> 00:43:32.870
And now let's consider we
have H equals to HA plus HB.

00:43:32.870 --> 00:43:34.800
But now they also
have interactions

00:43:34.800 --> 00:43:39.860
between the A and the B.
So A and B are coupled.

00:43:43.570 --> 00:43:47.910
Then in general, we can just
repeat what he said here.

00:43:47.910 --> 00:43:52.991
Then the ground state
is now entangled.

00:43:56.220 --> 00:44:02.840
OK, and now if you start
with the initial state,

00:44:02.840 --> 00:44:12.200
even if you start with the
initial unentangled state,

00:44:12.200 --> 00:44:16.150
it becomes entangled on
Hamiltonian evolution.

00:44:36.490 --> 00:44:40.980
So in general, we
will expect-- so just

00:44:40.980 --> 00:44:45.930
based on this
general expectation--

00:44:45.930 --> 00:44:48.760
so you would expect
the ground state

00:44:48.760 --> 00:44:54.808
of a typical many-body
system will be actually

00:44:54.808 --> 00:44:55.516
rather entangled.

00:44:59.380 --> 00:45:05.909
It will be rather entangled
unless the Hamiltonian factor

00:45:05.909 --> 00:45:07.200
rides into all three particles.

00:45:11.090 --> 00:45:13.840
The factor rides into
the Hilbert space

00:45:13.840 --> 00:45:17.082
at each point-- for
each degree of freedom.

00:45:25.170 --> 00:45:29.120
But now let's talk about
the many-body systems

00:45:29.120 --> 00:45:31.040
we're interested in.

00:45:31.040 --> 00:45:33.240
So they're either typical
condensed matter systems

00:45:33.240 --> 00:45:35.870
or quantum field series.

00:45:35.870 --> 00:45:44.810
So in typical condensed matter
systems, we face this problem.

00:45:44.810 --> 00:45:53.620
So in typical generic condensed
matter systems of quantum field

00:45:53.620 --> 00:46:08.200
theory, in general,
no matter how

00:46:08.200 --> 00:46:16.300
you divide A and B-- so
generic H AB is non-zero.

00:46:16.300 --> 00:46:18.105
But not only is
H AB is non-zero,

00:46:18.105 --> 00:46:27.250
it's in general, H AB-- so the
whole Hamiltonian, including

00:46:27.250 --> 00:46:30.962
H AB is local.

00:46:34.160 --> 00:46:37.050
This is a very
important concept.

00:46:37.050 --> 00:46:38.140
It's local.

00:46:38.140 --> 00:46:39.560
But local will
mean the following.

00:46:39.560 --> 00:46:41.226
For example, let me
give you an example.

00:46:43.440 --> 00:46:51.540
For example, one of the very
important condensed matter

00:46:51.540 --> 00:46:54.070
systems is, of course,
the Heisenberg model.

00:46:54.070 --> 00:46:56.480
So essentially, you
have the Hamiltonian.

00:46:56.480 --> 00:46:59.630
So you can see the lattice
in whatever dimension

00:46:59.630 --> 00:47:03.020
you're interested in whatever
kind of lattice you are.

00:47:05.550 --> 00:47:08.080
At each lattice,
there is a spin.

00:47:08.080 --> 00:47:11.030
So consider you have
a lattice of spins.

00:47:11.030 --> 00:47:14.500
And then you'll have nearest
neighbor interactions

00:47:14.500 --> 00:47:17.770
between the
neighboring spins, so

00:47:17.770 --> 00:47:19.310
some of the nearest neighbors.

00:47:28.710 --> 00:47:31.320
So you can imagine you
have a lattice system.

00:47:31.320 --> 00:47:35.210
And each point on the
lattice, you have a spin.

00:47:35.210 --> 00:47:40.160
And each spin interacts
only with the spins nearby.

00:47:40.160 --> 00:47:42.620
So this is a local interaction.

00:47:42.620 --> 00:47:49.270
So by local interaction, means
that the only direct couple

00:47:49.270 --> 00:47:53.830
to the spin falls
distance away from it.

00:47:53.830 --> 00:47:57.540
So this is an example
of a local system.

00:47:57.540 --> 00:48:00.180
And all our QFTs
are local systems.

00:48:06.540 --> 00:48:07.925
All our QFTs are local systems.

00:48:07.925 --> 00:48:09.800
For example, let me just
write down the phi 4

00:48:09.800 --> 00:48:24.690
theory, for example,
phi 4 theory.

00:48:24.690 --> 00:48:28.920
Then phi at each
point-- so now you

00:48:28.920 --> 00:48:31.870
have to go back to your first
day of your quantum field

00:48:31.870 --> 00:48:35.130
theory where you
emphasize locality.

00:48:35.130 --> 00:48:38.450
So the phi evaluated
at each point now

00:48:38.450 --> 00:48:40.960
is independent of
degrees of freedom.

00:48:40.960 --> 00:48:42.930
Under the quantum
field theory, it's

00:48:42.930 --> 00:48:44.750
a Hilbert space of
a tensor product

00:48:44.750 --> 00:48:47.017
of the degrees of freedom.

00:48:47.017 --> 00:48:47.516
OK?

00:48:50.110 --> 00:48:54.072
Under the form of the Lagrangian
say if you discretize it--

00:48:54.072 --> 00:48:55.530
and this is only
involving coupling

00:48:55.530 --> 00:48:57.738
between the neighboring
point, because the derivative

00:48:57.738 --> 00:49:00.460
is only dependent on
the second derivative.

00:49:00.460 --> 00:49:03.300
And this only depends
on the value of phi

00:49:03.300 --> 00:49:04.780
at the single point.

00:49:04.780 --> 00:49:07.610
So the typical
quantum field theory

00:49:07.610 --> 00:49:10.490
as far as you have found
a number of derivatives,

00:49:10.490 --> 00:49:11.890
it's all local.

00:49:11.890 --> 00:49:13.395
It's all OK intact.

00:49:13.395 --> 00:49:14.020
It's all local.

00:49:14.020 --> 00:49:14.520
OK?

00:49:21.010 --> 00:49:26.150
So in other words, in
these kind of systems,

00:49:26.150 --> 00:49:28.710
let me go back here.

00:49:28.710 --> 00:49:29.876
So this is very important.

00:49:32.520 --> 00:49:44.610
In other words, in
these kind of systems,

00:49:44.610 --> 00:49:50.270
so let me just
imagine-- so let me just

00:49:50.270 --> 00:49:51.800
say this box is the whole thing.

00:49:51.800 --> 00:49:56.000
It's the whole whatever
space I live in.

00:49:56.000 --> 00:50:01.020
And I divide it into A B.

00:50:01.020 --> 00:50:05.850
So this tells you that for
typical condensed matter

00:50:05.850 --> 00:50:15.035
or quantum field series,
the H AB is only supported.

00:50:17.950 --> 00:50:21.160
So let me call this epsilon.

00:50:21.160 --> 00:50:26.300
So that tells that H
AB is only supported

00:50:26.300 --> 00:50:29.610
near the boundary
between A and B,

00:50:29.610 --> 00:50:32.790
because there is only
local interactions.

00:50:32.790 --> 00:50:36.280
So the part of the Hamiltonian
which directly covers A and B

00:50:36.280 --> 00:50:37.760
is only within this part.

00:50:37.760 --> 00:50:43.200
And this epsilon is added
on the lattice spacing

00:50:43.200 --> 00:50:49.080
is order of lattice spacing,
which in the case of a lattice

00:50:49.080 --> 00:50:51.330
system or in a
quantum field theory,

00:50:51.330 --> 00:50:55.070
it would be your
short distance cutoff.

00:50:55.070 --> 00:50:57.760
So if you try to discretize
your field theory,

00:50:57.760 --> 00:50:59.653
then this will be a
short distance cutoff.

00:51:05.110 --> 00:51:10.340
So similarly, I can
consider another shape of A.

00:51:10.340 --> 00:51:13.820
So suppose I can
see the circular A.

00:51:13.820 --> 00:51:19.980
And then, again,
the part of which

00:51:19.980 --> 00:51:24.580
H AB is only supported in
the region between these two

00:51:24.580 --> 00:51:26.900
dashed lines.

00:51:26.900 --> 00:51:28.400
And those dashed
would be considered

00:51:28.400 --> 00:51:33.590
to be very close to the
A. I'm just going to pick.

00:51:33.590 --> 00:51:36.130
But this should be the
short distance cutoff.

00:51:36.130 --> 00:51:37.390
OK?

00:51:37.390 --> 00:51:41.120
And again, the H AB is
only supported in here.

00:51:43.860 --> 00:51:48.547
So this is an important
feature of which

00:51:48.547 --> 00:51:49.755
you have a local Hamiltonian.

00:51:54.720 --> 00:51:57.840
Let me just write one more word.

00:51:57.840 --> 00:52:06.915
So H AB only involves
degrees of freedom

00:52:06.915 --> 00:52:18.860
here-- the boundary of A. So
this means the boundary of A.

00:52:18.860 --> 00:52:23.800
So this turns out to have very,
very important implications--

00:52:23.800 --> 00:52:27.460
the fact that the
Hamiltonian is local.

00:52:27.460 --> 00:52:30.785
And that they only direct
and mediate coupling

00:52:30.785 --> 00:52:34.370
between the degrees of freedom
near the boundary of A.

00:52:34.370 --> 00:52:36.704
And they have very
important consequences.

00:52:42.390 --> 00:52:46.580
So now I'm just
telling you the result

00:52:46.580 --> 00:52:48.500
which is a result of
having accumulated

00:52:48.500 --> 00:52:55.450
for many, many years
since the early '90s.

00:52:55.450 --> 00:53:02.440
So then when I
find in the ground

00:53:02.440 --> 00:53:12.200
state of a local
Hamiltonian in general

00:53:12.200 --> 00:53:20.050
you may construct some kind
of evil counterexamples.

00:53:20.050 --> 00:53:21.740
But in general, in
the ground state

00:53:21.740 --> 00:53:23.810
of a local Hamiltonian--
in the ground

00:53:23.810 --> 00:53:33.750
state of a local
Hamiltonian-- any

00:53:33.750 --> 00:53:36.720
just choose any region
A-- you integrate out

00:53:36.720 --> 00:53:38.350
the degrees of
freedom outside of it.

00:53:46.180 --> 00:53:48.450
So let me emphasize
an important point.

00:53:48.450 --> 00:53:51.440
Let me just pause to
emphasize an important, which

00:53:51.440 --> 00:53:55.079
is only increasing
what I'm saying here.

00:53:55.079 --> 00:53:57.370
In this definition of the
entangled matrix which I just

00:53:57.370 --> 00:54:00.920
erased, you just need
to have a partition

00:54:00.920 --> 00:54:04.080
of a degree of freedom
between A and B.

00:54:04.080 --> 00:54:07.540
The way you partition
it does not matter.

00:54:07.540 --> 00:54:09.860
How you partition
it does not matter.

00:54:09.860 --> 00:54:13.470
You can partition it in
whatever way you want.

00:54:13.470 --> 00:54:16.550
But for typical this
kind of lattice system

00:54:16.550 --> 00:54:22.330
or for quantum field theory,
there's a very large partition.

00:54:22.330 --> 00:54:25.020
And the partition is just
based on the locality.

00:54:25.020 --> 00:54:29.240
It's just based on the degrees
of freedom at each point.

00:54:29.240 --> 00:54:31.500
You just factorize
your Hilbert space

00:54:31.500 --> 00:54:33.540
in terms of degrees of
freedom at each point.

00:54:33.540 --> 00:54:35.581
So of course in the lattice
system, it's obvious.

00:54:35.581 --> 00:54:37.180
You have a spin at each lattice.

00:54:37.180 --> 00:54:39.070
And for the quantum
field theory,

00:54:39.070 --> 00:54:44.020
you just factorize your
degrees of freedom.

00:54:44.020 --> 00:54:46.360
At each point, you
factorize them.

00:54:46.360 --> 00:54:52.590
And so this locality
provides a natural partition

00:54:52.590 --> 00:54:56.590
of your degrees of freedom.

00:54:56.590 --> 00:55:01.590
And when we talk
about A and the B,

00:55:01.590 --> 00:55:06.390
we always talk about in
terms of the space location.

00:55:06.390 --> 00:55:10.540
And it's natural to talk
about the partition in terms

00:55:10.540 --> 00:55:15.840
of just your metric locations.

00:55:15.840 --> 00:55:17.310
So is this point clear?

00:55:17.310 --> 00:55:18.816
I just want to
emphasize this point.

00:55:21.600 --> 00:55:26.180
So then you find that in
general for the ground

00:55:26.180 --> 00:55:28.550
state of a local Hamiltonian
satisfy so-called area

00:55:28.550 --> 00:55:34.060
law is that the leading term
of the entanglement entropy

00:55:34.060 --> 00:55:38.185
is given by the area
of the boundary of A--

00:55:38.185 --> 00:55:40.030
this applies no
matter which dimension

00:55:40.030 --> 00:55:45.410
you are in generic
dimensions-- and divide it

00:55:45.410 --> 00:55:52.540
by a short distance cutoff
of the lattice spacing.

00:55:52.540 --> 00:55:54.950
It's some number.

00:55:54.950 --> 00:55:59.260
So A is a dimensionless number.

00:55:59.260 --> 00:56:02.890
So area-- we can see the
d-dimensional theory--

00:56:02.890 --> 00:56:04.450
so d-dimensional system.

00:56:04.450 --> 00:56:07.030
Then the spatial dimension
will be d minus 1.

00:56:07.030 --> 00:56:09.580
And then you go to the
boundary of some region.

00:56:09.580 --> 00:56:13.540
The boundary of region A.
Then that will be a d minus 2

00:56:13.540 --> 00:56:15.330
dimensional surface.

00:56:15.330 --> 00:56:17.880
And then this would be
the area of that surface.

00:56:17.880 --> 00:56:19.900
So this may cover a
dimensionless number.

00:56:19.900 --> 00:56:22.730
And then you can have
some pre-factor here

00:56:22.730 --> 00:56:25.880
which depend on your systems.

00:56:25.880 --> 00:56:29.095
So this formula tells you
a very important physics.

00:56:32.720 --> 00:56:34.985
So this formula tells you
a very important physics.

00:56:41.280 --> 00:56:50.750
This tells you that
generically if you--

00:56:50.750 --> 00:56:55.370
in such kind of systems,
AB are entangled.

00:56:55.370 --> 00:57:05.120
But the entanglement between
AB between A is complementary,

00:57:05.120 --> 00:57:19.020
which I just call A
and B, are dominated

00:57:19.020 --> 00:57:33.720
by short-range entanglements
near the boundary of A

00:57:33.720 --> 00:57:37.200
where this H AB is supported.

00:57:44.301 --> 00:57:44.800
OK?

00:57:48.350 --> 00:57:53.250
So this is also very intuitive
once you load the answer--

00:57:53.250 --> 00:57:55.750
once you load the
answer, because we

00:57:55.750 --> 00:57:59.310
emphasized that H AB only
coupled to degrees of freedom

00:57:59.310 --> 00:58:00.820
nearby.

00:58:00.820 --> 00:58:03.540
And then we look
at the ground state

00:58:03.540 --> 00:58:09.150
you find most of the
degrees of freedom in here

00:58:09.150 --> 00:58:11.590
don't entangle with B. Only
the degrees of freedom them

00:58:11.590 --> 00:58:13.840
near the boundary of A are
entangled with B because

00:58:13.840 --> 00:58:15.320
of this interaction.

00:58:15.320 --> 00:58:17.295
This local interaction
directly leads

00:58:17.295 --> 00:58:20.820
to the entanglement between
the degrees of freedom nearby.

00:58:20.820 --> 00:58:24.290
And those degrees of freedom,
if they are far away,

00:58:24.290 --> 00:58:25.290
they might be entangled.

00:58:25.290 --> 00:58:28.200
But they are not dominant.

00:58:28.200 --> 00:58:29.650
OK?

00:58:29.650 --> 00:58:32.850
So I have put many dots here.

00:58:32.850 --> 00:58:37.140
So included in those dots
are possible long-range

00:58:37.140 --> 00:58:38.770
entanglements.

00:58:38.770 --> 00:58:40.560
Because when you find
the ground state,

00:58:40.560 --> 00:58:45.360
you really have to extremize
the energy of the total system.

00:58:45.360 --> 00:58:48.120
You don't only look
at the small part.

00:58:48.120 --> 00:58:51.700
So there's always some kind
of long-range entanglement

00:58:51.700 --> 00:58:55.080
beyond this H AB.

00:58:55.080 --> 00:58:57.915
But this formula tells you
that this short distance

00:58:57.915 --> 00:59:00.210
entanglement dominates.

00:59:00.210 --> 00:59:01.506
Yes?

00:59:01.506 --> 00:59:04.235
AUDIENCE: One question-- in QFT
in general, like [INAUDIBLE],

00:59:04.235 --> 00:59:05.693
there's really all
the interactions

00:59:05.693 --> 00:59:06.984
are perfectly local, I believe.

00:59:06.984 --> 00:59:10.053
But in string theory, are there
any non-local interactions

00:59:10.053 --> 00:59:11.265
that occur.

00:59:11.265 --> 00:59:13.090
Are there any interactions
with non-local?

00:59:13.090 --> 00:59:17.160
HONG LIU: Yeah, it will
depend on the scale.

00:59:17.160 --> 00:59:20.020
In string theory when
we say non-local,

00:59:20.020 --> 00:59:24.320
it's non-local at the
alpha prime scale.

00:59:24.320 --> 00:59:27.120
And that alpha prime can
perfectly be our short distance

00:59:27.120 --> 00:59:27.650
cutoff here.

00:59:27.650 --> 00:59:29.250
When I talk about
local and non-local,

00:59:29.250 --> 00:59:32.790
I'm talking about
infinite versus finite.

00:59:32.790 --> 00:59:34.520
For example, here,
when I talk about

00:59:34.520 --> 00:59:37.740
whether this is
a local system, I

00:59:37.740 --> 00:59:40.440
don't have to have
a nearest neighbor.

00:59:40.440 --> 00:59:49.620
As far as this matches
this point only directly

00:59:49.620 --> 00:59:52.810
coupled to the finite
distance neighbor.

00:59:52.810 --> 00:59:55.561
And this can see the
local Hamiltonian.

00:59:55.561 --> 00:59:57.560
But from the quantum field
theory point of view,

00:59:57.560 --> 00:59:59.018
this might be
considered non-local.

01:00:01.137 --> 01:00:01.970
You see what I mean?

01:00:01.970 --> 01:00:04.740
Depending on your scale.

01:00:04.740 --> 01:00:07.400
AUDIENCE: So is
there anything that

01:00:07.400 --> 01:00:09.980
appears in string theory which
is non-local at any scale?

01:00:09.980 --> 01:00:13.610
Or is the non-locality
just some--

01:00:13.610 --> 01:00:15.300
HONG LIU: Yeah, so
the string theory

01:00:15.300 --> 01:00:20.200
itself does not give
you that non-locality,

01:00:20.200 --> 01:00:21.817
but the black hole may.

01:00:21.817 --> 01:00:22.900
AUDIENCE: Oh, interesting.

01:00:22.900 --> 01:00:25.410
HONG LIU: The black
hole seems to give you

01:00:25.410 --> 01:00:29.060
some kind of non-locality which
we don't fully understand.

01:00:29.060 --> 01:00:31.390
So that's why the black
hole is so puzzling.

01:00:31.390 --> 01:00:34.650
And the [INAUDIBLE] of
string theory, you're

01:00:34.650 --> 01:00:38.010
non-local in the
scale of alpha prime.

01:00:38.010 --> 01:00:40.479
Yeah, if it's alpha
prime sufficient small,

01:00:40.479 --> 01:00:41.520
it's like a local theory.

01:00:41.520 --> 01:00:44.262
And even if alpha prime is
big, if you look at very much,

01:00:44.262 --> 01:00:45.220
it is still like local.

01:00:45.220 --> 01:00:46.210
AUDIENCE: I see.

01:00:46.210 --> 01:00:49.180
HONG LIU: Yeah, but black
hole can make things

01:00:49.180 --> 01:00:50.326
very long distance.

01:00:50.326 --> 01:00:52.200
So that's the tricky
thing about black holes.

01:00:55.060 --> 01:00:59.980
OK, so this is
something which are

01:00:59.980 --> 01:01:06.690
is interesting and was first
discovered in the early '90s.

01:01:06.690 --> 01:01:08.760
But in hindsight,
it's very intuitive.

01:01:08.760 --> 01:01:12.310
In hindsight, it's
very intuitive.

01:01:12.310 --> 01:01:16.420
So even though this formula,
even though this term

01:01:16.420 --> 01:01:20.020
is universal, but this
coefficient is actually

01:01:20.020 --> 01:01:22.450
highly non-universal.

01:01:22.450 --> 01:01:24.170
Normally, by saying
universal, we

01:01:24.170 --> 01:01:27.655
mean something which is
common to different systems.

01:01:27.655 --> 01:01:29.780
So this coefficient actually
depends on the details

01:01:29.780 --> 01:01:30.820
of individual systems.

01:01:30.820 --> 01:01:32.736
Say if you calculate for
the Heisenberg model,

01:01:32.736 --> 01:01:37.310
or if you calculate for the
free scale of field theory,

01:01:37.310 --> 01:01:40.800
or free fermion theory, and
then this coefficient in general

01:01:40.800 --> 01:01:45.040
is different, are depending
on how you define your cutoff.

01:01:45.040 --> 01:01:48.645
So even though this
formula is universal,

01:01:48.645 --> 01:01:52.470
but those pre-factors actually
are highly non-universal.

01:01:58.689 --> 01:02:02.951
But the partial
excitement-- there

01:02:02.951 --> 01:02:04.992
are many things people
are excited about in terms

01:02:04.992 --> 01:02:05.492
of entropy.

01:02:08.920 --> 01:02:11.800
This is one thing,
this area law,

01:02:11.800 --> 01:02:13.135
because the area law is nice.

01:02:19.050 --> 01:02:21.440
Then this can be used as
a distinguishing property

01:02:21.440 --> 01:02:23.381
of a ground state.

01:02:23.381 --> 01:02:24.880
If you go to a
highly excited state,

01:02:24.880 --> 01:02:27.730
then you find the generic
distribution inside

01:02:27.730 --> 01:02:29.639
and A and B will be entangled.

01:02:29.639 --> 01:02:30.930
And [INAUDIBLE] highly excited.

01:02:30.930 --> 01:02:33.890
But only in the ground state,
somehow only near the boundary

01:02:33.890 --> 01:02:35.290
they are entangled.

01:02:35.290 --> 01:02:38.020
This is also makes sense,
because if it were high energy

01:02:38.020 --> 01:02:40.270
then you can excite it,
because from the very far away

01:02:40.270 --> 01:02:45.123
from H AB, then of course,
everything would be entangled.

01:02:45.123 --> 01:02:48.460
AUDIENCE: So just
one other question.

01:02:48.460 --> 01:02:50.930
So why are you
dividing by epsilon?

01:02:50.930 --> 01:02:54.220
HONG LIU: No, this
estimation is a number.

01:02:54.220 --> 01:02:55.840
Making the dimension
is numbered.

01:02:55.840 --> 01:02:58.810
This is only a short-distance
information here.

01:02:58.810 --> 01:03:01.154
So that is spatial
short-distance cut off.

01:03:01.154 --> 01:03:02.883
AUDIENCE: Right,
the only reason I

01:03:02.883 --> 01:03:06.320
ask is that intuitively it
seems that the entropy should

01:03:06.320 --> 01:03:12.741
be proportional to this thin
volume around the boundary.

01:03:12.741 --> 01:03:14.240
But that doesn't--
OK, I don't know.

01:03:14.240 --> 01:03:16.760
HONG LIU: Yeah, you have to make
it into a dimensionless number.

01:03:16.760 --> 01:03:18.426
You have to be
proportional to the area.

01:03:21.260 --> 01:03:25.740
So there's a very simple way
to understand that thing.

01:03:25.740 --> 01:03:33.420
So the area divided by lattice
spacing-- so what is this guy?

01:03:33.420 --> 01:03:35.270
This is wonderfully
a lattice volume.

01:03:37.850 --> 01:03:42.520
That is the lattice
area on the surface.

01:03:42.520 --> 01:03:45.150
So essentially this tells you
how many degrees of freedom

01:03:45.150 --> 01:03:48.320
are lying on that boundary.

01:03:48.320 --> 01:03:49.460
AUDIENCE: OK, interesting.

01:03:49.460 --> 01:03:53.380
HONG LIU: Yeah, OK?

01:03:53.380 --> 01:03:54.680
So is this clear?

01:03:54.680 --> 01:03:58.950
OK, I think I'm spending
too much time on this.

01:03:58.950 --> 01:04:06.620
Anyway, so excited
in the last decade--

01:04:06.620 --> 01:04:08.550
the main excitement
about entangled

01:04:08.550 --> 01:04:09.920
entropy in the last decade.

01:04:09.920 --> 01:04:11.719
The one thing is
about this behavior,

01:04:11.719 --> 01:04:14.010
then you can use it as a
characterization of the ground

01:04:14.010 --> 01:04:15.270
state.

01:04:15.270 --> 01:04:18.230
But a lot of the
excitement about this thing

01:04:18.230 --> 01:04:21.050
is on those dot, dot, dots.

01:04:21.050 --> 01:04:27.570
So people actually find
that the subleading term--

01:04:27.570 --> 01:04:30.450
some excitement.

01:04:30.450 --> 01:04:38.550
Yeah, I'm just saying--
from last decade people

01:04:38.550 --> 01:04:48.820
have discover that the
subleading terms in entangled

01:04:48.820 --> 01:04:54.520
entropy other than
this area term which

01:04:54.520 --> 01:04:59.338
come in from long-range
entanglements.

01:05:02.330 --> 01:05:06.470
So this captures the
short-range entanglement

01:05:06.470 --> 01:05:11.800
from the long-range
entanglement can actually

01:05:11.800 --> 01:05:17.365
provide important
characterizations of a system.

01:05:35.180 --> 01:05:38.560
So let me just
mention two examples.

01:05:38.560 --> 01:05:40.770
So it will be little
bit fast because we

01:05:40.770 --> 01:05:48.753
are a little bit-- do you
want a break for the last day?

01:05:51.900 --> 01:05:53.840
AUDIENCE: Well, when
you say it like that--

01:05:53.840 --> 01:05:54.676
HONG LIU: OK, good.

01:05:59.210 --> 01:06:00.030
OK, thanks.

01:06:15.130 --> 01:06:17.540
So let me just quickly
talk about two examples

01:06:17.540 --> 01:06:19.735
because I want to talk
about the holographic case.

01:06:22.590 --> 01:06:25.550
So the first thing is something
called the topological can

01:06:25.550 --> 01:06:41.670
be used to characterize
so-called topological order

01:06:41.670 --> 01:06:43.083
in 2 plus 1 dimensions.

01:06:53.360 --> 01:06:58.310
So since the '90s, mid
'80s, and the '980s,

01:06:58.310 --> 01:07:02.420
people discovered
that if you look

01:07:02.420 --> 01:07:11.490
at the typical gapped
system-- so when

01:07:11.490 --> 01:07:15.330
we say a gapped system means
that system between the ground

01:07:15.330 --> 01:07:20.130
state and the first excited
state have a finite energy gap.

01:07:22.800 --> 01:07:27.460
So our general understanding
of a gap system is that

01:07:27.460 --> 01:07:31.930
in the ground state because
you have a finite gap,

01:07:31.930 --> 01:07:33.430
and there's sufficient
low energies,

01:07:33.430 --> 01:07:35.730
you simply cannot
excite anything.

01:07:35.730 --> 01:07:38.430
You have low energy because you
have a finite gap to the first

01:07:38.430 --> 01:07:39.070
excited state.

01:07:39.070 --> 01:07:40.462
You cannot excite anything.

01:07:40.462 --> 01:07:42.920
So essentially, when you look
at the correlation functions,

01:07:42.920 --> 01:07:45.110
we can look at
all the variables.

01:07:45.110 --> 01:07:47.566
They all only have
short-range correlations.

01:07:47.566 --> 01:07:49.190
The cannot have
long-range correlation,

01:07:49.190 --> 01:07:52.534
because there's nothing to
propagate you to long distance.

01:07:52.534 --> 01:07:53.950
And because long
distance requires

01:07:53.950 --> 01:07:55.430
massive degrees of freedom.

01:07:55.430 --> 01:07:58.055
And because you have a gap, you
don't have such massive degrees

01:07:58.055 --> 01:07:58.890
of freedom.

01:07:58.890 --> 01:08:03.610
So the typical conventional
idea about the gap system

01:08:03.610 --> 01:08:05.840
is that all correlations
should be short-ranged.

01:08:08.680 --> 01:08:12.500
And so this is a
condition that meets them.

01:08:12.500 --> 01:08:14.730
But in the late
'80s and early '90s,

01:08:14.730 --> 01:08:17.930
our colleague, Xiao-Gang
Wen, he introduced

01:08:17.930 --> 01:08:24.870
this notion of a
topological ordered system

01:08:24.870 --> 01:08:27.375
based on fractional
quantum Hall effect.

01:08:29.910 --> 01:08:37.439
So he actually reasoned that
there can be gap systems even

01:08:37.439 --> 01:08:39.979
though I have a finite gap,
but in the ground state

01:08:39.979 --> 01:08:42.850
actually contain
long-range correlations--

01:08:42.850 --> 01:08:45.520
nontrivial, long-range
correlations.

01:08:45.520 --> 01:08:49.080
And those long-range
correlations cannot be seen

01:08:49.080 --> 01:08:50.547
using the standard observables.

01:08:50.547 --> 01:08:52.630
So if you are using the
standard observables-- say

01:08:52.630 --> 01:08:56.100
correlation functions of local
operators, you don't see them.

01:08:56.100 --> 01:08:59.060
You only see the
short-range correlations.

01:08:59.060 --> 01:09:00.560
And those long-range
correlations

01:09:00.560 --> 01:09:03.100
are topological in nature.

01:09:03.100 --> 01:09:04.880
You have to see it
in some subtle ways.

01:09:04.880 --> 01:09:07.770
And then in the '80s, except
in the late '80s, early '90s,

01:09:07.770 --> 01:09:11.010
he was trying to argue
in some [INAUDIBLE] way

01:09:11.010 --> 01:09:14.550
to argue there is such kind
of subtle correlations.

01:09:14.550 --> 01:09:19.670
But then around 2000,
maybe 2005, then

01:09:19.670 --> 01:09:22.760
he realized actually you can
see it very easily directly

01:09:22.760 --> 01:09:24.852
from the ground
state wave function.

01:09:24.852 --> 01:09:26.560
You just calculate
the entangled entropy.

01:09:29.750 --> 01:09:31.840
You just calculate
entangled entropy.

01:09:31.840 --> 01:09:34.479
So in entangled entropy,
you always have this--

01:09:34.479 --> 01:09:37.930
so this is a 2 plus 1
system, and then the boundary

01:09:37.930 --> 01:09:39.702
is just a line.

01:09:39.702 --> 01:09:41.410
So essentially, you
just have a boundary.

01:09:41.410 --> 01:09:45.500
It's a number-- the
length of the boundary

01:09:45.500 --> 01:09:48.929
divided by your
cutoff, but then it

01:09:48.929 --> 01:09:51.970
turns out for these kind of
topological ordered system,

01:09:51.970 --> 01:09:55.310
there is a finite constant
as a subleading term.

01:09:58.190 --> 01:10:04.170
And this finite constant is
independent of the shape of A

01:10:04.170 --> 01:10:06.740
and independent of length
of A and independent

01:10:06.740 --> 01:10:09.970
of whatever A is purely
topological in nature.

01:10:09.970 --> 01:10:15.080
And so if you have
achieved a gap system then

01:10:15.080 --> 01:10:16.500
this gamma will be 0.

01:10:16.500 --> 01:10:20.210
So you look at a free
massive scale of field--

01:10:20.210 --> 01:10:21.740
then this gamma will be 0.

01:10:21.740 --> 01:10:23.840
But for those topological
ordered systems,

01:10:23.840 --> 01:10:25.440
this gamma would be non-zero.

01:10:25.440 --> 01:10:30.020
And it tells you actually there
is long-range correlations

01:10:30.020 --> 01:10:35.070
which are encoded in the
entanglement entropy,

01:10:35.070 --> 01:10:37.910
because this cannot come from
the local thing because this

01:10:37.910 --> 01:10:41.250
does not depend on the shape of
A, does not depend on N sub A,

01:10:41.250 --> 01:10:43.600
does not depend
on anything of A.

01:10:43.600 --> 01:10:45.400
So this is really
topological in nature.

01:10:45.400 --> 01:10:48.320
It tells you this can only
be some long-range, subtle

01:10:48.320 --> 01:10:50.390
correlation, which
is only capture

01:10:50.390 --> 01:10:51.765
by these entangled
entropy rather

01:10:51.765 --> 01:10:53.556
than by the standard
correlation functions.

01:10:53.556 --> 01:10:54.180
OK

01:10:54.180 --> 01:11:00.270
So this is one discovery which
is generated a lot of interest

01:11:00.270 --> 01:11:02.670
in the entangled entropy
because now this can

01:11:02.670 --> 01:11:05.720
be used to define the phases.

01:11:05.720 --> 01:11:08.010
OK, you can actually
define non-trivial phases

01:11:08.010 --> 01:11:11.179
using this number.

01:11:11.179 --> 01:11:12.470
And so this is the first thing.

01:11:15.280 --> 01:11:18.860
So this is about in
condensed matter systems.

01:11:18.860 --> 01:11:23.330
In quantum field theories ,
this can be used to characterize

01:11:23.330 --> 01:11:33.670
the number of degrees
of freedom of a QFT.

01:11:38.180 --> 01:11:40.950
Again, this is a long story.

01:11:40.950 --> 01:11:43.670
But I don't have time.

01:11:43.670 --> 01:11:45.990
So let me just tell
you a short story.

01:11:45.990 --> 01:11:50.906
So first let me look at 1
plus 1 dimensions, a CFT.

01:11:56.170 --> 01:12:00.640
So for 1 plus 1 dimension
of CFT or important

01:12:00.640 --> 01:12:04.470
the quantity is so-called
essential charge.

01:12:04.470 --> 01:12:08.940
So have you all heard of
essential charge or not really?

01:12:08.940 --> 01:12:12.500
It just says, imagine you have
a 1 plus 1 dimensional CFT.

01:12:12.500 --> 01:12:16.450
And for each CFT, you can
define a single number, which

01:12:16.450 --> 01:12:18.240
is called the center charge.

01:12:18.240 --> 01:12:20.720
And this center charge
is very important

01:12:20.720 --> 01:12:23.680
because it controls
the asymptotic density

01:12:23.680 --> 01:12:25.320
of state of the system.

01:12:25.320 --> 01:12:27.611
So if you look at the system
go to very high energies,

01:12:27.611 --> 01:12:29.110
and the density of
state essentially

01:12:29.110 --> 01:12:31.440
is controlled by this number C.

01:12:31.440 --> 01:12:33.950
So essentially C can
be used to characterize

01:12:33.950 --> 01:12:35.770
a number of degrees of freedom.

01:12:35.770 --> 01:12:38.400
And the C will appear
in many other places.

01:12:38.400 --> 01:12:40.800
But I don't have
time to explain.

01:12:40.800 --> 01:12:43.480
But anyway, there's something
called the central charge,

01:12:43.480 --> 01:12:46.222
which can be used to
characterize numbers of degrees

01:12:46.222 --> 01:12:47.820
of freedom of the system.

01:12:47.820 --> 01:12:51.040
And thus, if you look at the
entanglement entropy for 1

01:12:51.040 --> 01:12:54.239
plus 1 dimensional system for
1 plus 1 dimension of CFT.

01:12:54.239 --> 01:12:55.655
So in 1 plus 1
dimension, you only

01:12:55.655 --> 01:12:57.630
have a line in the
spatial direction.

01:12:57.630 --> 01:13:03.480
So then this takes A just to
be a sacrament of length l.

01:13:03.480 --> 01:13:05.250
Then you find when
you integrate out

01:13:05.250 --> 01:13:07.440
everything else, the
entangled entropy

01:13:07.440 --> 01:13:13.210
for the A given by C divided
by 3 log L divided by epsilon.

01:13:13.210 --> 01:13:16.040
Again, epsilon is a
short distance cutoff.

01:13:16.040 --> 01:13:18.440
And the C is your center charge.

01:13:18.440 --> 01:13:24.520
So you see that the
entanglement entropy-- first you

01:13:24.520 --> 01:13:27.650
notice that formula
becomes degenerate when

01:13:27.650 --> 01:13:30.780
you go to the two dimensions.

01:13:30.780 --> 01:13:32.224
It's equal to 1
plus 1 dimensions.

01:13:32.224 --> 01:13:34.640
1 plus 1 dimension the were
played by something like this.

01:13:34.640 --> 01:13:37.750
You have log epsilon.

01:13:37.750 --> 01:13:41.140
So the key important
thing is not the prefactor

01:13:41.140 --> 01:13:41.990
because of the log.

01:13:41.990 --> 01:13:43.850
The prefactor is
actually reversal.

01:13:43.850 --> 01:13:46.620
And it's actually controlled
by the central charge.

01:13:46.620 --> 01:13:48.750
So that saves you from
entangled entropy.

01:13:48.750 --> 01:13:50.840
You can actually read
the central charge

01:13:50.840 --> 01:13:52.306
and actually read what
are the numbers of degrees

01:13:52.306 --> 01:13:53.514
of was freedom of the system.

01:13:57.240 --> 01:14:00.110
So the fact that the central
charge in the 1 plus 1

01:14:00.110 --> 01:14:03.720
dimensional CFT captures the
number of degrees of freedom

01:14:03.720 --> 01:14:06.730
was known-- known
before people thought

01:14:06.730 --> 01:14:10.070
about entangled entropy.

01:14:10.070 --> 01:14:15.014
But it turned out-- and
people spent many years

01:14:15.014 --> 01:14:17.180
trying too hard to generalize
this concept to higher

01:14:17.180 --> 01:14:18.017
dimensional series.

01:14:20.485 --> 01:14:22.610
And it turned out they are
very hard to generalize.

01:14:22.610 --> 01:14:24.890
People didn't really
know how to do it.

01:14:24.890 --> 01:14:27.170
But now we know
because you can just

01:14:27.170 --> 01:14:30.090
generalize the entangled
entropy to higher dimensions.

01:14:30.090 --> 01:14:33.640
And then you find the same
kind of central charge

01:14:33.640 --> 01:14:35.720
which appeared.

01:14:35.720 --> 01:14:38.780
You find again
things can be defined

01:14:38.780 --> 01:14:41.180
as analog of the single
charge in 1 plus 1 dimension

01:14:41.180 --> 01:14:43.800
appears in the entangled entropy
in the higher dimensions.

01:14:43.800 --> 01:14:45.970
And then they can be
used to characterize

01:14:45.970 --> 01:14:49.110
the number degrees of freedom.

01:14:49.110 --> 01:14:51.610
And so in some sense, the
entangled entropy really

01:14:51.610 --> 01:14:55.354
provides a unified way to
think about this central charge

01:14:55.354 --> 01:14:57.520
and to characterize the
number of degrees of freedom

01:14:57.520 --> 01:15:01.960
across all dimensions--
across all dimensions.

01:15:01.960 --> 01:15:03.575
And so this is a key formula.

01:15:06.790 --> 01:15:11.160
Actually, I first realized by
our old friend Frank Wilczek

01:15:11.160 --> 01:15:14.650
when they studied
free field theory.

01:15:14.650 --> 01:15:17.190
And people later generalized
to show this actually

01:15:17.190 --> 01:15:17.940
works for any CFT.

01:15:22.700 --> 01:15:24.009
Good.

01:15:24.009 --> 01:15:26.300
So now finally, we can talk
about holographic entangled

01:15:26.300 --> 01:15:30.120
entropy with lots
of preparations.

01:15:30.120 --> 01:15:33.700
So this just gives
you some taste

01:15:33.700 --> 01:15:38.690
of actually why we are actually
interested in entangled entropy

01:15:38.690 --> 01:15:43.230
and why this actually not only
people in quantum computing

01:15:43.230 --> 01:15:47.920
or quantum information
people are interested in it.

01:15:47.920 --> 01:15:49.670
People are doing
condensed matter-- people

01:15:49.670 --> 01:15:53.620
doing quantum field theory
are also interested in it.

01:15:56.980 --> 01:15:58.730
And now people who are
doing string theory

01:15:58.730 --> 01:16:01.040
are also interested
in it because

01:16:01.040 --> 01:16:02.830
of this holographic
entanglement entropy.

01:16:16.070 --> 01:16:18.070
So suppose we have our CFT.

01:16:21.150 --> 01:16:22.840
So a two-dimensional
CFT with a gravity

01:16:22.840 --> 01:16:32.350
dual-- so a dual to some
theory in d plus 1 dimensional

01:16:32.350 --> 01:16:35.330
[INAUDIBLE] space time.

01:16:35.330 --> 01:16:41.540
So let me give you origin A. How
do I find the entangled-- what

01:16:41.540 --> 01:16:43.050
is the counterpart
of this entangled

01:16:43.050 --> 01:16:46.555
entropy on the gravity side?

01:16:46.555 --> 01:16:47.554
So this is the question.

01:16:50.420 --> 01:16:52.460
So let me just draw this figure.

01:16:52.460 --> 01:16:53.920
Suppose this is
the box again that

01:16:53.920 --> 01:16:56.340
represents your total system.

01:16:56.340 --> 01:16:59.260
And then this carves
out of region A.

01:16:59.260 --> 01:17:01.846
Then the question what
does this translate into?

01:17:05.500 --> 01:17:06.612
And the graph decides.

01:17:06.612 --> 01:17:08.070
OK, so now in the
boundary, we have

01:17:08.070 --> 01:17:12.080
some region A is equal to 0.

01:17:21.060 --> 01:17:31.980
So for this question, in
some sense it's difficult

01:17:31.980 --> 01:17:33.770
because it's not
like other questions

01:17:33.770 --> 01:17:35.090
we've talked about before.

01:17:35.090 --> 01:17:37.890
So we also know
partition function.

01:17:37.890 --> 01:17:40.580
You can say the partition
function cannot be the same.

01:17:40.580 --> 01:17:42.850
Those things appear
to be natural.

01:17:42.850 --> 01:17:46.300
Here, there is no natural
thing you can think about,

01:17:46.300 --> 01:17:49.730
because you cannot
define entangled entropy

01:17:49.730 --> 01:17:52.313
as a partition function.

01:17:52.313 --> 01:17:55.460
Entangled entropy have to
involve in so-called trace

01:17:55.460 --> 01:17:56.750
out the degrees of freedom.

01:17:56.750 --> 01:17:59.610
Then take the logarithm.

01:17:59.610 --> 01:18:03.040
It's highly non-local and
a complicated procedure.

01:18:03.040 --> 01:18:06.980
And it's obvious how you would
define on the gravity side--

01:18:06.980 --> 01:18:10.080
how you would guess
on the gravity side.

01:18:10.080 --> 01:18:13.710
So it's quite remarkable
and the Ryu-Takayanagi,

01:18:13.710 --> 01:18:15.990
they just made the guess.

01:18:15.990 --> 01:18:18.570
And then worked.

01:18:18.570 --> 01:18:34.250
So the proposal is just find
the minimal area surface,

01:18:34.250 --> 01:18:50.070
let me call gamma A,
which extends in the back

01:18:50.070 --> 01:18:52.660
and with the boundary
of A as the boundary.

01:18:57.050 --> 01:18:58.910
You first find the surface.

01:18:58.910 --> 01:19:01.600
And then you say the
entangled entropy for A

01:19:01.600 --> 01:19:07.070
is just the area of this
gamma A divided by 40 Newton.

01:19:11.450 --> 01:19:12.850
And this is the Ryu-Takayanagi.

01:19:22.720 --> 01:19:27.730
So the idea is that you find
a surface of a minimal area

01:19:27.730 --> 01:19:31.120
is going into the back, but
ending on the boundary of A.

01:19:31.120 --> 01:19:35.740
So this is your gamma A. And
you find such a minimal surface.

01:19:35.740 --> 01:19:38.756
Then you divide by 40 Newton.

01:19:38.756 --> 01:19:39.255
OK?

01:19:45.280 --> 01:19:48.182
So let me just
mention one thing.

01:19:58.540 --> 01:20:03.390
Let me just note a simple
scene, because again,

01:20:03.390 --> 01:20:08.390
the A of the boundary, this
is d minus 2 dimensional.

01:20:11.680 --> 01:20:15.540
And A and the gamma A are
all d minus 1 dimensional.

01:20:19.650 --> 01:20:21.450
So this is d minus
2 dimension is

01:20:21.450 --> 01:20:23.270
what we said before
because A is just

01:20:23.270 --> 01:20:25.430
a region in your
spatial section.

01:20:25.430 --> 01:20:27.430
So A is a d minus 1 dimensional.

01:20:27.430 --> 01:20:29.990
And the boundary A will
d minus 2 dimensional.

01:20:29.990 --> 01:20:31.740
And the gamma A
would be a surface

01:20:31.740 --> 01:20:37.280
which is ending on the boundary
of A. This ends on that thing.

01:20:37.280 --> 01:20:40.050
So this is also the d
minus 1 dimensional.

01:20:40.050 --> 01:20:41.660
And this is d minus
1 dimensional.

01:20:41.660 --> 01:20:44.070
So this area have
dimension d minus 1.

01:20:44.070 --> 01:20:45.940
And that's precisely
the dimension

01:20:45.940 --> 01:20:49.760
of G Newton in the d plus
1 dimensional [INAUDIBLE]

01:20:49.760 --> 01:20:50.770
of space time.

01:20:50.770 --> 01:20:52.850
And then this is a
dimensionless number.

01:20:52.850 --> 01:20:54.930
OK, because we did it
the SA with dimensions.

01:20:59.350 --> 01:21:01.640
So you can say this is
a very difficult guess.

01:21:01.640 --> 01:21:07.950
You can also say this is a very
simple guess, because clearly

01:21:07.950 --> 01:21:10.560
this formula is motivated by
the black hole entropy formula.

01:21:13.930 --> 01:21:16.110
You may say, ah,
black hole entropy

01:21:16.110 --> 01:21:18.570
is something divided 4GN.

01:21:18.570 --> 01:21:21.260
If it is entangled
entropy, it's also entropy.

01:21:21.260 --> 01:21:24.250
Maybe it's also
something divided by 4GN.

01:21:24.250 --> 01:21:26.710
And maybe it's some
surface divided by 4GN.

01:21:26.710 --> 01:21:30.300
And then the special surface
would be the minimal surface.

01:21:30.300 --> 01:21:33.630
So in some sense, it's
a very naive guess.

01:21:33.630 --> 01:21:36.610
And you could say it's
a very simple guess.

01:21:36.610 --> 01:21:39.330
But as I said, it's also
a very difficult guess,

01:21:39.330 --> 01:21:42.030
because essentially,
other than that, you don't

01:21:42.030 --> 01:21:44.230
have other starting points.

01:21:44.230 --> 01:21:44.960
Yes?

01:21:44.960 --> 01:21:46.840
AUDIENCE: So [INAUDIBLE].

01:21:46.840 --> 01:21:49.570
So we have A in the CFT.

01:21:49.570 --> 01:21:53.515
Where do we know exactly
where to place A in the AdS?

01:21:53.515 --> 01:21:55.790
Because it's not--

01:21:55.790 --> 01:21:58.076
HONG LIU: This is
a good question.

01:21:58.076 --> 01:22:00.010
This is what I'm
going to say next.

01:22:00.010 --> 01:22:02.370
Yeah, but let me
just emphasize that.

01:22:02.370 --> 01:22:04.657
So when we talk
about A, we're always

01:22:04.657 --> 01:22:06.240
talking about the
constant time slice,

01:22:06.240 --> 01:22:10.230
because you have to
specify some time.

01:22:10.230 --> 01:22:12.240
And actually, when we
consider the ground state

01:22:12.240 --> 01:22:14.480
or typical state does
not depend on time.

01:22:14.480 --> 01:22:17.180
And it actually does not
matter which slice we choose.

01:22:17.180 --> 01:22:18.860
You just choose a time slice.

01:22:18.860 --> 01:22:21.064
And then you specify
region A, because to talk

01:22:21.064 --> 01:22:22.480
about degrees of
freedom, you only

01:22:22.480 --> 01:22:24.380
talk about it in
the spatial section.

01:22:29.720 --> 01:22:33.040
So if you have a ground state
or have any state which does not

01:22:33.040 --> 01:22:35.140
depend on time, then
the gravity side

01:22:35.140 --> 01:22:36.400
also does not depend on time.

01:22:36.400 --> 01:22:38.440
It's a time-dependent geometry.

01:22:38.440 --> 01:22:41.190
So time slicing in the
field series naturally

01:22:41.190 --> 01:22:43.310
extends in the gravity side.

01:22:43.310 --> 01:22:45.480
So in the gravity
side, this surface

01:22:45.480 --> 01:22:48.252
would be in the constant
time slicing, which

01:22:48.252 --> 01:22:49.460
would go to the gravity side.

01:22:52.080 --> 01:22:54.320
AUDIENCE: Professor,
one more thing.

01:22:54.320 --> 01:22:57.980
With the CFT and
AdS-- I know we like

01:22:57.980 --> 01:23:00.840
to think one existing on
the boundary of the other.

01:23:00.840 --> 01:23:04.765
But how does-- that's sort
of an abstraction, right?

01:23:04.765 --> 01:23:06.640
HONG LIU: No, that you
should really consider

01:23:06.640 --> 01:23:09.300
as a real thing-- real stuff.

01:23:09.300 --> 01:23:14.050
Yeah, if you just think
of that as an abstraction,

01:23:14.050 --> 01:23:16.770
then you miss a
lot of intuition,

01:23:16.770 --> 01:23:19.170
because you use that
as a genuine boundary,

01:23:19.170 --> 01:23:20.870
will give you a
lot of intuition.

01:23:20.870 --> 01:23:24.997
And many things
become very natural.

01:23:24.997 --> 01:23:26.330
Many things become very natural.

01:23:26.330 --> 01:23:27.775
Yes?

01:23:27.775 --> 01:23:29.775
AUDIENCE: So comparing
to the other formula that

01:23:29.775 --> 01:23:33.160
also has an area
of the boundary,

01:23:33.160 --> 01:23:35.877
all this tells us
is just it helps

01:23:35.877 --> 01:23:38.130
us count the number of
degrees of freedom in the CFT.

01:23:38.130 --> 01:23:39.546
So it all it tells
us is basically

01:23:39.546 --> 01:23:40.837
the epsilon in the denominator?

01:23:40.837 --> 01:23:44.990
HONG LIU: No, no, they have a
complete different dimension.

01:23:44.990 --> 01:23:47.476
No, this one higher dimension.

01:23:47.476 --> 01:23:49.542
AUDIENCE: So that's
another strange--

01:23:49.542 --> 01:23:51.250
HONG LIU: No, this is
not strange at all,

01:23:51.250 --> 01:23:53.200
because they're not
supposed to be the same.

01:23:53.200 --> 01:23:56.050
No, this is the
way it should be.

01:23:56.050 --> 01:23:57.690
So this is d minus
1 dimensional.

01:23:57.690 --> 01:24:00.540
And that is the d
minus 2 dimensional.

01:24:00.540 --> 01:24:02.785
It's just completely different.

01:24:02.785 --> 01:24:05.130
AUDIENCE: Well, right,
but they are not-- there

01:24:05.130 --> 01:24:07.972
is something the
same about them.

01:24:07.972 --> 01:24:10.250
HONG LIU: No, there is
something to same about them.

01:24:10.250 --> 01:24:12.910
But these two have
completely different physics.

01:24:12.910 --> 01:24:15.310
So this has a close analog
with black hole physics.

01:24:18.884 --> 01:24:20.800
Don't think they're
motivated by that formula.

01:24:20.800 --> 01:24:23.250
I think they're motivated
by the black hole formula.

01:24:23.250 --> 01:24:25.420
And people also try to connect
that formula to the black hole

01:24:25.420 --> 01:24:25.730
formula.

01:24:25.730 --> 01:24:26.659
That's another story.

01:24:26.659 --> 01:24:27.742
Anyway, it doesn't matter.

01:24:30.290 --> 01:24:31.140
OK, good.

01:24:31.140 --> 01:24:32.050
So this is a guess.

01:24:32.050 --> 01:24:33.300
Yes?

01:24:33.300 --> 01:24:36.830
AUDIENCE: So is this result for
the ground state on the CFT?

01:24:36.830 --> 01:24:38.500
HONG LIU: No, this is valid.

01:24:38.500 --> 01:24:42.570
Yeah, for the
time-independent state.

01:24:42.570 --> 01:24:45.533
The eigenstate is not
invoked with time.

01:24:45.533 --> 01:24:47.116
AUDIENCE: The thing
I don't understand

01:24:47.116 --> 01:24:50.374
is on the site of the
CFT, there is this state

01:24:50.374 --> 01:24:51.915
that you did as an
input to calculate

01:24:51.915 --> 01:24:53.890
the entanglement entropy.

01:24:53.890 --> 01:24:56.850
And what's the corresponding
thing on the AdS side?

01:24:56.850 --> 01:24:59.120
HONG LIU: The AdS side is
the corresponding geometry.

01:24:59.120 --> 01:25:02.710
So it's as said before, each
state on the field theory

01:25:02.710 --> 01:25:04.440
should correspond
to some geometry

01:25:04.440 --> 01:25:07.050
with normalizable modes.

01:25:07.050 --> 01:25:09.510
Yeah, so it's the
corresponding geometry.

01:25:09.510 --> 01:25:13.810
And the ground state, then
this will be just pure ideas.

01:25:13.810 --> 01:25:16.130
And if you look at the
final temperature state,

01:25:16.130 --> 01:25:18.190
then this will be
the black hole.

01:25:18.190 --> 01:25:20.960
And then if you are able to
construct some other geometry

01:25:20.960 --> 01:25:24.640
corresponding to some other
state, you can use that.

01:25:24.640 --> 01:25:29.170
AUDIENCE: How about this entropy
on independent on which state

01:25:29.170 --> 01:25:30.950
we choose for CFT?

01:25:30.950 --> 01:25:33.400
HONG LIU: No, of
course it depends.

01:25:33.400 --> 01:25:35.700
No this formula does not.

01:25:35.700 --> 01:25:37.500
But the geometry does.

01:25:37.500 --> 01:25:39.380
The geometry does.

01:25:39.380 --> 01:25:44.020
This is a minimal surface in
whatever [INAUDIBLE] geometry.

01:25:44.020 --> 01:25:47.040
AUDIENCE: Does the region--
after we choose the region,

01:25:47.040 --> 01:25:48.350
we can choose any state--

01:25:48.350 --> 01:25:49.532
HONG LIU: Yes, yes.

01:25:49.532 --> 01:25:50.990
AUDIENCE: And as
a minimal surface,

01:25:50.990 --> 01:25:52.965
is it independent on
which state we choose?

01:25:52.965 --> 01:25:55.090
HONG LIU: Of course, it
would depend on which state

01:25:55.090 --> 01:25:58.050
you choose, because
each state corresponds

01:25:58.050 --> 01:25:59.480
to a different geometry.

01:25:59.480 --> 01:26:03.370
And the minimal surface
depends on your geometry.

01:26:03.370 --> 01:26:06.540
So each state-- so
remember previously,

01:26:06.540 --> 01:26:10.480
the state on the
CFT side is mapped

01:26:10.480 --> 01:26:15.110
to the geometry with
normalizable boundary

01:26:15.110 --> 01:26:17.740
conditions with
normalizable modes.

01:26:17.740 --> 01:26:20.120
Yeah, anyway, just let
me say just geometry.

01:26:20.120 --> 01:26:21.500
So a state with the geometry.

01:26:21.500 --> 01:26:24.060
If you have a different state,
you have a different geometry.

01:26:24.060 --> 01:26:26.670
But by definition,
all this geometry

01:26:26.670 --> 01:26:28.430
would be asymptotic AdS.

01:26:28.430 --> 01:26:32.450
So in a boundary, they
should all be asymptotic AdS.

01:26:32.450 --> 01:26:38.180
And so that what means to
have [INAUDIBLE] mode excited.

01:26:38.180 --> 01:26:39.270
OK?

01:26:39.270 --> 01:26:39.770
Yes?

01:26:39.770 --> 01:26:41.700
AUDIENCE: So this
entropy formula

01:26:41.700 --> 01:26:45.540
takes into account the entropy
of gravitational excitations

01:26:45.540 --> 01:26:47.470
in AdS, if you can
put it that way.

01:26:47.470 --> 01:26:48.760
But black hole entropy--

01:26:48.760 --> 01:26:55.480
HONG LIU: No, let's try
not to jump too fast.

01:26:55.480 --> 01:27:00.230
And I think you are trying
to extrapolate too fast.

01:27:00.230 --> 01:27:03.290
This formula is
supposed to calculate

01:27:03.290 --> 01:27:06.330
the entangled entropy
in the field series

01:27:06.330 --> 01:27:09.900
side in the [INAUDIBLE]
limit and the notch

01:27:09.900 --> 01:27:12.330
lambda toward coupling limit.

01:27:12.330 --> 01:27:15.160
And we can talk about the
gravitational interpretation

01:27:15.160 --> 01:27:15.770
later.

01:27:15.770 --> 01:27:18.830
But this formula is about
the field series entanglement

01:27:18.830 --> 01:27:19.330
entropy.

01:27:19.330 --> 01:27:19.829
OK?

01:27:22.280 --> 01:27:22.780
Good.

01:27:22.780 --> 01:27:24.350
Any other questions?

01:27:27.250 --> 01:27:30.190
OK, good.

01:27:30.190 --> 01:27:33.160
So there are many support of it.

01:27:33.160 --> 01:27:34.900
So this is a rule of the game.

01:27:34.900 --> 01:27:36.510
You make a guess.

01:27:36.510 --> 01:27:39.390
Then you do a check.

01:27:39.390 --> 01:27:44.480
If the check worked, then
that gives you confidence,

01:27:44.480 --> 01:27:46.910
and then you do another check.

01:27:46.910 --> 01:27:49.727
And here you find it will fail.

01:27:49.727 --> 01:27:51.560
And if you do a sufficient
number of checks,

01:27:51.560 --> 01:27:52.852
then people believed you.

01:27:52.852 --> 01:27:54.060
Then people will believe you.

01:27:54.060 --> 01:27:56.020
And everybody
started checking it.

01:27:56.020 --> 01:27:59.730
And then sooner or
later we will see

01:27:59.730 --> 01:28:03.450
whether this fails or works.

01:28:03.450 --> 01:28:07.882
Anyway, so this is nice because
this is simple to compute,

01:28:07.882 --> 01:28:10.090
because finding a minimal
surface on the gravity side

01:28:10.090 --> 01:28:18.220
is in some sense, it's a
straightforward a mathematical

01:28:18.220 --> 01:28:20.655
problem conceptually,
even though technically it

01:28:20.655 --> 01:28:21.800
may not be simple.

01:28:21.800 --> 01:28:25.210
But it's straightforward
conceptually.

01:28:25.210 --> 01:28:29.747
Anyway, so you can use these
to calculate many quantities.

01:28:32.300 --> 01:28:42.180
Let me just say in
support of this--

01:28:42.180 --> 01:28:45.540
let me just say it in
words just to save time.

01:28:45.540 --> 01:28:47.430
So the first thing
you should check

01:28:47.430 --> 01:28:51.710
is that this satisfies the
subadditive condition which

01:28:51.710 --> 01:28:53.440
we wrote down earlier,
because that's

01:28:53.440 --> 01:28:55.360
a very non-trivial condition.

01:28:55.360 --> 01:28:57.370
It should be satisfied
by the entropy.

01:28:57.370 --> 01:29:01.500
So first, you should check that,
because if you violate that,

01:29:01.500 --> 01:29:04.840
than this proposal is gone.

01:29:04.840 --> 01:29:06.840
And then there are
other things that you

01:29:06.840 --> 01:29:10.680
can try to use to reproduce
all the known behavior we know

01:29:10.680 --> 01:29:12.920
about entangled
entropy, for example,

01:29:12.920 --> 01:29:16.890
this behavior and this
behavior-- this area along.

01:29:21.470 --> 01:29:24.120
And then we actually
build up confidence

01:29:24.120 --> 01:29:26.250
by checking all
those known results.

01:29:26.250 --> 01:29:28.950
And then you can try
to derive [INAUDIBLE]

01:29:28.950 --> 01:29:32.300
and to see whether [INAUDIBLE]
makes physical sense.

01:29:32.300 --> 01:29:36.500
So essentially,
that's how it works.

01:29:36.500 --> 01:29:39.620
And this is nice also
at the tactical level,

01:29:39.620 --> 01:29:43.530
because entangled entropy is
something very hard to compute.

01:29:43.530 --> 01:29:44.990
I don't know.

01:29:44.990 --> 01:29:48.110
You guys may not have experience
of calculating entangled

01:29:48.110 --> 01:29:48.930
entropy.

01:29:48.930 --> 01:29:50.650
But ask Frank Wilczek.

01:29:50.650 --> 01:29:52.500
He was one of the
first few people

01:29:52.500 --> 01:29:54.530
to do it in some
free field theory.

01:29:54.530 --> 01:29:58.260
Even for free field
theory, people could mostly

01:29:58.260 --> 01:30:00.330
do it in 1 plus 1 dimension.

01:30:00.330 --> 01:30:04.540
Going beyond 1 plus 1 dimension
free scalar field theory in 2

01:30:04.540 --> 01:30:07.540
plus 1 dimension
3 plus 1 dimension

01:30:07.540 --> 01:30:10.480
for some simple region
like a circle or sphere

01:30:10.480 --> 01:30:12.810
you have to do a
numerical calculation.

01:30:12.810 --> 01:30:15.310
You have to discretize
the field theory

01:30:15.310 --> 01:30:17.500
to do very massive
numerical calculation.

01:30:17.500 --> 01:30:19.890
It's not easy to compute.

01:30:19.890 --> 01:30:21.220
It's not easy to compute.

01:30:21.220 --> 01:30:25.210
But this guy is actually easy
to compute in comparison.

01:30:25.210 --> 01:30:27.940
So this guy at the
technical level

01:30:27.940 --> 01:30:32.245
provides a huge convenience
to calculate many things.

01:30:34.950 --> 01:30:36.740
Yeah, so this is a side remark.

01:30:36.740 --> 01:30:42.540
So now let's just do
some calculations.

01:30:42.540 --> 01:30:44.290
So first, let me show
that this actually

01:30:44.290 --> 01:30:45.723
satisfies strong subaddivity.

01:30:56.850 --> 01:31:03.380
So remember, the formula
we had before is S(AC),

01:31:03.380 --> 01:31:04.880
unfortunately I erased it.

01:31:13.710 --> 01:31:16.430
OK, so this is one
of the inequalities.

01:31:16.430 --> 01:31:18.910
So we are just drawing
it 1 plus 1 dimension

01:31:18.910 --> 01:31:20.950
because it's easy to draw.

01:31:20.950 --> 01:31:23.590
But a similar thing can
be easily generalized

01:31:23.590 --> 01:31:25.180
in a higher dimension.

01:31:25.180 --> 01:31:28.330
So let's look at just the
line in 1 plus 1 dimension.

01:31:28.330 --> 01:31:31.335
So this can call
this region A. This

01:31:31.335 --> 01:31:34.460
is region C. This is region B.

01:31:34.460 --> 01:31:36.335
You can also make them separate.

01:31:36.335 --> 01:31:37.820
It doesn't matter.

01:31:37.820 --> 01:31:39.500
You can also make them separate.

01:31:39.500 --> 01:31:41.530
It's easy to work out.

01:31:41.530 --> 01:31:45.060
So I'm just giving
you one cases.

01:31:45.060 --> 01:31:49.270
So suppose a minimal service
for the AC reading is like this.

01:31:49.270 --> 01:31:50.960
And the minimal
surface the BC reading

01:31:50.960 --> 01:31:54.370
would be like this-- so
minimal surface like that.

01:31:57.950 --> 01:32:03.980
So let me call this curve gamma
AC and this curve gamma BC.

01:32:03.980 --> 01:32:10.290
OK, I could not find
the colored chalk today.

01:32:10.290 --> 01:32:14.070
So I did not have
the colored chalk.

01:32:14.070 --> 01:32:17.447
Let me see whether
this chalk works.

01:32:17.447 --> 01:32:18.530
Is this the colored chalk?

01:32:18.530 --> 01:32:19.030
Yes.

01:32:21.740 --> 01:32:24.440
So now, let me define
two other surfaces.

01:32:24.440 --> 01:32:36.436
Let me call this one gamma
1 and this one gamma 2.

01:32:36.436 --> 01:32:38.810
In principle, I should use
two different chalks for that.

01:32:43.720 --> 01:32:47.240
You think it looks
like something?

01:32:47.240 --> 01:32:52.850
OK, anyway, so we have
gamma AC plus gamma

01:32:52.850 --> 01:32:57.240
BC equals to gamma
1 plus gamma 2

01:32:57.240 --> 01:33:02.730
because-- gamma AC-- this
is length-- gamma BC is

01:33:02.730 --> 01:33:03.980
equal to gamma 1 plus gamma 2.

01:33:12.820 --> 01:33:17.000
And then just from
the definition

01:33:17.000 --> 01:33:18.840
of the minimal
surface, then the gamma

01:33:18.840 --> 01:33:23.700
1 must be greater than gamma
C in terms of the length

01:33:23.700 --> 01:33:27.360
because the minimal
surface associated with C

01:33:27.360 --> 01:33:30.740
must be the minimal area.

01:33:30.740 --> 01:33:33.320
It must be smaller than gamma 1.

01:33:33.320 --> 01:33:39.410
Under the area of gamma 2 must
be greater than gamma ABC,

01:33:39.410 --> 01:33:44.850
because the gamma 2 is
the boundary is the ABC.

01:33:44.850 --> 01:33:48.060
And the gamma 2 must be
greater than gamma BC

01:33:48.060 --> 01:33:50.310
because this is supposed
to be the minimal surface.

01:33:54.200 --> 01:33:59.310
So now we conclude that
gamma AC plus gamma

01:33:59.310 --> 01:34:05.730
BC is greater than
gamma ABC plus gamma C.

01:34:05.730 --> 01:34:08.070
And then this translates
into that formula.

01:34:08.070 --> 01:34:08.830
OK?

01:34:08.830 --> 01:34:15.362
Very simple, and elegant proof--
very simple and elegant proof.

01:34:15.362 --> 01:34:17.070
And now you can prove
another inequality.

01:34:20.546 --> 01:34:21.920
You can prove
another inequality,

01:34:21.920 --> 01:34:29.675
which is S(AC) plus S(BC)
greater than S(A) and S(B).

01:34:32.410 --> 01:34:37.700
So now, I will define
another surface.

01:34:37.700 --> 01:34:39.640
OK?

01:34:39.640 --> 01:34:47.860
So now, let me just
redraw it-- A, C, B.

01:34:47.860 --> 01:34:50.000
So I have AC like this.

01:34:50.000 --> 01:34:51.980
I have BC like this.

01:34:51.980 --> 01:35:02.216
And now, let me call this one
gamma 1 tilde-- this one gamma

01:35:02.216 --> 01:35:02.893
2 tilde.

01:35:12.410 --> 01:35:14.590
This is annoying.

01:35:14.590 --> 01:35:15.460
OK.

01:35:15.460 --> 01:35:20.420
And again, gamma AB plus
gamma AC plus gamma BC

01:35:20.420 --> 01:35:26.670
equals to gamma 1 tilde
plus gamma 2 tilde.

01:35:26.670 --> 01:35:30.790
So the gamma 1 tilde
ends in A. Gamma

01:35:30.790 --> 01:35:32.950
2 tilde ends in B.
That means gamma

01:35:32.950 --> 01:35:38.150
1 tilde must be greater than
gamma A and gamma 2 tilde

01:35:38.150 --> 01:35:41.310
must be greater
than gamma B. OK?

01:35:41.310 --> 01:35:43.500
So now you show that
gamma AC plus gamma

01:35:43.500 --> 01:35:49.630
BC is greater than gamma A plus
gamma B. So this is very easy.

01:35:49.630 --> 01:35:52.020
So this is a very
simple and elegant.

01:35:52.020 --> 01:35:58.250
So if you want to really
to be impressed by this,

01:35:58.250 --> 01:36:03.640
I urge you to look at the
proof of the strong subaddivity

01:36:03.640 --> 01:36:10.210
itself in say in some textbook
or in the original papers.

01:36:10.210 --> 01:36:12.220
It's actually
highly non-trivial.

01:36:12.220 --> 01:36:15.350
You need some double,
double concave functions.

01:36:15.350 --> 01:36:19.900
It's quite a height involved.

01:36:19.900 --> 01:36:27.550
OK, so this is a
great confidence boost

01:36:27.550 --> 01:36:30.345
that satisfies the strong
subadditivity condition.

01:36:36.372 --> 01:36:37.830
So now, let's look
at the last one.

01:36:37.830 --> 01:36:39.620
Let's try to reproduce
this formula.

01:36:39.620 --> 01:36:41.190
Let's try to do a
simple calculation

01:36:41.190 --> 01:36:42.680
to reproduce this formula.

01:36:42.680 --> 01:36:46.880
So this formula is supposed
to be true for any CFT.

01:36:46.880 --> 01:36:54.020
This should be also applied
for the holographic CFT

01:36:54.020 --> 01:36:56.185
and to check whether
this formula works.

01:36:59.310 --> 01:37:05.940
The reason you would like to
choose an example which is new.

01:37:05.940 --> 01:37:07.639
But this example
is the simplest.

01:37:07.639 --> 01:37:09.430
So that's the reason
I choose this example,

01:37:09.430 --> 01:37:13.590
even though it's just
reproduced all the results.

01:37:13.590 --> 01:37:18.680
So let's now look at the
entangled entropy of 1

01:37:18.680 --> 01:37:19.982
plus 1 dimensional CFT.

01:37:24.390 --> 01:37:27.850
So here, now you
have a CFT 1 plus 1

01:37:27.850 --> 01:37:29.150
should be dual to the AdS3.

01:37:33.660 --> 01:37:36.330
You will do some theory in AdS3.

01:37:36.330 --> 01:37:41.460
So the S squared-- so let's
only look at the vacuum

01:37:41.460 --> 01:37:44.650
then we just work with AdS3.

01:37:44.650 --> 01:37:46.299
So let me write down
the AdS3 metric.

01:37:51.950 --> 01:37:53.803
So dx is the
boundary directions.

01:37:56.760 --> 01:37:59.140
So as I said, each
CFT is characterized

01:37:59.140 --> 01:38:00.090
by a central charge.

01:38:04.230 --> 01:38:05.690
And you can obtain
a central charge

01:38:05.690 --> 01:38:07.720
from various different ways.

01:38:07.720 --> 01:38:11.170
And again, from the
holographic, you

01:38:11.170 --> 01:38:12.710
can try to express
the single charge

01:38:12.710 --> 01:38:14.506
in terms of gravity quantities.

01:38:14.506 --> 01:38:17.130
And there is many ways to derive
it that we will not go through

01:38:17.130 --> 01:38:18.570
with that calculation.

01:38:18.570 --> 01:38:21.330
So let me just write
down the results.

01:38:21.330 --> 01:38:28.330
So far the holographic
CFTs, the single charge

01:38:28.330 --> 01:38:31.000
is related to the
gravity quantity

01:38:31.000 --> 01:38:34.160
by 3R divided by 2 G Newton.

01:38:36.740 --> 01:38:38.860
So this is a result
which I just caught.

01:38:38.860 --> 01:38:42.330
I will not derive it for you.

01:38:42.330 --> 01:38:45.090
But you can derive it
in many different ways.

01:38:45.090 --> 01:38:50.150
So for CFT, to do
a gravity idea 3,

01:38:50.150 --> 01:38:52.360
its central charge
is given by 3R,

01:38:52.360 --> 01:38:56.820
which is the AdS radius divided
by 2 times the Newton constant.

01:38:56.820 --> 01:38:59.760
And the Newton constant
in three dimensions

01:38:59.760 --> 01:39:00.980
has mass dimension 1.

01:39:00.980 --> 01:39:04.110
So this is a
dimensionless number--

01:39:04.110 --> 01:39:06.850
a dimensionless number.

01:39:06.850 --> 01:39:10.694
So now let's calculate
the entanglement entropy

01:39:10.694 --> 01:39:12.116
using that formula.

01:39:15.750 --> 01:39:18.260
So this now reduced
your calculation

01:39:18.260 --> 01:39:23.770
very similar to our
Wilson loop calculation.

01:39:23.770 --> 01:39:25.340
So let's call this
L divided by 2.

01:39:25.340 --> 01:39:27.320
So lets call this X direction.

01:39:27.320 --> 01:39:29.040
So this is a V direction.

01:39:29.040 --> 01:39:30.760
So this is L divided by 2.

01:39:30.760 --> 01:39:32.050
This is minus L divided by 2.

01:39:32.050 --> 01:39:34.780
This region is A.
Then we need to find

01:39:34.780 --> 01:39:40.470
a curve, because this now 1 plus
1 dimensions-- one dimensions.

01:39:40.470 --> 01:39:44.130
So we need to find the curve
which end on this segment of A.

01:39:44.130 --> 01:39:46.340
OK?

01:39:46.340 --> 01:39:47.090
So is this clear?

01:39:49.650 --> 01:39:51.380
So this is the x direction.

01:39:51.380 --> 01:39:55.530
And so this is 0, x equals to 0.

01:39:55.530 --> 01:39:57.350
Yes?

01:39:57.350 --> 01:39:58.690
OK.

01:39:58.690 --> 01:40:04.150
So what we need to find is find
the minimal surface connecting

01:40:04.150 --> 01:40:07.270
these two points-- a minimal
curve-- a minimal length

01:40:07.270 --> 01:40:08.862
curve connecting
these two points.

01:40:11.960 --> 01:40:15.180
So it's as we said
before, we should

01:40:15.180 --> 01:40:18.780
look at the constant time slice
because this is a vacuum which

01:40:18.780 --> 01:40:19.720
is time independent.

01:40:19.720 --> 01:40:21.810
Let's look at the
constant time slice.

01:40:21.810 --> 01:40:24.240
The constant time
slice, the metric

01:40:24.240 --> 01:40:27.440
becomes-- so in this
space, the metric becomes r

01:40:27.440 --> 01:40:30.414
square divided by z square
because the time does not

01:40:30.414 --> 01:40:30.913
change.

01:40:34.490 --> 01:40:40.850
So now if I treat the x as a
function of z, and then these

01:40:40.850 --> 01:40:41.630
will become.

01:40:41.630 --> 01:40:43.200
So this curve lets
me parametrize

01:40:43.200 --> 01:40:44.760
x as a function of z.

01:40:44.760 --> 01:40:48.140
Then this just becomes r
square divided by z square 1

01:40:48.140 --> 01:40:52.110
plus x prime square dz square.

01:40:56.400 --> 01:41:01.040
And so this is dl square.

01:41:01.040 --> 01:41:02.910
So the length of
the curve square

01:41:02.910 --> 01:41:05.150
would be just
parametrized by this guy.

01:41:05.150 --> 01:41:05.650
OK?

01:41:10.630 --> 01:41:13.310
And we have to satisfy
the boundary condition--

01:41:13.310 --> 01:41:17.340
by symmetry, we only need to
consider the right stuff OK,

01:41:17.340 --> 01:41:18.990
because it's symmetric.

01:41:18.990 --> 01:41:21.820
So we impose the boundary
condition at the x

01:41:21.820 --> 01:41:23.420
z equal to 0.

01:41:23.420 --> 01:41:24.990
It's L divided by 2.

01:41:27.885 --> 01:41:29.220
We can see the right half.

01:41:42.655 --> 01:41:45.155
So now you can just write down
the entanglement entropy now.

01:41:52.110 --> 01:41:55.000
So S(A) will be 1 over 4GN.

01:41:58.250 --> 01:42:01.270
So I integrate
half of the curve.

01:42:01.270 --> 01:42:03.537
Let me just call
here z is equal to 0.

01:42:03.537 --> 01:42:04.620
Let me call this point z0.

01:42:07.610 --> 01:42:12.790
So I just integrate
from 0 to z0.

01:42:12.790 --> 01:42:17.300
Z0 we will find out when you
find out this minimal length

01:42:17.300 --> 01:42:17.800
curve.

01:42:17.800 --> 01:42:22.480
So this one will be r/z dz.

01:42:22.480 --> 01:42:25.320
Then just taking the
square root of this guy--

01:42:25.320 --> 01:42:27.680
the square of 1
over x prime square.

01:42:27.680 --> 01:42:31.500
So this is the length
and the times 2,

01:42:31.500 --> 01:42:34.530
because we only can see
that this is half of it.

01:42:38.520 --> 01:42:41.110
And now you extremize it.

01:42:41.110 --> 01:42:44.010
Then you find the
minimal length curve.

01:42:44.010 --> 01:42:45.510
And then you plug
the solution here.

01:42:45.510 --> 01:42:47.096
Then you find the action.

01:42:47.096 --> 01:42:48.720
Actually we don't
need to extremize it,

01:42:48.720 --> 01:42:50.920
because this is a
well-known problem.

01:42:50.920 --> 01:42:53.450
And this is called
the Poincare disk.

01:42:53.450 --> 01:42:57.160
So this is a prototype of a
two-dimensional hyperbolic

01:42:57.160 --> 01:42:58.390
space.

01:42:58.390 --> 01:43:03.003
And the minimal length curve
inside your hyperbolic space

01:43:03.003 --> 01:43:07.949
is maybe a 16th century problem.

01:43:07.949 --> 01:43:08.740
Yeah, I don't know.

01:43:08.740 --> 01:43:10.800
Actually, maybe 17th century.

01:43:10.800 --> 01:43:13.990
Anyway, so the answer
is actually very long.

01:43:13.990 --> 01:43:19.150
But you can also easily
work out yourself.

01:43:19.150 --> 01:43:22.880
So the result is actually
this is just exactly a circle.

01:43:22.880 --> 01:43:24.300
OK?

01:43:24.300 --> 01:43:25.650
Exactly a half circle.

01:43:31.120 --> 01:43:32.800
So that's actually
a half circle.

01:43:32.800 --> 01:43:35.260
That means that x equals 2.

01:43:35.260 --> 01:43:37.712
The half circle with a
radius L divided by 2.

01:43:37.712 --> 01:43:39.520
So this has just
become L squared

01:43:39.520 --> 01:43:41.280
divided by 4 minus d square.

01:43:44.590 --> 01:43:48.030
In particular, z0 corresponding
to the point which

01:43:48.030 --> 01:43:51.360
x equal to 0 just L divided 2.

01:43:54.740 --> 01:43:59.000
So now you just
plug this into here.

01:43:59.000 --> 01:44:00.900
You just plug this here.

01:44:00.900 --> 01:44:05.250
Let me just write it a
little bit fast, because we

01:44:05.250 --> 01:44:08.660
are a little bit out of time.

01:44:08.660 --> 01:44:12.090
So you plug this in.

01:44:12.090 --> 01:44:14.080
Then you can scale the z out.

01:44:14.080 --> 01:44:18.619
You can scale z to be scaled
L divided by 2 factors out.

01:44:18.619 --> 01:44:19.660
And let me just scale it.

01:44:19.660 --> 01:44:21.730
You can rewrite the
integral as follows.

01:44:21.730 --> 01:44:25.820
So after the scaling
it's say 2z scale

01:44:25.820 --> 01:44:30.020
with L divided by 2z
to do the scaling.

01:44:30.020 --> 01:44:33.380
Then you find this
expression becomes 2r/4GN.

01:44:37.420 --> 01:44:40.840
And the upper limit becomes 1.

01:44:40.840 --> 01:44:43.032
And the lower limit you see.

01:44:43.032 --> 01:44:44.990
When you plug this in,
actually the lower limit

01:44:44.990 --> 01:44:47.970
is divergent because
there's 1 over z here.

01:44:47.970 --> 01:44:49.710
As always, we need
to put the cutoff.

01:44:49.710 --> 01:44:53.300
And also, this divergence
is expected because of below

01:44:53.300 --> 01:44:55.700
that here it depends
on short distance

01:44:55.700 --> 01:44:57.900
cutoff it would be divergent.

01:44:57.900 --> 01:45:00.110
So we need to for the
short-distance cutoff here.

01:45:00.110 --> 01:45:04.430
So let me put it here epsilon.

01:45:04.430 --> 01:45:07.895
And this will be 2
epsilon divided by L

01:45:07.895 --> 01:45:10.360
when I do this scaling.

01:45:10.360 --> 01:45:15.880
And then you have
dz/z 1 minus d square.

01:45:15.880 --> 01:45:18.420
So the integral reduces
to something like this.

01:45:18.420 --> 01:45:21.900
And now you can evaluate
this integral trivially.

01:45:21.900 --> 01:45:28.040
Then you find the epsilon
goes to the limit.

01:45:28.040 --> 01:45:35.280
Then you find the one term
survives 2r divided by 4GN log

01:45:35.280 --> 01:45:36.500
L divided by epsilon.

01:45:39.360 --> 01:45:41.420
So now let's try
to rewrite in terms

01:45:41.420 --> 01:45:46.290
of this C-- we
rewrite in terms of C.

01:45:46.290 --> 01:45:55.920
So this is 1/3 3r divided by
2GN log L divided by epsilon.

01:45:55.920 --> 01:45:59.924
And so this is precisely
that formula C divided by 3

01:45:59.924 --> 01:46:03.830
log L of epsilon.

01:46:03.830 --> 01:46:07.030
And with that, you just count
from the minimal surface

01:46:07.030 --> 01:46:08.565
in the hyperbolic
space is a circle.

01:46:12.050 --> 01:46:15.320
And it takes a lot of effort
to calculate this thing,

01:46:15.320 --> 01:46:19.622
even to do a free field
series calculation.

01:46:19.622 --> 01:46:21.330
But in gravity you
can do it very easily.

01:46:24.490 --> 01:46:27.310
So now let me just
quickly mention

01:46:27.310 --> 01:46:31.430
what happens if you do
at a finite temperature.

01:46:31.430 --> 01:46:34.610
So you can also do this
at a finite temperature.

01:46:38.924 --> 01:46:41.340
So since we're out of time,
we'll be doing it a little bit

01:46:41.340 --> 01:46:42.950
fast.

01:46:42.950 --> 01:46:47.230
So first doing it at a
finite temperature is easier.

01:46:47.230 --> 01:46:53.510
Let me just say finite T.
And then I will connect

01:46:53.510 --> 01:46:55.600
to the black hole entropy.

01:46:55.600 --> 01:46:59.189
I will show that
this formula actually

01:46:59.189 --> 01:47:01.230
reduces to the black hole
entropy in some limits.

01:47:15.709 --> 01:47:17.250
So for these problems,
actually, it's

01:47:17.250 --> 01:47:19.306
easier to consider
CFT on the circle.

01:47:26.700 --> 01:47:27.906
OK.

01:47:27.906 --> 01:47:29.280
So when the CFT
is on the circle,

01:47:29.280 --> 01:47:31.030
the boundary will be
a circle because this

01:47:31.030 --> 01:47:34.070
is 1 plus 1 dimension.

01:47:34.070 --> 01:47:37.320
So remember, in
the global ideas,

01:47:37.320 --> 01:47:40.750
it's like sitting
there in the boundary.

01:47:40.750 --> 01:47:43.360
And in the 1 plus 1 dimensional
case and the boundary

01:47:43.360 --> 01:47:44.390
is just really a circle.

01:47:44.390 --> 01:47:46.860
And this is the bark

01:47:46.860 --> 01:47:50.560
And a finite temperature,
you put the black hole there.

01:47:50.560 --> 01:47:52.670
Let me just put the
black hole here.

01:47:52.670 --> 01:47:53.572
OK?

01:47:53.572 --> 01:47:54.925
Just put the black hole there.

01:47:57.820 --> 01:48:01.390
So now you can ask the
following question.

01:48:01.390 --> 01:48:04.990
So now let's consider
some reaching A. So

01:48:04.990 --> 01:48:08.200
let's consider A is very small.

01:48:08.200 --> 01:48:10.160
Yo can work out the
minimal surface.

01:48:10.160 --> 01:48:13.250
So it acts the same as
our Wilson loop story.

01:48:13.250 --> 01:48:14.880
So you find a minimal surface.

01:48:14.880 --> 01:48:17.660
But if A is sufficiently
small, then the geometry around

01:48:17.660 --> 01:48:21.130
here is still AdS.

01:48:21.130 --> 01:48:23.370
And then you just
get a minimal surface

01:48:23.370 --> 01:48:26.030
around here-- so a
small deformation

01:48:26.030 --> 01:48:28.810
from the vacuum behavior.

01:48:28.810 --> 01:48:31.320
So we'll make A
larger and larger.

01:48:34.140 --> 01:48:40.750
So this z0 depends on L. If
we make L larger and larger,

01:48:40.750 --> 01:48:46.070
this is z0 will be
deeper in the bark.

01:48:46.070 --> 01:48:49.752
So if you make A larger, so
this will go more to the bark

01:48:49.752 --> 01:48:51.460
and then will be
deformed, and then we'll

01:48:51.460 --> 01:48:54.574
see the black hole geometry.

01:48:54.574 --> 01:48:55.990
But now to answer
the question how

01:48:55.990 --> 01:49:02.410
about if I make A to be
larger-- this region to be

01:49:02.410 --> 01:49:08.110
A to be larger than the half
of the size of the circle.

01:49:08.110 --> 01:49:11.960
So the minimal surface
should be able to be

01:49:11.960 --> 01:49:14.050
to be smoothly
deformed back into A.

01:49:14.050 --> 01:49:15.800
So now you have to do
something like this.

01:49:15.800 --> 01:49:19.338
It turns out the minimal surface
want to go around the horizon

01:49:19.338 --> 01:49:20.796
and then doing
something like this.

01:49:24.940 --> 01:49:27.065
AUDIENCE: Why don't you do
it the other way around?

01:49:27.065 --> 01:49:29.116
HONG LIU: No, it's
because the other won't.

01:49:29.116 --> 01:49:31.750
It's because A is like this.

01:49:31.750 --> 01:49:34.580
And if you do
something like this,

01:49:34.580 --> 01:49:37.110
then it can be
deformed back into A.

01:49:37.110 --> 01:49:39.160
It's mostly without
crossing the black hole.

01:49:39.160 --> 01:49:42.200
AUDIENCE: Yes, or it could be
like a phase jump or something

01:49:42.200 --> 01:49:44.854
at some point when one
surface becomes longer than

01:49:44.854 --> 01:49:45.710
[INAUDIBLE].

01:49:45.710 --> 01:49:49.750
HONG LIU: No, in this
situation, it's not that thing.

01:49:49.750 --> 01:49:51.090
It's something like this.

01:49:51.090 --> 01:49:57.040
Now, if you make A even
longer, eventually,

01:49:57.040 --> 01:49:58.997
let's take the A even longer.

01:49:58.997 --> 01:50:00.580
And that would be
something like this.

01:50:03.730 --> 01:50:06.270
Yeah, because you always
want to hug near the horizon.

01:50:06.270 --> 01:50:09.284
And eventually, we'll
make A to be tiny,

01:50:09.284 --> 01:50:10.950
then it will become
something like this.

01:50:10.950 --> 01:50:13.260
You have something
close on the horizon.

01:50:13.260 --> 01:50:14.996
Then you have
something like this.

01:50:14.996 --> 01:50:16.370
It's roughly
something like this.

01:50:21.530 --> 01:50:25.421
Yeah, is it clear to you?

01:50:25.421 --> 01:50:26.170
It doesn't matter.

01:50:26.170 --> 01:50:28.142
[LAUGHTER]

01:50:34.560 --> 01:50:38.810
So now let's consider--
anyway, yeah,

01:50:38.810 --> 01:50:40.100
let me just draw this again.

01:50:40.100 --> 01:50:41.929
It actually does matter.

01:50:41.929 --> 01:50:42.428
Sorry.

01:50:47.930 --> 01:50:50.870
So let's take A
to be very small.

01:50:50.870 --> 01:50:52.150
Let's take A to be a very big.

01:50:52.150 --> 01:50:54.890
This part will be very small.

01:50:54.890 --> 01:50:59.070
So we are finally we
will go like this.

01:50:59.070 --> 01:51:03.080
Anyway, something like this
when you hug the black hole.

01:51:03.080 --> 01:51:05.360
And eventually, when
these two points shrink,

01:51:05.360 --> 01:51:08.370
so take A with the total region.

01:51:08.370 --> 01:51:10.880
Take A to be the total space.

01:51:10.880 --> 01:51:12.460
What do you get?

01:51:12.460 --> 01:51:16.710
Then what you see that the
minimal surface essentially

01:51:16.710 --> 01:51:19.490
is just essentially a surface
hugger on the horizon.

01:51:27.100 --> 01:51:29.350
And this is a precise
black hole formula,

01:51:29.350 --> 01:51:31.640
because then this is a
imprecise a black hole formula.

01:51:31.640 --> 01:51:34.400
And this is a situation you
expect it to be black hole

01:51:34.400 --> 01:51:46.540
formula, because if you take
the A to be the host space,

01:51:46.540 --> 01:51:50.776
then rho A is just equal to
the rho of the whole system.

01:51:53.580 --> 01:51:58.020
And the S(A) by definition
is just equal to the S

01:51:58.020 --> 01:52:00.310
of the whole system.

01:52:00.310 --> 01:52:04.280
And this is just a thermal
density, a thermal entropy.

01:52:04.280 --> 01:52:07.500
And this is given by
the black hole formula.

01:52:07.500 --> 01:52:10.080
It's given by the
black hole formula.

01:52:10.080 --> 01:52:12.470
And this is given
by [INAUDIBLE].

01:52:12.470 --> 01:52:14.382
So you find in these
special cases, actually,

01:52:14.382 --> 01:52:15.840
you recover the
black hole formula.

01:52:19.480 --> 01:52:21.467
So I have some small things.

01:52:23.580 --> 01:52:24.830
We can talk a little bit more.

01:52:24.830 --> 01:52:26.610
But I think we'll skip it.

01:52:26.610 --> 01:52:30.840
You can also show that actually
for general dimensions,

01:52:30.840 --> 01:52:33.180
you always recover the area
more without the details

01:52:33.180 --> 01:52:38.380
of the geometry, because no
matter what kind of dimension

01:52:38.380 --> 01:52:41.000
you look at, the
typical state which will

01:52:41.000 --> 01:52:42.430
be normalizable to geometry.

01:52:42.430 --> 01:52:45.360
When you approach the AdS--
when you approach the boundary,

01:52:45.360 --> 01:52:48.330
it's always like AdS.

01:52:48.330 --> 01:52:50.010
And then you find in
a higher dimension,

01:52:50.010 --> 01:52:52.394
it's always like this.

01:52:52.394 --> 01:52:53.810
It says, when this
minimal surface

01:52:53.810 --> 01:52:56.500
is close to the boundary,
it becomes perpendicular

01:52:56.500 --> 01:52:58.406
to the boundary.

01:52:58.406 --> 01:52:59.780
Here is a circle,
then of course,

01:52:59.780 --> 01:53:01.238
it's perpendicular
to the boundary.

01:53:01.238 --> 01:53:02.870
But you see this
feature is actually

01:53:02.870 --> 01:53:06.930
generalized to higher
dimensions-- to any dimensions.

01:53:06.930 --> 01:53:12.210
So what you see is that
in the general dimension,

01:53:12.210 --> 01:53:14.751
a minimal surface will be
just near the boundary will

01:53:14.751 --> 01:53:16.917
be just perpendicular to
going down from everywhere.

01:53:19.306 --> 01:53:20.680
And then from
there, you can show

01:53:20.680 --> 01:53:25.640
that you always have the
area law from this behavior.

01:53:28.190 --> 01:53:30.540
You always have the area law.

01:53:30.540 --> 01:53:37.010
So this can give you an
exercise for yourself.

01:53:37.010 --> 01:53:38.980
Yeah, so let me just
finish by saying

01:53:38.980 --> 01:53:42.490
a couple of philosophical
remarks, and then we'll finish.

01:53:48.680 --> 01:53:54.620
So this RT formula
is really remarkable,

01:53:54.620 --> 01:53:58.110
because it's actually,
as I said, technically

01:53:58.110 --> 01:54:02.390
provides a very simple way to
calculate entangled entropy.

01:54:02.390 --> 01:54:09.670
But actually, conceptually,
it's even more profound

01:54:09.670 --> 01:54:15.830
because it tells you that
the space time is related

01:54:15.830 --> 01:54:21.950
to the entanglement, because
it simulates something which

01:54:21.950 --> 01:54:28.000
is very subtle from the field
theory side, because this S(A),

01:54:28.000 --> 01:54:30.480
as we said, you have to do
some complicated procedure

01:54:30.480 --> 01:54:33.400
to trace out degrees of
freedom-- take the log.

01:54:33.400 --> 01:54:36.060
And that turns out to be
related to the geometry

01:54:36.060 --> 01:54:37.292
in a very simple way.

01:54:37.292 --> 01:54:39.500
So essentially, you see that
entanglement-- it's just

01:54:39.500 --> 01:54:41.110
captured by the space time.

01:54:41.110 --> 01:54:43.130
And in particular,
it's something

01:54:43.130 --> 01:54:45.660
you can say the
geometry is essentially

01:54:45.660 --> 01:54:47.035
equal to the
quantum information.

01:54:56.020 --> 01:54:59.130
So this is a very exciting
also from the gravity

01:54:59.130 --> 01:55:02.470
perspective, because if they
ask you if you really understand

01:55:02.470 --> 01:55:05.930
this duality, then you can
actually using the quantum

01:55:05.930 --> 01:55:10.610
information of the field theory
to understand the subtle things

01:55:10.610 --> 01:55:13.050
about the geometry.

01:55:13.050 --> 01:55:14.906
It just tells you the
geometry is actually

01:55:14.906 --> 01:55:16.280
equal to the
quantum information.

01:55:19.660 --> 01:55:27.570
So for many years, actually not
that many years, so previously,

01:55:27.570 --> 01:55:32.160
we have people doing
quantum information.

01:55:32.160 --> 01:55:36.250
And we have people doing
this matter, which is

01:55:36.250 --> 01:55:41.390
the quantum many-body system.

01:55:41.390 --> 01:55:45.870
And this matter, or QCD, so this
is kind of quantum many-body.

01:55:49.470 --> 01:55:55.462
And then we have people doing
black hole, string theory,

01:55:55.462 --> 01:55:56.128
and the gravity.

01:56:00.170 --> 01:56:04.490
So they are all very
different fields.

01:56:04.490 --> 01:56:08.454
But now, they're all connected
by this holographic duality.

01:56:12.050 --> 01:56:14.500
Somehow essentially,
they connect all of them.

01:56:14.500 --> 01:56:17.440
So now, we essentially
just see a unified picture,

01:56:17.440 --> 01:56:23.570
a unified paradigm, to
understand all quantum systems,

01:56:23.570 --> 01:56:27.710
including quantum systems
without gravity, with gravity.

01:56:27.710 --> 01:56:29.480
So I will stop here.

01:56:29.480 --> 01:56:31.930
[APPLAUSE]