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

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00:00:23,490 --> 00:00:26,630
this Hawking-Page transition
we talked about last time

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00:00:26,630 --> 00:00:32,479
because we were running
out of time, so it

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00:00:32,479 --> 00:00:35,556
was a little bit hurried.

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00:00:35,556 --> 00:00:37,097
Do you have any
questions about that?

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00:00:39,447 --> 00:00:41,530
AUDIENCE: So what's the
original thermal AdS stat?

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00:00:44,230 --> 00:00:48,630
Did you put some random
[INAUDIBLE] onto that metric,

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00:00:48,630 --> 00:00:50,614
and it will generate
some [INAUDIBLE]?

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00:00:50,614 --> 00:00:51,280
HONG LIU: Sorry?

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00:00:51,280 --> 00:00:52,280
Say it again?

18
00:00:52,280 --> 00:00:57,780
AUDIENCE: So what's the
original thermal AdS state?

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00:00:57,780 --> 00:01:05,810
HONG LIU: The thermal
AdS state-- you couple

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00:01:05,810 --> 00:01:09,030
AdS to a thermal bath.

21
00:01:09,030 --> 00:01:14,070
And then whatever
excitation will

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00:01:14,070 --> 00:01:17,940
be generated by that thermal
bath will be generated.

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00:01:17,940 --> 00:01:20,720
So mostly it's the graviton gas.

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00:01:20,720 --> 00:01:26,760
It's the gas of the massless
particles inside AdS.

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00:01:26,760 --> 00:01:28,810
So that essentially.

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00:01:28,810 --> 00:01:31,620
The field theory procedure
to go to cleaning space

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00:01:31,620 --> 00:01:35,060
and to make the
cleaning time periodic.

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00:01:35,060 --> 00:01:39,444
And that physically should
be interpreted as you

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00:01:39,444 --> 00:01:41,360
just a coupled decisions
to your thermal bath.

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00:01:47,782 --> 00:01:51,021
AUDIENCE: I have a question
not really related to this.

31
00:01:51,021 --> 00:01:52,562
But I'm just wondering
why do we want

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00:01:52,562 --> 00:01:56,170
to consider [INAUDIBLE]
on a sphere,

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00:01:56,170 --> 00:01:59,260
because previously we
discussed about [INAUDIBLE] 4,

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00:01:59,260 --> 00:02:01,460
which is reality.

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00:02:01,460 --> 00:02:02,220
What is this?

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00:02:12,780 --> 00:02:14,840
HONG LIU: So normally when
we look at the theory,

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00:02:14,840 --> 00:02:17,110
you want to look at the
theory from as many angles as

38
00:02:17,110 --> 00:02:18,460
possible.

39
00:02:18,460 --> 00:02:22,550
So some of them may not be
able to directly realize it

40
00:02:22,550 --> 00:02:24,060
in the experiment.

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00:02:24,060 --> 00:02:27,130
But still, it's useful from
a theoretical perspective

42
00:02:27,130 --> 00:02:29,160
because that gives you
additional insights.

43
00:02:29,160 --> 00:02:31,640
So we see that the
physics on the sphere

44
00:02:31,640 --> 00:02:33,020
is actually quite rich.

45
00:02:33,020 --> 00:02:37,200
So that actually gives you
some insight into the dynamics

46
00:02:37,200 --> 00:02:39,371
and also into the
duality itself.

47
00:02:39,371 --> 00:02:39,870
Yeah.

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00:02:39,870 --> 00:02:42,070
AUDIENCE: So there's no
condensed matter system

49
00:02:42,070 --> 00:02:43,991
essentially like that.

50
00:02:43,991 --> 00:02:46,240
HONG LIU: Oh, if you're
talking about condensed matter

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00:02:46,240 --> 00:02:50,170
applications, you may even
imagine in some systems,

52
00:02:50,170 --> 00:02:52,420
you may be able to
put it on a sphere.

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00:02:52,420 --> 00:02:55,100
Certainly, you can put it
on a two-dimensional sphere.

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00:02:55,100 --> 00:03:00,610
And I can imagine you can put
it-- you can manipulate say,

55
00:03:00,610 --> 00:03:03,380
and put some spins on the
two-dimensional sphere.

56
00:03:03,380 --> 00:03:05,280
Yeah, you might be
able to do that.

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00:03:07,920 --> 00:03:11,260
Any other questions?

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00:03:11,260 --> 00:03:12,250
Yes?

59
00:03:12,250 --> 00:03:13,982
AUDIENCE: [INAUDIBLE]
you said there

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00:03:13,982 --> 00:03:20,480
are two kinds of
[INAUDIBLE] connected

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00:03:20,480 --> 00:03:21,490
on the same [INAUDIBLE].

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00:03:21,490 --> 00:03:23,127
HONG LIU: Yes.

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00:03:23,127 --> 00:03:24,460
AUDIENCE: So it's like two CFTs?

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00:03:24,460 --> 00:03:26,390
HONG LIU: No, it's a single CFT.

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00:03:26,390 --> 00:03:30,282
It's just a different sector
of the CFT contributes.

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00:03:30,282 --> 00:03:34,530
AUDIENCE: So it's like 0 quantum
critical point [INAUDIBLE]?

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00:03:34,530 --> 00:03:36,990
HONG LIU: No, this is a
first-order phase transition.

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00:03:36,990 --> 00:03:39,920
So it's not the
quantum critical point.

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00:03:39,920 --> 00:03:42,760
So the picture is that
you have a temperature.

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00:03:42,760 --> 00:03:44,330
So it's some TC.

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00:03:44,330 --> 00:03:46,860
So below TC, you
have a phase which

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00:03:46,860 --> 00:03:48,950
we call the thermal AdS phase.

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00:03:48,950 --> 00:03:51,850
And in this you have a
big black hole phase.

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00:03:51,850 --> 00:03:54,600
So translated from the
field theory side-- so this

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00:03:54,600 --> 00:04:01,500
goes running to the states which
energy scales into the power 0

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00:04:01,500 --> 00:04:03,020
contributes.

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00:04:03,020 --> 00:04:08,300
And here, it's dominated by the
state of energy O (N square).

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00:04:11,880 --> 00:04:14,160
Of course, in the
thermal ensemble,

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00:04:14,160 --> 00:04:16,370
every state in
principle contributes.

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00:04:16,370 --> 00:04:20,079
And in this phase, it's
dominated by the contribution

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00:04:20,079 --> 00:04:22,650
of the no energy state.

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00:04:22,650 --> 00:04:25,780
And here, it's dominated
by the high-energy state.

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00:04:25,780 --> 00:04:28,820
And really the high-energy state
also has a much higher entropy.

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00:04:28,820 --> 00:04:30,695
And then they dominate
your thermal ensemble.

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00:04:34,710 --> 00:04:36,954
Does this answer your question?

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00:04:36,954 --> 00:04:40,363
AUDIENCE: Yeah, this looks
like in any transition,

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00:04:40,363 --> 00:04:46,210
you have [INAUDIBLE] carry
energy and also [INAUDIBLE].

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00:04:46,210 --> 00:04:48,420
HONG LIU: Yeah, it is a
little bit similar to that.

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00:04:48,420 --> 00:04:51,230
There's also an
entropy thing there.

90
00:04:51,230 --> 00:04:53,460
Yeah, it's a balance
between two things.

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00:04:53,460 --> 00:04:55,370
But the details
is very different.

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00:04:55,370 --> 00:04:56,970
The details are different.

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00:04:56,970 --> 00:04:58,493
There's some
qualitative similarity.

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00:05:03,770 --> 00:05:08,716
AUDIENCE: So I read on
the paper [INAUDIBLE].

95
00:05:08,716 --> 00:05:13,870
He said that the Hawking-Page
transition on the field theory

96
00:05:13,870 --> 00:05:18,045
side is due to some
kind of deconfinement

97
00:05:18,045 --> 00:05:21,600
and confinement of QCD.

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00:05:21,600 --> 00:05:26,500
HONG LIU: Yeah, so that's a
heuristic way to say about it.

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00:05:26,500 --> 00:05:30,890
So the key thing is that here,
it's not really a confinement

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00:05:30,890 --> 00:05:33,780
or not confined because
in the [INAUDIBLE] series,

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00:05:33,780 --> 00:05:35,700
there's no confinement.

102
00:05:35,700 --> 00:05:38,080
It's just scaled in
[? Warren's ?] theory.

103
00:05:38,080 --> 00:05:42,230
And so in the sense he says
this is confined-- so he

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00:05:42,230 --> 00:05:45,720
called this a confined phase.

105
00:05:45,720 --> 00:05:50,540
And this is a deconfined phase--
it's based on the following.

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00:05:50,540 --> 00:05:53,720
It just refers to
these two behaviors.

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00:05:53,720 --> 00:05:55,720
It just refers to
these two behaviors.

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00:05:55,720 --> 00:06:05,290
So in QCD, which will generating
confinement or deconfinement--

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00:06:05,290 --> 00:06:08,660
so in QCD, we have
[INAUDIBLE] 3.

110
00:06:08,660 --> 00:06:11,390
But if you scale
into infinity, then

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00:06:11,390 --> 00:06:15,080
you will find in QCD
in the confined phase,

112
00:06:15,080 --> 00:06:18,610
the free energy will
scale as N with N power 0.

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00:06:18,610 --> 00:06:20,090
But in the deconfined
phase, which

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00:06:20,090 --> 00:06:23,760
is scale-- obviously
the N square.

115
00:06:23,760 --> 00:06:26,370
So that aspect of
scaling is the same

116
00:06:26,370 --> 00:06:29,446
as a serious risk confinement.

117
00:06:32,980 --> 00:06:35,150
So that aspect is similar.

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00:06:35,150 --> 00:06:39,820
So he essentially
refers to this aspect.

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00:06:39,820 --> 00:06:44,520
But on the sphere, every
state has to be a singlet.

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00:06:44,520 --> 00:06:46,770
So in some sense,
this notion there

121
00:06:46,770 --> 00:06:49,360
is no genuinely
notion of confinement

122
00:06:49,360 --> 00:06:54,364
as we say in the QCD
in the flat space case.

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00:06:54,364 --> 00:06:55,780
If you read his
paper, he actually

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00:06:55,780 --> 00:06:56,946
described this very clearly.

125
00:07:01,514 --> 00:07:03,680
Yeah, he just called this
a deconfinement transition

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00:07:03,680 --> 00:07:04,960
the heuristic way.

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00:07:04,960 --> 00:07:06,298
Yes?

128
00:07:06,298 --> 00:07:09,770
AUDIENCE: I have a question
about the sum by which we

129
00:07:09,770 --> 00:07:11,782
calculate partition function.

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00:07:11,782 --> 00:07:14,872
The previous class on
Monday, you wrote it as sum

131
00:07:14,872 --> 00:07:15,842
over energies.

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00:07:15,842 --> 00:07:19,660
The degeneracy of that
[INAUDIBLE] factor.

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00:07:19,660 --> 00:07:21,280
And you noticed
that both of these

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00:07:21,280 --> 00:07:24,940
grew at exponential to
something times N squared.

135
00:07:24,940 --> 00:07:28,096
One was positive,
and one was negative.

136
00:07:28,096 --> 00:07:30,794
I was wondering if it's
possible that sum diverges?

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00:07:30,794 --> 00:07:32,460
HONG LIU: No, that
sum does not diverge.

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00:07:32,460 --> 00:07:34,360
That sum is always
proportional to n square.

139
00:07:37,190 --> 00:07:39,690
Yeah, depending on what
you mean the sum diverges.

140
00:07:39,690 --> 00:07:45,320
So that's just from the
definition of your partition

141
00:07:45,320 --> 00:07:48,090
function.

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00:07:48,090 --> 00:07:50,810
So it's just the sum
of all possible states.

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00:07:50,810 --> 00:07:53,584
And then you can just
reach this-- yeah, when

144
00:07:53,584 --> 00:07:55,000
you have a large
number of states,

145
00:07:55,000 --> 00:08:00,700
you can just roughly write it as
a continuous integral and then

146
00:08:00,700 --> 00:08:02,470
with the density of states.

147
00:08:06,420 --> 00:08:09,860
And so the key is
that what happens

148
00:08:09,860 --> 00:08:12,250
to the density of states--
so you're asking maybe

149
00:08:12,250 --> 00:08:14,910
whether this
integral will diverge

150
00:08:14,910 --> 00:08:16,410
when it equals infinity.

151
00:08:16,410 --> 00:08:17,910
So when it equals
infinity, you will

152
00:08:17,910 --> 00:08:24,829
see that say supposing
E scale as N cubed.

153
00:08:24,829 --> 00:08:26,620
Then you will see that
the density of state

154
00:08:26,620 --> 00:08:30,300
actually does not
grow that fast.

155
00:08:30,300 --> 00:08:36,900
And actually, it's only
for the E of O(N squared),

156
00:08:36,900 --> 00:08:39,164
then they scale in the same way.

157
00:08:39,164 --> 00:08:40,289
They scale in the same way.

158
00:08:40,289 --> 00:08:42,320
So that's why you
have this balance.

159
00:08:42,320 --> 00:08:45,200
And if you are not in the
scaling with [INAUDIBLE].

160
00:08:45,200 --> 00:08:49,167
If you have N cubed, and
then this will dominate.

161
00:08:49,167 --> 00:08:50,583
So that factor
will be suppressed.

162
00:08:56,940 --> 00:08:57,773
Any other questions?

163
00:09:02,590 --> 00:09:03,450
OK, very good.

164
00:09:03,450 --> 00:09:05,440
So now let's go to entanglement.

165
00:09:40,320 --> 00:09:42,980
So first, let me say a few
words about entanglement entropy

166
00:09:42,980 --> 00:09:43,480
itself.

167
00:09:48,750 --> 00:09:51,620
So this is very
elementary stuff,

168
00:09:51,620 --> 00:09:58,560
even though maybe
not everybody knows.

169
00:09:58,560 --> 00:10:01,397
So let's consider a quantum
system, just a general quantum

170
00:10:01,397 --> 00:10:01,897
system.

171
00:10:06,434 --> 00:10:08,100
So let's separate the
degrees of freedom

172
00:10:08,100 --> 00:10:13,607
into two parts, say A
plus B. So AB together

173
00:10:13,607 --> 00:10:14,440
is the whole system.

174
00:10:14,440 --> 00:10:18,052
We just separated the
degrees of freedom.

175
00:10:18,052 --> 00:10:19,760
So essentially by
definition, the Hilbert

176
00:10:19,760 --> 00:10:28,030
space of the system-- then
we'll have a tensor structure.

177
00:10:28,030 --> 00:10:31,350
So the full Hilbert space
will be a tensor product

178
00:10:31,350 --> 00:10:33,990
over the Hilbert
space for the A part

179
00:10:33,990 --> 00:10:37,057
and tensored with the
Hilbert space in the B part.

180
00:10:37,057 --> 00:10:37,556
OK?

181
00:10:47,030 --> 00:10:49,220
And consider, of
course, in general, we

182
00:10:49,220 --> 00:10:51,250
have an infinite
dimensional system.

183
00:10:51,250 --> 00:10:54,150
But in the case if you have
say a [INAUDIBLE] Hilbert

184
00:10:54,150 --> 00:10:57,290
space-- so i f this M
dimension, this is N dimension,

185
00:10:57,290 --> 00:10:59,835
then the total Hilbert space
will be M times N dimension.

186
00:11:02,558 --> 00:11:13,520
And the typical wave function
will have the form-- will be

187
00:11:13,520 --> 00:11:19,880
some sum-- say you can write
it in some basis in A and times

188
00:11:19,880 --> 00:11:23,730
some wave function
writing some bases in B.

189
00:11:23,730 --> 00:11:27,020
And this actually means that
the Hilbert space is a tensor

190
00:11:27,020 --> 00:11:27,550
product.

191
00:11:27,550 --> 00:11:29,800
It's just your wave form can
typically have this form.

192
00:11:35,920 --> 00:11:47,570
So we say that AB-- so we say
that in the states psi that AB

193
00:11:47,570 --> 00:12:11,830
are entangled if psi cannot be
written as a single product--

194
00:12:11,830 --> 00:12:15,920
rather than the sum of product
and if you just have a single

195
00:12:15,920 --> 00:12:16,420
product.

196
00:12:22,480 --> 00:12:25,980
So for a simple state,
it might be easy to see.

197
00:12:25,980 --> 00:12:27,960
But if I write the very
complicated state which

198
00:12:27,960 --> 00:12:33,100
many-- in principle,
write the state

199
00:12:33,100 --> 00:12:36,070
in some bases which look
at the complicated sum.

200
00:12:36,070 --> 00:12:39,160
But they may be some other
bases and be a simple product.

201
00:12:39,160 --> 00:12:41,140
So in general, it's
actually hard to tell.

202
00:12:41,140 --> 00:12:43,640
In general, it's hard to tell,
even though it's easy to say.

203
00:12:43,640 --> 00:12:46,580
But in general,
it's hard to say.

204
00:12:46,580 --> 00:12:51,990
So the entangled
entropy-- so let me just

205
00:12:51,990 --> 00:12:54,870
call it EE just to save space.

206
00:12:54,870 --> 00:13:02,990
EE essentially
provides a measure

207
00:13:02,990 --> 00:13:23,210
to quantify the entanglement
between A and B--

208
00:13:23,210 --> 00:13:25,940
because even in the case
which you know this state is

209
00:13:25,940 --> 00:13:28,190
in entangled state,
you may still

210
00:13:28,190 --> 00:13:30,305
want to ask how much
they are entangled.

211
00:13:34,910 --> 00:13:38,055
Entanglement entropy provides
the way to quantify it.

212
00:13:41,600 --> 00:13:46,950
So the definitions
are very simple.

213
00:13:46,950 --> 00:13:50,280
So first we look at
this the density matrix

214
00:13:50,280 --> 00:13:53,360
for the total system.

215
00:13:53,360 --> 00:13:55,910
So if the system is
in the state psi,

216
00:13:55,910 --> 00:13:58,110
then the density matrix
for the total system

217
00:13:58,110 --> 00:14:02,115
would be just on the psi
conjugate-- psi itself.

218
00:14:04,650 --> 00:14:06,640
So in this basic matrix,
we just trace out

219
00:14:06,640 --> 00:14:13,640
all the degrees of freedom in
B. So since the Hilbert space

220
00:14:13,640 --> 00:14:17,090
have a tensor product structure,
you can always do this.

221
00:14:17,090 --> 00:14:19,470
Then what you get is you
get this the density matrix.

222
00:14:19,470 --> 00:14:23,440
And then here, you
get a density matrix

223
00:14:23,440 --> 00:14:28,037
which only depends on the
degree of freedom in A.

224
00:14:28,037 --> 00:14:30,120
And then we just calculate
the Von Neumann entropy

225
00:14:30,120 --> 00:14:33,025
corresponding to this row
A. So entangled entropy

226
00:14:33,025 --> 00:14:41,460
is just defined to be
[INAUDIBLE] entropy associates

227
00:14:41,460 --> 00:14:45,169
resistance in the matrix
rho A. So this rho A

228
00:14:45,169 --> 00:14:46,960
is often called the
reduced density matrix.

229
00:14:58,990 --> 00:15:01,490
And the Von Neumann entropy
associated with the reduced

230
00:15:01,490 --> 00:15:05,642
density matrix is defined
as the entanglement entropy.

231
00:15:05,642 --> 00:15:06,142
OK?

232
00:15:10,390 --> 00:15:13,010
So this provides a
very good the measure

233
00:15:13,010 --> 00:15:26,725
because say if SA is equal to
0, from our knowledge of the Von

234
00:15:26,725 --> 00:15:30,340
Neumann entropy,
you can immediately

235
00:15:30,340 --> 00:15:37,620
deduce where the SA is equal
to 0 if only if the rhos A,

236
00:15:37,620 --> 00:15:40,030
this density matrix
is a pure state.

237
00:15:45,590 --> 00:15:46,472
It's a pure state.

238
00:15:51,500 --> 00:15:56,482
And then from here, you can also
deduce rho A is a pure state.

239
00:15:56,482 --> 00:15:58,690
This reduced density matrix
comes from this procedure

240
00:15:58,690 --> 00:16:00,060
as a pure state.

241
00:16:00,060 --> 00:16:08,470
Only if the psi can be
written as a simple product.

242
00:16:22,900 --> 00:16:31,190
So this tells you as
when SA is non-zero,

243
00:16:31,190 --> 00:16:36,496
then you can be sure this
state must be non-entangled.

244
00:16:40,760 --> 00:16:44,740
So whenever SA is
equal to 0, you

245
00:16:44,740 --> 00:16:46,460
can be sure this
is non-entangled.

246
00:16:46,460 --> 00:16:49,880
When SA is not
equal to 0, you know

247
00:16:49,880 --> 00:16:56,800
for sure A and B must be
entangled in this state.

248
00:16:56,800 --> 00:16:58,540
So that's why this
is a good measure.

249
00:16:58,540 --> 00:16:59,800
OK?

250
00:16:59,800 --> 00:17:02,990
And then the value of
SA then will tell you

251
00:17:02,990 --> 00:17:04,736
how entangled the system is.

252
00:17:12,390 --> 00:17:15,200
And also, you may
immediately ask

253
00:17:15,200 --> 00:17:18,500
the question what happens
instead of defining the rho A?

254
00:17:18,500 --> 00:17:20,630
Of course, you can
also do the same thing.

255
00:17:20,630 --> 00:17:25,154
You trace out A to define
out rho B and then the

256
00:17:25,154 --> 00:17:27,750
define the entropy
for the rho B. OK?

257
00:17:27,750 --> 00:17:30,860
Define entropy for the
rho B. But you can easily

258
00:17:30,860 --> 00:17:42,970
show for any pure state
psi SA always equals to SB.

259
00:17:42,970 --> 00:17:45,530
So it doesn't matter
which are symmetric.

260
00:17:45,530 --> 00:17:49,840
So it doesn't matter which
one you're looking at.

261
00:17:53,040 --> 00:17:58,480
So this is very easy
to prove, essentially

262
00:17:58,480 --> 00:18:02,750
just following from something
called a Schmidt decomposition.

263
00:18:02,750 --> 00:18:06,520
And essentially right to this
state in terms of Schmidt

264
00:18:06,520 --> 00:18:09,560
decomposition between the
degrees of freedom rho A and B.

265
00:18:09,560 --> 00:18:12,710
And then you can show that
rho A and rho B essentially

266
00:18:12,710 --> 00:18:14,690
have the same eigenvalues.

267
00:18:14,690 --> 00:18:17,780
And if rho A and rho B have the
same eigenvalues, of course,

268
00:18:17,780 --> 00:18:20,440
then the entropy will be the
same, because the entropy only

269
00:18:20,440 --> 00:18:21,700
depends on the eigenvalues.

270
00:18:21,700 --> 00:18:22,200
OK?

271
00:18:24,710 --> 00:18:25,825
Any questions about this?

272
00:18:31,010 --> 00:18:32,350
Good.

273
00:18:32,350 --> 00:18:36,860
So here is what a mixed
state-- about a pure state,

274
00:18:36,860 --> 00:18:43,410
let me just say a side remark.

275
00:18:43,410 --> 00:18:56,510
So if AB, the total system is
in a mixed state-- so far always

276
00:18:56,510 --> 00:18:57,830
is in the pure state.

277
00:18:57,830 --> 00:18:59,840
But suppose it's
in the mixed state.

278
00:18:59,840 --> 00:19:03,960
Suppose the system itself is
described by a density matrix.

279
00:19:03,960 --> 00:19:08,130
So the total system
itself is spread.

280
00:19:08,130 --> 00:19:10,330
In general, SA is
not equal to SB.

281
00:19:12,970 --> 00:19:14,510
OK?

282
00:19:14,510 --> 00:19:19,440
Not only in general,
it's not equal to-- SA

283
00:19:19,440 --> 00:19:20,500
is not equal to SB.

284
00:19:23,960 --> 00:19:36,570
And in such a case, the
entanglement entropy

285
00:19:36,570 --> 00:19:55,750
also contains classical
statistical correlations

286
00:19:55,750 --> 00:20:08,010
of the mixed state--
of the mixed state--

287
00:20:08,010 --> 00:20:10,070
so in addition to
quantum correlations.

288
00:20:13,250 --> 00:20:17,950
It's almost trivial to see
because the people suppose

289
00:20:17,950 --> 00:20:19,180
here is not a pure state.

290
00:20:19,180 --> 00:20:23,690
So here you replace it
by the density matrix.

291
00:20:23,690 --> 00:20:27,500
And when you trace all the B
in this density matrix, then,

292
00:20:27,500 --> 00:20:31,280
of course, there's still
some original uncertainty

293
00:20:31,280 --> 00:20:34,200
in the previous density
matrix remaining in A.

294
00:20:34,200 --> 00:20:41,770
And then they will come into
this entropy-- so as defined,

295
00:20:41,770 --> 00:20:45,370
will depend on the
classical uncertainties--

296
00:20:45,370 --> 00:20:47,300
classical statistical
uncertainties

297
00:20:47,300 --> 00:20:51,520
of your original density matrix.

298
00:20:51,520 --> 00:20:55,440
So for a mixed state-- so
that's why for a mixed state,

299
00:20:55,440 --> 00:20:59,840
the internal entropy is not a
very good measure of quantum

300
00:20:59,840 --> 00:21:03,860
entanglements, because
it's contaminated

301
00:21:03,860 --> 00:21:07,240
by classical
statistical information.

302
00:21:10,350 --> 00:21:12,060
But for today, we
all need to talk

303
00:21:12,060 --> 00:21:14,820
about the pure-- mostly
talk about the pure.

304
00:21:14,820 --> 00:21:19,190
Yeah, we actually
talk about in general.

305
00:21:19,190 --> 00:21:23,100
But I want you to
keep this in mind.

306
00:21:23,100 --> 00:21:27,625
So any questions so far?

307
00:21:27,625 --> 00:21:29,750
So just to give you an a
little bit more intuition,

308
00:21:29,750 --> 00:21:34,949
let's look at this
very simple example

309
00:21:34,949 --> 00:21:36,240
to calculate entangled entropy.

310
00:21:36,240 --> 00:21:37,740
So let's consider
a two-spin system.

311
00:21:42,310 --> 00:21:45,840
So let's consider you
have two spins, OK?

312
00:21:45,840 --> 00:21:51,430
So this is my A and B. So this
has a two-dimensional Hilbert

313
00:21:51,430 --> 00:21:51,930
space.

314
00:21:51,930 --> 00:21:53,600
This has a two-dimensional
Hilbert space.

315
00:21:53,600 --> 00:21:55,849
Altogether, you have a
four-dimensional Hilbert space.

316
00:21:58,570 --> 00:22:02,520
So for example, so let
me consider such a state.

317
00:22:18,310 --> 00:22:21,780
So this looks like
a complicated state.

318
00:22:21,780 --> 00:22:25,190
But actually, this can be
written as a single product,

319
00:22:25,190 --> 00:22:41,670
because you it can
be written at OK,

320
00:22:41,670 --> 00:22:43,190
I hope this notation is for me.

321
00:22:43,190 --> 00:22:44,510
It's OK with you.

322
00:22:44,510 --> 00:22:50,300
You just A and B--
A spin and B spin.

323
00:22:50,300 --> 00:22:54,375
So even though in this
space, this is written as

324
00:22:54,375 --> 00:22:55,480
say a sum of state.

325
00:22:55,480 --> 00:22:57,230
It looks the entangleds.

326
00:22:57,230 --> 00:23:01,340
But in fact, it's not
because you can write it

327
00:23:01,340 --> 00:23:04,230
in terms of a simple product.

328
00:23:04,230 --> 00:23:05,970
So this is another
entangled state.

329
00:23:13,350 --> 00:23:16,100
Of course, for the simple
system, it's very easy to tell.

330
00:23:16,100 --> 00:23:18,430
But to give you a
complicated system

331
00:23:18,430 --> 00:23:20,630
with many, many,
degrees of freedom,

332
00:23:20,630 --> 00:23:22,595
then it becomes very hard.

333
00:23:22,595 --> 00:23:24,865
And then entangled
entropy becomes useful.

334
00:23:28,026 --> 00:23:30,400
So now let me give you an
example to calculated entangled

335
00:23:30,400 --> 00:23:31,150
entropy.

336
00:23:31,150 --> 00:23:36,591
So let's consider
a state like this.

337
00:23:45,970 --> 00:23:49,560
So let's see some parameter.

338
00:23:49,560 --> 00:23:53,920
So clearly this state is
entangled because you cannot

339
00:23:53,920 --> 00:23:57,280
write it as a simple product.

340
00:23:57,280 --> 00:23:59,680
So now, let's check it.

341
00:23:59,680 --> 00:24:01,390
Now, let's check it.

342
00:24:01,390 --> 00:24:08,630
So you can look at the full
density matrix of the system.

343
00:24:08,630 --> 00:24:13,850
You just look at this--
the bra and the ket itself.

344
00:24:13,850 --> 00:24:17,560
And then just do the product.

345
00:24:17,560 --> 00:24:34,300
And you get the quotient theta--
under the [INAUDIBLE] terms.

346
00:25:01,900 --> 00:25:07,990
So you just take the
product with itself.

347
00:25:07,990 --> 00:25:11,520
And then let's try
to find what's rho A.

348
00:25:11,520 --> 00:25:13,650
So you trace out
degrees of freedom of B.

349
00:25:13,650 --> 00:25:17,340
So this is our B,
the second spin.

350
00:25:17,340 --> 00:25:19,420
And now let's trace out
degrees of freedom B--

351
00:25:19,420 --> 00:25:26,690
trace out the second spin--
so rho A. Goes one into we

352
00:25:26,690 --> 00:25:29,580
trace out the B in here.

353
00:25:29,580 --> 00:25:32,480
So when you trace
out the B, you just

354
00:25:32,480 --> 00:25:35,030
take the end product
between these two.

355
00:25:35,030 --> 00:25:36,980
So these are the same.

356
00:25:36,980 --> 00:25:39,940
So this will remain.

357
00:25:39,940 --> 00:25:41,351
And so you have this.

358
00:25:44,510 --> 00:25:47,410
And similarly, this
one you have that.

359
00:25:47,410 --> 00:25:48,490
These two are the same.

360
00:25:48,490 --> 00:25:50,323
So when you take the
trace, this is nonzero.

361
00:25:58,030 --> 00:26:00,160
But this too will give
you 0, because this one is

362
00:26:00,160 --> 00:26:01,650
orthogonal to this one.

363
00:26:01,650 --> 00:26:03,677
And this spin is
orthogonal to that spin.

364
00:26:03,677 --> 00:26:04,635
So that's what you get.

365
00:26:07,020 --> 00:26:09,395
And then now you can usually
just write down the entropy.

366
00:26:30,890 --> 00:26:33,910
So this is the entangled
entropy for these two states--

367
00:26:33,910 --> 00:26:37,220
for this two spin system
as a general function

368
00:26:37,220 --> 00:26:40,560
of this parameter theta.

369
00:26:40,560 --> 00:26:43,882
So now let's plot these
as a function of S theta.

370
00:26:47,120 --> 00:26:49,320
So clearly, this is
a period function.

371
00:26:49,320 --> 00:26:51,177
We only need to go to pi over 2.

372
00:26:53,920 --> 00:26:57,187
And then you have
something like this.

373
00:26:57,187 --> 00:26:58,770
And you can easily
plot that function.

374
00:26:58,770 --> 00:27:00,230
You will see
something like this.

375
00:27:00,230 --> 00:27:02,580
So when theta is equal to 0.

376
00:27:02,580 --> 00:27:06,220
This is equal to 0
because this term is 0.

377
00:27:06,220 --> 00:27:08,510
This term is also 0 because
of the quotient theta

378
00:27:08,510 --> 00:27:10,770
square is equal to 1.

379
00:27:10,770 --> 00:27:13,710
When theta equals to
pi over 2, this term

380
00:27:13,710 --> 00:27:15,710
becomes 0 and this
term also becomes,

381
00:27:15,710 --> 00:27:18,670
because it's [INAUDIBLE] 0.

382
00:27:18,670 --> 00:27:24,300
But this maximum in the
pi over 2 or pi over 4.

383
00:27:24,300 --> 00:27:25,590
So I had a pi over 4.

384
00:27:46,910 --> 00:27:50,310
Of course, you can also go
to the pi equals to minus 4.

385
00:27:50,310 --> 00:27:53,409
OK, you can go do plus minus 4.

386
00:27:53,409 --> 00:27:54,700
I didn't go to the [INAUDIBLE].

387
00:27:54,700 --> 00:27:57,936
Yeah, anyway, so these are
called the maximum entangled

388
00:27:57,936 --> 00:27:58,435
states.

389
00:28:07,690 --> 00:28:10,299
This is a maximum entangled.

390
00:28:10,299 --> 00:28:12,673
So this is a state in which
you have the highest entropy.

391
00:28:19,860 --> 00:28:20,960
Any questions so far?

392
00:28:25,164 --> 00:28:26,613
AUDIENCE: You switched it.

393
00:28:26,613 --> 00:28:27,579
It should be up down.

394
00:28:27,579 --> 00:28:29,520
It's down.

395
00:28:29,520 --> 00:28:31,130
HONG LIU: Right, that's right.

396
00:28:34,490 --> 00:28:35,150
OK, good.

397
00:28:41,120 --> 00:28:44,710
So let me also say a few
things about the properties

398
00:28:44,710 --> 00:28:47,420
of the entanglement entropy.

399
00:28:47,420 --> 00:28:51,090
So there are many properties
you can derive from here.

400
00:28:51,090 --> 00:28:56,436
Let me only say a
few important ones.

401
00:29:08,850 --> 00:29:16,790
OK, so one property of
the entangled entropy

402
00:29:16,790 --> 00:29:18,670
is called subadditivity
condition.

403
00:29:26,250 --> 00:29:30,080
So if you can see that
the two systems A and B,

404
00:29:30,080 --> 00:29:33,550
then you can show that
S(AB), the entropy

405
00:29:33,550 --> 00:29:37,590
for the total system,
is smaller than the sum

406
00:29:37,590 --> 00:29:38,633
of the separate system.

407
00:29:41,390 --> 00:29:44,370
So when I write AB, I
mean the combined system.

408
00:29:44,370 --> 00:29:45,760
OK?

409
00:29:45,760 --> 00:29:49,338
And this is greater than
the difference between them.

410
00:29:54,750 --> 00:29:57,640
So this is so-called the
subadditive condition.

411
00:29:57,640 --> 00:30:02,560
And then intuitively,
you can understand.

412
00:30:02,560 --> 00:30:05,230
So if you have entropy
of A and entropy of B,

413
00:30:05,230 --> 00:30:09,630
you add them together, then it's
larger than the entropy of AB

414
00:30:09,630 --> 00:30:11,110
because there is
some redundancy.

415
00:30:11,110 --> 00:30:14,380
There might be some
redundancy here.

416
00:30:14,380 --> 00:30:17,280
Yeah, because when
you add AB together,

417
00:30:17,280 --> 00:30:19,740
there may be some common
correlation between them.

418
00:30:19,740 --> 00:30:22,000
And so this is
greater than that.

419
00:30:22,000 --> 00:30:24,321
OK, intuitively that's
what this inequality means.

420
00:30:28,920 --> 00:30:31,575
And also there are some property
for the strong subadditivity

421
00:30:31,575 --> 00:30:32,075
condition.

422
00:30:39,161 --> 00:30:40,661
AUDIENCE: I don't
quite understand--

423
00:30:43,540 --> 00:30:45,424
HONG LIU: Yes?

424
00:30:45,424 --> 00:30:48,360
AUDIENCE: What S of
A and S of B means?

425
00:30:48,360 --> 00:30:52,510
HONG LIU: So this is
the entropy for the A.

426
00:30:52,510 --> 00:30:55,290
This is the entropy for the B.

427
00:30:55,290 --> 00:30:57,660
AUDIENCE: Don't we
need this to partition

428
00:30:57,660 --> 00:31:00,270
system A into two subsystems?

429
00:31:00,270 --> 00:31:04,310
HONG LIU: No, you don't--
let me explain my notation.

430
00:31:04,310 --> 00:31:10,770
So S(A) means the
entropy equals 1

431
00:31:10,770 --> 00:31:14,030
if you integrate out
everything else except A.

432
00:31:14,030 --> 00:31:15,930
And S(B) means you
integrate everything

433
00:31:15,930 --> 00:31:19,000
else except B.
And S(AB) be means

434
00:31:19,000 --> 00:31:22,820
to integrate everything
else except A and B.

435
00:31:22,820 --> 00:31:25,760
And this is S(AB).

436
00:31:25,760 --> 00:31:29,694
AUDIENCE: But in our case, A
and B is all that there is.

437
00:31:29,694 --> 00:31:30,630
HONG LIU: No, no, no.

438
00:31:30,630 --> 00:31:33,582
Now I'm just doing generally.

439
00:31:33,582 --> 00:31:40,960
Once I have this definition,
so this can be-- even

440
00:31:40,960 --> 00:31:46,070
AB is a total-- yeah, so this
can apply both to the case

441
00:31:46,070 --> 00:31:49,070
I said earlier-- say
if you divide-- here I

442
00:31:49,070 --> 00:31:53,090
just-- so this AB does not
have to be the same as that AB.

443
00:31:53,090 --> 00:31:56,810
Here I'm just talking
about only two subsystems.

444
00:32:01,310 --> 00:32:04,290
AUDIENCE: So it's
for example, S(A)

445
00:32:04,290 --> 00:32:09,630
is like-- A and the supplement
as two parts of the system.

446
00:32:09,630 --> 00:32:11,310
And they could have
some entanglement.

447
00:32:11,310 --> 00:32:13,500
S(A) is just an
entanglement field.

448
00:32:13,500 --> 00:32:16,330
HONG LIU: A, S(A)
is the entanglement

449
00:32:16,330 --> 00:32:18,850
of A with the rest.

450
00:32:18,850 --> 00:32:24,940
And S(B) is the entanglement
of B between B and the rest.

451
00:32:24,940 --> 00:32:28,540
And the S(AB) is
entangled between the AB

452
00:32:28,540 --> 00:32:32,200
together with the rest.

453
00:32:32,200 --> 00:32:32,930
Yes.

454
00:32:32,930 --> 00:32:35,290
AUDIENCE: So to clarify one
thing, when you say S of A,

455
00:32:35,290 --> 00:32:36,430
we have this whole system.

456
00:32:36,430 --> 00:32:38,380
It means that trace
everything which is not A?

457
00:32:38,380 --> 00:32:39,936
HONG LIU: Yeah, that's right.

458
00:32:47,630 --> 00:32:51,090
Any more questions about this?

459
00:32:51,090 --> 00:32:53,580
So here when I write
to those expressions,

460
00:32:53,580 --> 00:32:57,902
I assume A and B don't
have interceptions, OK?

461
00:32:57,902 --> 00:32:59,652
I assume A and B don't
have interceptions.

462
00:33:04,000 --> 00:33:06,860
You can also have the strong
subbadditivity condition.

463
00:33:06,860 --> 00:33:09,595
So this is pretty easy to prove.

464
00:33:13,020 --> 00:33:17,010
If we have time, it takes
five minutes to prove.

465
00:33:17,010 --> 00:33:21,750
But the strong subadditivity
which I'm going to write down.

466
00:33:21,750 --> 00:33:26,320
So strong additivity is
involving three systems.

467
00:33:26,320 --> 00:33:28,790
Now, you have to add
the three systems.

468
00:33:28,790 --> 00:33:48,177
And then greater than ABC-- OK?

469
00:34:00,550 --> 00:34:04,050
So, again, intuitively, the
meaning of that inequality

470
00:34:04,050 --> 00:34:07,130
is clear.

471
00:34:07,130 --> 00:34:11,070
So the first
inequality just says

472
00:34:11,070 --> 00:34:12,720
that I have two systems here.

473
00:34:16,090 --> 00:34:20,524
And the sum of
these two systems--

474
00:34:20,524 --> 00:34:21,940
the entropy of
these two systems--

475
00:34:21,940 --> 00:34:27,320
is greater than the sum
between the combined

476
00:34:27,320 --> 00:34:30,610
system and the intersection
between the two systems,

477
00:34:30,610 --> 00:34:33,360
OK, because C in the
section between the two.

478
00:34:33,360 --> 00:34:35,640
And ABC is the whole
thing combined.

479
00:34:38,239 --> 00:34:42,679
And similarly here, he said
they have S(A) and S(B).

480
00:34:42,679 --> 00:34:45,600
I have two A and B. So
I trace out the system.

481
00:34:45,600 --> 00:34:49,960
Outside A, I get entropy for A.
I trace the system outside B,

482
00:34:49,960 --> 00:34:57,990
anyway for B. This
inequality says if now you

483
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.

484
00:35:04,740 --> 00:35:07,990
And then the resulting system
will be larger than the B form.

485
00:35:07,990 --> 00:35:11,130
So if you add something,
we increase the entropy.

486
00:35:11,130 --> 00:35:12,830
That, of course, is
intuitively clear,

487
00:35:12,830 --> 00:35:19,240
because if you have entropy
essentially parameterized

488
00:35:19,240 --> 00:35:21,190
the unknown part of the system.

489
00:35:21,190 --> 00:35:24,960
And if you add the third
system, and then this

490
00:35:24,960 --> 00:35:29,460
just increase your unknown and
then increase your entropy.

491
00:35:33,300 --> 00:35:37,590
So this strong
subadditivity condition

492
00:35:37,590 --> 00:35:39,980
is actually very hard to prove.

493
00:35:39,980 --> 00:35:41,265
It's very hard to prove.

494
00:35:43,910 --> 00:35:52,490
It can get rather mathematical
and requires some effort

495
00:35:52,490 --> 00:35:53,850
to do it.

496
00:35:53,850 --> 00:35:56,110
But this one is
pretty easy to do.

497
00:35:56,110 --> 00:35:56,610
Yes?

498
00:35:56,610 --> 00:35:59,885
AUDIENCE: Can [INAUDIBLE] about
the first of the two equations?

499
00:35:59,885 --> 00:36:02,420
For strong subadditivity,
what's the first equation?

500
00:36:02,420 --> 00:36:05,670
HONG LIU: Yeah,
so this means-- so

501
00:36:05,670 --> 00:36:08,490
look at this as a single system.

502
00:36:08,490 --> 00:36:10,260
This is a single system.

503
00:36:10,260 --> 00:36:12,950
That means that the sum of these
two-- the entropy of this two

504
00:36:12,950 --> 00:36:17,390
system is greater than
the sum of the combined

505
00:36:17,390 --> 00:36:19,815
system under the intersection.

506
00:36:28,960 --> 00:36:30,434
Any questions about this?

507
00:36:33,006 --> 00:36:33,506
Good.

508
00:36:41,040 --> 00:36:43,200
So this concludes the
very simple introduction

509
00:36:43,200 --> 00:36:44,680
to the entangled entropy.

510
00:36:44,680 --> 00:36:46,054
AUDIENCE: Does
the first one have

511
00:36:46,054 --> 00:36:50,664
any implications of topology?

512
00:36:50,664 --> 00:36:52,580
HONG LIU: Sorry, what
do you mean by topology?

513
00:36:52,580 --> 00:37:00,842
AUDIENCE: Because it's like the
intersection of the combined

514
00:37:00,842 --> 00:37:03,720
thing.

515
00:37:03,720 --> 00:37:08,400
HONG LIU: Yeah, so
as classical entropy,

516
00:37:08,400 --> 00:37:10,640
this is a very simple thing.

517
00:37:10,640 --> 00:37:12,540
You can easily convince
yourself if ABC

518
00:37:12,540 --> 00:37:16,620
are classical systems, and
just classical distributions,

519
00:37:16,620 --> 00:37:19,220
describe classical
probabilities,

520
00:37:19,220 --> 00:37:21,200
statistical distributions.

521
00:37:21,200 --> 00:37:24,590
And then with those things, you
can understand this inequality

522
00:37:24,590 --> 00:37:27,620
just by drawing this kind of
standard diagram corresponding

523
00:37:27,620 --> 00:37:28,659
to the different sets.

524
00:37:28,659 --> 00:37:30,950
The quantum mechanics proving
those things are actually

525
00:37:30,950 --> 00:37:33,500
not trivial are not trivial.

526
00:37:33,500 --> 00:37:35,430
Quantum mechanically,
they no longer come in

527
00:37:35,430 --> 00:37:37,380
a very intuitive way.

528
00:37:37,380 --> 00:37:38,634
Yes?

529
00:37:38,634 --> 00:37:40,540
AUDIENCE: I was noticing
something curious

530
00:37:40,540 --> 00:37:44,194
about the last line classically.

531
00:37:44,194 --> 00:37:51,120
If A and B are the same, I think
the equation will still hold.

532
00:37:51,120 --> 00:37:53,619
But quantumly--

533
00:37:53,619 --> 00:37:54,660
HONG LIU: It still holds.

534
00:37:54,660 --> 00:37:58,180
AUDIENCE: Doesn't it
not hold quantumly?

535
00:37:58,180 --> 00:38:01,190
HONG LIU: No, they are supposed
to hold quantum mechanically.

536
00:38:01,190 --> 00:38:05,504
AUDIENCE: But we are under the
assumption that A, B, and C--

537
00:38:05,504 --> 00:38:07,900
they don't overlap, right?

538
00:38:07,900 --> 00:38:10,790
I'm saying if A and B
are totally overlapping.

539
00:38:10,790 --> 00:38:14,990
HONG LIU: No, but
this inequality

540
00:38:14,990 --> 00:38:17,850
I'm writing in a way which
they are not intersecting.

541
00:38:17,850 --> 00:38:20,190
You can write them in a
way which they intersect.

542
00:38:20,190 --> 00:38:22,620
And I'm just
writing in this way.

543
00:38:22,620 --> 00:38:26,190
So the condition for
this particular form ABC

544
00:38:26,190 --> 00:38:29,222
is not supposed to intersect.

545
00:38:29,222 --> 00:38:33,870
AUDIENCE: Right, I'm saying
if we set A equals to B.

546
00:38:33,870 --> 00:38:37,740
HONG LIU: No, no, then you
cannot use this equation.

547
00:38:37,740 --> 00:38:39,215
Then you have to
write the equation

548
00:38:39,215 --> 00:38:42,560
in somewhat different way, which
apply to the intersection case.

549
00:38:42,560 --> 00:38:46,550
You can do them just choosing
to write in the way which

550
00:38:46,550 --> 00:38:47,936
did only intersect.

551
00:38:47,936 --> 00:38:50,580
AUDIENCE: OK, but the
thing I want to say

552
00:38:50,580 --> 00:38:53,130
is that we can still keep the
bottom equation classically,

553
00:38:53,130 --> 00:38:54,649
even if we took--

554
00:38:54,649 --> 00:38:56,440
HONG LIU: No, both are
applied classically.

555
00:38:56,440 --> 00:38:58,850
Both are trivial classically.

556
00:38:58,850 --> 00:39:00,930
You can understand very
easily classically.

557
00:39:00,930 --> 00:39:04,260
It's just quantum mechanically,
it's no longer trivial.

558
00:39:04,260 --> 00:39:06,410
Quantum mechanics is
no longer trivial.

559
00:39:06,410 --> 00:39:09,900
And there are various ways
you can write those equations.

560
00:39:09,900 --> 00:39:13,750
I'm just choosing to write the
way which ABC don't intersect.

561
00:39:13,750 --> 00:39:16,344
And you can also write
the way, just rename them

562
00:39:16,344 --> 00:39:18,099
so that they intersect.

563
00:39:18,099 --> 00:39:20,494
AUDIENCE: So [INAUDIBLE].

564
00:39:20,494 --> 00:39:24,280
So this is true for any
entanglement state we choose?

565
00:39:24,280 --> 00:39:26,667
HONG LIU: Any state,
yeah, density matrix.

566
00:39:26,667 --> 00:39:28,500
Whether it's a pure
state or density matrix,

567
00:39:28,500 --> 00:39:34,490
it doesn't matter--
a general statement.

568
00:39:34,490 --> 00:39:36,180
OK, good.

569
00:39:36,180 --> 00:39:39,550
So this concludes the
short introduction

570
00:39:39,550 --> 00:39:43,630
to the entangled entropy itself.

571
00:39:43,630 --> 00:39:47,450
And you may know that the
entanglement entropy actually

572
00:39:47,450 --> 00:39:48,810
plays a very important role.

573
00:39:48,810 --> 00:39:50,791
Entanglement and
entanglement entropy

574
00:39:50,791 --> 00:39:53,165
itself plays a very important
role in quantum information

575
00:39:53,165 --> 00:39:57,390
and quantum computing because
of quantum entanglement

576
00:39:57,390 --> 00:39:59,750
is the kind of
quantum correlations

577
00:39:59,750 --> 00:40:03,680
which you don't
have classically.

578
00:40:03,680 --> 00:40:07,464
So this EPR paradox and
the [INAUDIBLE] inequality,

579
00:40:07,464 --> 00:40:09,130
and the teleportation--
all those things

580
00:40:09,130 --> 00:40:12,895
that take advantage of.

581
00:40:13,870 --> 00:40:15,940
But that's not our
main point here.

582
00:40:15,940 --> 00:40:18,170
So what I'm going
to talk about next

583
00:40:18,170 --> 00:40:20,560
is actually entangled
entropy is also

584
00:40:20,560 --> 00:40:22,810
starting playing a
very important role

585
00:40:22,810 --> 00:40:27,640
in our understanding of
many-body physics and the space

586
00:40:27,640 --> 00:40:28,140
time.

587
00:40:43,900 --> 00:40:51,390
So now, let's talk about
the entanglement entropy

588
00:40:51,390 --> 00:40:55,660
in many-body systems in
quantum many-body systems.

589
00:41:02,360 --> 00:41:06,000
So I hope you're familiar with
this word many-body systems.

590
00:41:06,000 --> 00:41:08,810
It's just a system of a
large number of particles

591
00:41:08,810 --> 00:41:12,050
or a large number
of constituents.

592
00:41:12,050 --> 00:41:16,390
So quantum field theory issue
is a many-bodied system.

593
00:41:16,390 --> 00:41:19,910
Any quantum field theory
is a many-body system.

594
00:41:19,910 --> 00:41:23,740
But this also includes
many other lattice systems,

595
00:41:23,740 --> 00:41:26,380
which condensed
matter people use,

596
00:41:26,380 --> 00:41:30,847
which are not necessary
quantum fields-- can be written

597
00:41:30,847 --> 00:41:31,930
as a quantum field theory.

598
00:41:31,930 --> 00:41:34,520
OK?

599
00:41:34,520 --> 00:41:38,780
So now let's again consider
just a simplified case.

600
00:41:38,780 --> 00:41:46,390
Let's consider a system which
is composed of A and B--

601
00:41:46,390 --> 00:41:54,910
so A plus B. And then I will
say some trivial statements.

602
00:41:54,910 --> 00:42:00,485
So if H is HA plus HB.

603
00:42:00,485 --> 00:42:01,860
So now we talk
about Hamiltonian.

604
00:42:07,260 --> 00:42:10,140
So far, I'm not using
actually many-body systems.

605
00:42:10,140 --> 00:42:14,420
What I'm talking about are
pretty generic systems.

606
00:42:14,420 --> 00:42:16,910
So even the
Hamiltonian is actually

607
00:42:16,910 --> 00:42:18,800
just a direct sum
of the Hamiltonian

608
00:42:18,800 --> 00:42:21,670
for A and Hamiltonian for B.

609
00:42:21,670 --> 00:42:28,250
If they don't couple, then of
course, the [INAUDIBLE] state

610
00:42:28,250 --> 00:42:31,070
or, in particular, the
ground state is unentangled.

611
00:42:34,694 --> 00:42:37,500
So you just find the ground
state of each system.

612
00:42:37,500 --> 00:42:39,755
And then that's the ground
state of the total system.

613
00:42:42,870 --> 00:42:46,880
And also, in general, if you
start with the initial stage,

614
00:42:46,880 --> 00:42:58,300
which is unentangled, with
the initial state, which

615
00:42:58,300 --> 00:43:08,980
is unentangled, it will
remain unentangled.

616
00:43:14,810 --> 00:43:17,060
So if you started with
an unentangled state,

617
00:43:17,060 --> 00:43:19,770
you wove it using this
Hamiltonian, of course,

618
00:43:19,770 --> 00:43:22,090
it will just act on
the specific part.

619
00:43:22,090 --> 00:43:24,040
And then you will
remain unentangled.

620
00:43:24,040 --> 00:43:26,810
OK?

621
00:43:26,810 --> 00:43:32,870
And now let's consider we
have H equals to HA plus HB.

622
00:43:32,870 --> 00:43:34,800
But now they also
have interactions

623
00:43:34,800 --> 00:43:39,860
between the A and the B.
So A and B are coupled.

624
00:43:43,570 --> 00:43:47,910
Then in general, we can just
repeat what he said here.

625
00:43:47,910 --> 00:43:52,991
Then the ground state
is now entangled.

626
00:43:56,220 --> 00:44:02,840
OK, and now if you start
with the initial state,

627
00:44:02,840 --> 00:44:12,200
even if you start with the
initial unentangled state,

628
00:44:12,200 --> 00:44:16,150
it becomes entangled on
Hamiltonian evolution.

629
00:44:36,490 --> 00:44:40,980
So in general, we
will expect-- so just

630
00:44:40,980 --> 00:44:45,930
based on this
general expectation--

631
00:44:45,930 --> 00:44:48,760
so you would expect
the ground state

632
00:44:48,760 --> 00:44:54,808
of a typical many-body
system will be actually

633
00:44:54,808 --> 00:44:55,516
rather entangled.

634
00:44:59,380 --> 00:45:05,909
It will be rather entangled
unless the Hamiltonian factor

635
00:45:05,909 --> 00:45:07,200
rides into all three particles.

636
00:45:11,090 --> 00:45:13,840
The factor rides into
the Hilbert space

637
00:45:13,840 --> 00:45:17,082
at each point-- for
each degree of freedom.

638
00:45:25,170 --> 00:45:29,120
But now let's talk about
the many-body systems

639
00:45:29,120 --> 00:45:31,040
we're interested in.

640
00:45:31,040 --> 00:45:33,240
So they're either typical
condensed matter systems

641
00:45:33,240 --> 00:45:35,870
or quantum field series.

642
00:45:35,870 --> 00:45:44,810
So in typical condensed matter
systems, we face this problem.

643
00:45:44,810 --> 00:45:53,620
So in typical generic condensed
matter systems of quantum field

644
00:45:53,620 --> 00:46:08,200
theory, in general,
no matter how

645
00:46:08,200 --> 00:46:16,300
you divide A and B-- so
generic H AB is non-zero.

646
00:46:16,300 --> 00:46:18,105
But not only is
H AB is non-zero,

647
00:46:18,105 --> 00:46:27,250
it's in general, H AB-- so the
whole Hamiltonian, including

648
00:46:27,250 --> 00:46:30,962
H AB is local.

649
00:46:34,160 --> 00:46:37,050
This is a very
important concept.

650
00:46:37,050 --> 00:46:38,140
It's local.

651
00:46:38,140 --> 00:46:39,560
But local will
mean the following.

652
00:46:39,560 --> 00:46:41,226
For example, let me
give you an example.

653
00:46:43,440 --> 00:46:51,540
For example, one of the very
important condensed matter

654
00:46:51,540 --> 00:46:54,070
systems is, of course,
the Heisenberg model.

655
00:46:54,070 --> 00:46:56,480
So essentially, you
have the Hamiltonian.

656
00:46:56,480 --> 00:46:59,630
So you can see the lattice
in whatever dimension

657
00:46:59,630 --> 00:47:03,020
you're interested in whatever
kind of lattice you are.

658
00:47:05,550 --> 00:47:08,080
At each lattice,
there is a spin.

659
00:47:08,080 --> 00:47:11,030
So consider you have
a lattice of spins.

660
00:47:11,030 --> 00:47:14,500
And then you'll have nearest
neighbor interactions

661
00:47:14,500 --> 00:47:17,770
between the
neighboring spins, so

662
00:47:17,770 --> 00:47:19,310
some of the nearest neighbors.

663
00:47:28,710 --> 00:47:31,320
So you can imagine you
have a lattice system.

664
00:47:31,320 --> 00:47:35,210
And each point on the
lattice, you have a spin.

665
00:47:35,210 --> 00:47:40,160
And each spin interacts
only with the spins nearby.

666
00:47:40,160 --> 00:47:42,620
So this is a local interaction.

667
00:47:42,620 --> 00:47:49,270
So by local interaction, means
that the only direct couple

668
00:47:49,270 --> 00:47:53,830
to the spin falls
distance away from it.

669
00:47:53,830 --> 00:47:57,540
So this is an example
of a local system.

670
00:47:57,540 --> 00:48:00,180
And all our QFTs
are local systems.

671
00:48:06,540 --> 00:48:07,925
All our QFTs are local systems.

672
00:48:07,925 --> 00:48:09,800
For example, let me just
write down the phi 4

673
00:48:09,800 --> 00:48:24,690
theory, for example,
phi 4 theory.

674
00:48:24,690 --> 00:48:28,920
Then phi at each
point-- so now you

675
00:48:28,920 --> 00:48:31,870
have to go back to your first
day of your quantum field

676
00:48:31,870 --> 00:48:35,130
theory where you
emphasize locality.

677
00:48:35,130 --> 00:48:38,450
So the phi evaluated
at each point now

678
00:48:38,450 --> 00:48:40,960
is independent of
degrees of freedom.

679
00:48:40,960 --> 00:48:42,930
Under the quantum
field theory, it's

680
00:48:42,930 --> 00:48:44,750
a Hilbert space of
a tensor product

681
00:48:44,750 --> 00:48:47,017
of the degrees of freedom.

682
00:48:47,017 --> 00:48:47,516
OK?

683
00:48:50,110 --> 00:48:54,072
Under the form of the Lagrangian
say if you discretize it--

684
00:48:54,072 --> 00:48:55,530
and this is only
involving coupling

685
00:48:55,530 --> 00:48:57,738
between the neighboring
point, because the derivative

686
00:48:57,738 --> 00:49:00,460
is only dependent on
the second derivative.

687
00:49:00,460 --> 00:49:03,300
And this only depends
on the value of phi

688
00:49:03,300 --> 00:49:04,780
at the single point.

689
00:49:04,780 --> 00:49:07,610
So the typical
quantum field theory

690
00:49:07,610 --> 00:49:10,490
as far as you have found
a number of derivatives,

691
00:49:10,490 --> 00:49:11,890
it's all local.

692
00:49:11,890 --> 00:49:13,395
It's all OK intact.

693
00:49:13,395 --> 00:49:14,020
It's all local.

694
00:49:14,020 --> 00:49:14,520
OK?

695
00:49:21,010 --> 00:49:26,150
So in other words, in
these kind of systems,

696
00:49:26,150 --> 00:49:28,710
let me go back here.

697
00:49:28,710 --> 00:49:29,876
So this is very important.

698
00:49:32,520 --> 00:49:44,610
In other words, in
these kind of systems,

699
00:49:44,610 --> 00:49:50,270
so let me just
imagine-- so let me just

700
00:49:50,270 --> 00:49:51,800
say this box is the whole thing.

701
00:49:51,800 --> 00:49:56,000
It's the whole whatever
space I live in.

702
00:49:56,000 --> 00:50:01,020
And I divide it into A B.

703
00:50:01,020 --> 00:50:05,850
So this tells you that for
typical condensed matter

704
00:50:05,850 --> 00:50:15,035
or quantum field series,
the H AB is only supported.

705
00:50:17,950 --> 00:50:21,160
So let me call this epsilon.

706
00:50:21,160 --> 00:50:26,300
So that tells that H
AB is only supported

707
00:50:26,300 --> 00:50:29,610
near the boundary
between A and B,

708
00:50:29,610 --> 00:50:32,790
because there is only
local interactions.

709
00:50:32,790 --> 00:50:36,280
So the part of the Hamiltonian
which directly covers A and B

710
00:50:36,280 --> 00:50:37,760
is only within this part.

711
00:50:37,760 --> 00:50:43,200
And this epsilon is added
on the lattice spacing

712
00:50:43,200 --> 00:50:49,080
is order of lattice spacing,
which in the case of a lattice

713
00:50:49,080 --> 00:50:51,330
system or in a
quantum field theory,

714
00:50:51,330 --> 00:50:55,070
it would be your
short distance cutoff.

715
00:50:55,070 --> 00:50:57,760
So if you try to discretize
your field theory,

716
00:50:57,760 --> 00:50:59,653
then this will be a
short distance cutoff.

717
00:51:05,110 --> 00:51:10,340
So similarly, I can
consider another shape of A.

718
00:51:10,340 --> 00:51:13,820
So suppose I can
see the circular A.

719
00:51:13,820 --> 00:51:19,980
And then, again,
the part of which

720
00:51:19,980 --> 00:51:24,580
H AB is only supported in
the region between these two

721
00:51:24,580 --> 00:51:26,900
dashed lines.

722
00:51:26,900 --> 00:51:28,400
And those dashed
would be considered

723
00:51:28,400 --> 00:51:33,590
to be very close to the
A. I'm just going to pick.

724
00:51:33,590 --> 00:51:36,130
But this should be the
short distance cutoff.

725
00:51:36,130 --> 00:51:37,390
OK?

726
00:51:37,390 --> 00:51:41,120
And again, the H AB is
only supported in here.

727
00:51:43,860 --> 00:51:48,547
So this is an important
feature of which

728
00:51:48,547 --> 00:51:49,755
you have a local Hamiltonian.

729
00:51:54,720 --> 00:51:57,840
Let me just write one more word.

730
00:51:57,840 --> 00:52:06,915
So H AB only involves
degrees of freedom

731
00:52:06,915 --> 00:52:18,860
here-- the boundary of A. So
this means the boundary of A.

732
00:52:18,860 --> 00:52:23,800
So this turns out to have very,
very important implications--

733
00:52:23,800 --> 00:52:27,460
the fact that the
Hamiltonian is local.

734
00:52:27,460 --> 00:52:30,785
And that they only direct
and mediate coupling

735
00:52:30,785 --> 00:52:34,370
between the degrees of freedom
near the boundary of A.

736
00:52:34,370 --> 00:52:36,704
And they have very
important consequences.

737
00:52:42,390 --> 00:52:46,580
So now I'm just
telling you the result

738
00:52:46,580 --> 00:52:48,500
which is a result of
having accumulated

739
00:52:48,500 --> 00:52:55,450
for many, many years
since the early '90s.

740
00:52:55,450 --> 00:53:02,440
So then when I
find in the ground

741
00:53:02,440 --> 00:53:12,200
state of a local
Hamiltonian in general

742
00:53:12,200 --> 00:53:20,050
you may construct some kind
of evil counterexamples.

743
00:53:20,050 --> 00:53:21,740
But in general, in
the ground state

744
00:53:21,740 --> 00:53:23,810
of a local Hamiltonian--
in the ground

745
00:53:23,810 --> 00:53:33,750
state of a local
Hamiltonian-- any

746
00:53:33,750 --> 00:53:36,720
just choose any region
A-- you integrate out

747
00:53:36,720 --> 00:53:38,350
the degrees of
freedom outside of it.

748
00:53:46,180 --> 00:53:48,450
So let me emphasize
an important point.

749
00:53:48,450 --> 00:53:51,440
Let me just pause to
emphasize an important, which

750
00:53:51,440 --> 00:53:55,079
is only increasing
what I'm saying here.

751
00:53:55,079 --> 00:53:57,370
In this definition of the
entangled matrix which I just

752
00:53:57,370 --> 00:54:00,920
erased, you just need
to have a partition

753
00:54:00,920 --> 00:54:04,080
of a degree of freedom
between A and B.

754
00:54:04,080 --> 00:54:07,540
The way you partition
it does not matter.

755
00:54:07,540 --> 00:54:09,860
How you partition
it does not matter.

756
00:54:09,860 --> 00:54:13,470
You can partition it in
whatever way you want.

757
00:54:13,470 --> 00:54:16,550
But for typical this
kind of lattice system

758
00:54:16,550 --> 00:54:22,330
or for quantum field theory,
there's a very large partition.

759
00:54:22,330 --> 00:54:25,020
And the partition is just
based on the locality.

760
00:54:25,020 --> 00:54:29,240
It's just based on the degrees
of freedom at each point.

761
00:54:29,240 --> 00:54:31,500
You just factorize
your Hilbert space

762
00:54:31,500 --> 00:54:33,540
in terms of degrees of
freedom at each point.

763
00:54:33,540 --> 00:54:35,581
So of course in the lattice
system, it's obvious.

764
00:54:35,581 --> 00:54:37,180
You have a spin at each lattice.

765
00:54:37,180 --> 00:54:39,070
And for the quantum
field theory,

766
00:54:39,070 --> 00:54:44,020
you just factorize your
degrees of freedom.

767
00:54:44,020 --> 00:54:46,360
At each point, you
factorize them.

768
00:54:46,360 --> 00:54:52,590
And so this locality
provides a natural partition

769
00:54:52,590 --> 00:54:56,590
of your degrees of freedom.

770
00:54:56,590 --> 00:55:01,590
And when we talk
about A and the B,

771
00:55:01,590 --> 00:55:06,390
we always talk about in
terms of the space location.

772
00:55:06,390 --> 00:55:10,540
And it's natural to talk
about the partition in terms

773
00:55:10,540 --> 00:55:15,840
of just your metric locations.

774
00:55:15,840 --> 00:55:17,310
So is this point clear?

775
00:55:17,310 --> 00:55:18,816
I just want to
emphasize this point.

776
00:55:21,600 --> 00:55:26,180
So then you find that in
general for the ground

777
00:55:26,180 --> 00:55:28,550
state of a local Hamiltonian
satisfy so-called area

778
00:55:28,550 --> 00:55:34,060
law is that the leading term
of the entanglement entropy

779
00:55:34,060 --> 00:55:38,185
is given by the area
of the boundary of A--

780
00:55:38,185 --> 00:55:40,030
this applies no
matter which dimension

781
00:55:40,030 --> 00:55:45,410
you are in generic
dimensions-- and divide it

782
00:55:45,410 --> 00:55:52,540
by a short distance cutoff
of the lattice spacing.

783
00:55:52,540 --> 00:55:54,950
It's some number.

784
00:55:54,950 --> 00:55:59,260
So A is a dimensionless number.

785
00:55:59,260 --> 00:56:02,890
So area-- we can see the
d-dimensional theory--

786
00:56:02,890 --> 00:56:04,450
so d-dimensional system.

787
00:56:04,450 --> 00:56:07,030
Then the spatial dimension
will be d minus 1.

788
00:56:07,030 --> 00:56:09,580
And then you go to the
boundary of some region.

789
00:56:09,580 --> 00:56:13,540
The boundary of region A.
Then that will be a d minus 2

790
00:56:13,540 --> 00:56:15,330
dimensional surface.

791
00:56:15,330 --> 00:56:17,880
And then this would be
the area of that surface.

792
00:56:17,880 --> 00:56:19,900
So this may cover a
dimensionless number.

793
00:56:19,900 --> 00:56:22,730
And then you can have
some pre-factor here

794
00:56:22,730 --> 00:56:25,880
which depend on your systems.

795
00:56:25,880 --> 00:56:29,095
So this formula tells you
a very important physics.

796
00:56:32,720 --> 00:56:34,985
So this formula tells you
a very important physics.

797
00:56:41,280 --> 00:56:50,750
This tells you that
generically if you--

798
00:56:50,750 --> 00:56:55,370
in such kind of systems,
AB are entangled.

799
00:56:55,370 --> 00:57:05,120
But the entanglement between
AB between A is complementary,

800
00:57:05,120 --> 00:57:19,020
which I just call A
and B, are dominated

801
00:57:19,020 --> 00:57:33,720
by short-range entanglements
near the boundary of A

802
00:57:33,720 --> 00:57:37,200
where this H AB is supported.

803
00:57:44,301 --> 00:57:44,800
OK?

804
00:57:48,350 --> 00:57:53,250
So this is also very intuitive
once you load the answer--

805
00:57:53,250 --> 00:57:55,750
once you load the
answer, because we

806
00:57:55,750 --> 00:57:59,310
emphasized that H AB only
coupled to degrees of freedom

807
00:57:59,310 --> 00:58:00,820
nearby.

808
00:58:00,820 --> 00:58:03,540
And then we look
at the ground state

809
00:58:03,540 --> 00:58:09,150
you find most of the
degrees of freedom in here

810
00:58:09,150 --> 00:58:11,590
don't entangle with B. Only
the degrees of freedom them

811
00:58:11,590 --> 00:58:13,840
near the boundary of A are
entangled with B because

812
00:58:13,840 --> 00:58:15,320
of this interaction.

813
00:58:15,320 --> 00:58:17,295
This local interaction
directly leads

814
00:58:17,295 --> 00:58:20,820
to the entanglement between
the degrees of freedom nearby.

815
00:58:20,820 --> 00:58:24,290
And those degrees of freedom,
if they are far away,

816
00:58:24,290 --> 00:58:25,290
they might be entangled.

817
00:58:25,290 --> 00:58:28,200
But they are not dominant.

818
00:58:28,200 --> 00:58:29,650
OK?

819
00:58:29,650 --> 00:58:32,850
So I have put many dots here.

820
00:58:32,850 --> 00:58:37,140
So included in those dots
are possible long-range

821
00:58:37,140 --> 00:58:38,770
entanglements.

822
00:58:38,770 --> 00:58:40,560
Because when you find
the ground state,

823
00:58:40,560 --> 00:58:45,360
you really have to extremize
the energy of the total system.

824
00:58:45,360 --> 00:58:48,120
You don't only look
at the small part.

825
00:58:48,120 --> 00:58:51,700
So there's always some kind
of long-range entanglement

826
00:58:51,700 --> 00:58:55,080
beyond this H AB.

827
00:58:55,080 --> 00:58:57,915
But this formula tells you
that this short distance

828
00:58:57,915 --> 00:59:00,210
entanglement dominates.

829
00:59:00,210 --> 00:59:01,506
Yes?

830
00:59:01,506 --> 00:59:04,235
AUDIENCE: One question-- in QFT
in general, like [INAUDIBLE],

831
00:59:04,235 --> 00:59:05,693
there's really all
the interactions

832
00:59:05,693 --> 00:59:06,984
are perfectly local, I believe.

833
00:59:06,984 --> 00:59:10,053
But in string theory, are there
any non-local interactions

834
00:59:10,053 --> 00:59:11,265
that occur.

835
00:59:11,265 --> 00:59:13,090
Are there any interactions
with non-local?

836
00:59:13,090 --> 00:59:17,160
HONG LIU: Yeah, it will
depend on the scale.

837
00:59:17,160 --> 00:59:20,020
In string theory when
we say non-local,

838
00:59:20,020 --> 00:59:24,320
it's non-local at the
alpha prime scale.

839
00:59:24,320 --> 00:59:27,120
And that alpha prime can
perfectly be our short distance

840
00:59:27,120 --> 00:59:27,650
cutoff here.

841
00:59:27,650 --> 00:59:29,250
When I talk about
local and non-local,

842
00:59:29,250 --> 00:59:32,790
I'm talking about
infinite versus finite.

843
00:59:32,790 --> 00:59:34,520
For example, here,
when I talk about

844
00:59:34,520 --> 00:59:37,740
whether this is
a local system, I

845
00:59:37,740 --> 00:59:40,440
don't have to have
a nearest neighbor.

846
00:59:40,440 --> 00:59:49,620
As far as this matches
this point only directly

847
00:59:49,620 --> 00:59:52,810
coupled to the finite
distance neighbor.

848
00:59:52,810 --> 00:59:55,561
And this can see the
local Hamiltonian.

849
00:59:55,561 --> 00:59:57,560
But from the quantum field
theory point of view,

850
00:59:57,560 --> 00:59:59,018
this might be
considered non-local.

851
01:00:01,137 --> 01:00:01,970
You see what I mean?

852
01:00:01,970 --> 01:00:04,740
Depending on your scale.

853
01:00:04,740 --> 01:00:07,400
AUDIENCE: So is
there anything that

854
01:00:07,400 --> 01:00:09,980
appears in string theory which
is non-local at any scale?

855
01:00:09,980 --> 01:00:13,610
Or is the non-locality
just some--

856
01:00:13,610 --> 01:00:15,300
HONG LIU: Yeah, so
the string theory

857
01:00:15,300 --> 01:00:20,200
itself does not give
you that non-locality,

858
01:00:20,200 --> 01:00:21,817
but the black hole may.

859
01:00:21,817 --> 01:00:22,900
AUDIENCE: Oh, interesting.

860
01:00:22,900 --> 01:00:25,410
HONG LIU: The black
hole seems to give you

861
01:00:25,410 --> 01:00:29,060
some kind of non-locality which
we don't fully understand.

862
01:00:29,060 --> 01:00:31,390
So that's why the black
hole is so puzzling.

863
01:00:31,390 --> 01:00:34,650
And the [INAUDIBLE] of
string theory, you're

864
01:00:34,650 --> 01:00:38,010
non-local in the
scale of alpha prime.

865
01:00:38,010 --> 01:00:40,479
Yeah, if it's alpha
prime sufficient small,

866
01:00:40,479 --> 01:00:41,520
it's like a local theory.

867
01:00:41,520 --> 01:00:44,262
And even if alpha prime is
big, if you look at very much,

868
01:00:44,262 --> 01:00:45,220
it is still like local.

869
01:00:45,220 --> 01:00:46,210
AUDIENCE: I see.

870
01:00:46,210 --> 01:00:49,180
HONG LIU: Yeah, but black
hole can make things

871
01:00:49,180 --> 01:00:50,326
very long distance.

872
01:00:50,326 --> 01:00:52,200
So that's the tricky
thing about black holes.

873
01:00:55,060 --> 01:00:59,980
OK, so this is
something which are

874
01:00:59,980 --> 01:01:06,690
is interesting and was first
discovered in the early '90s.

875
01:01:06,690 --> 01:01:08,760
But in hindsight,
it's very intuitive.

876
01:01:08,760 --> 01:01:12,310
In hindsight, it's
very intuitive.

877
01:01:12,310 --> 01:01:16,420
So even though this formula,
even though this term

878
01:01:16,420 --> 01:01:20,020
is universal, but this
coefficient is actually

879
01:01:20,020 --> 01:01:22,450
highly non-universal.

880
01:01:22,450 --> 01:01:24,170
Normally, by saying
universal, we

881
01:01:24,170 --> 01:01:27,655
mean something which is
common to different systems.

882
01:01:27,655 --> 01:01:29,780
So this coefficient actually
depends on the details

883
01:01:29,780 --> 01:01:30,820
of individual systems.

884
01:01:30,820 --> 01:01:32,736
Say if you calculate for
the Heisenberg model,

885
01:01:32,736 --> 01:01:37,310
or if you calculate for the
free scale of field theory,

886
01:01:37,310 --> 01:01:40,800
or free fermion theory, and
then this coefficient in general

887
01:01:40,800 --> 01:01:45,040
is different, are depending
on how you define your cutoff.

888
01:01:45,040 --> 01:01:48,645
So even though this
formula is universal,

889
01:01:48,645 --> 01:01:52,470
but those pre-factors actually
are highly non-universal.

890
01:01:58,689 --> 01:02:02,951
But the partial
excitement-- there

891
01:02:02,951 --> 01:02:04,992
are many things people
are excited about in terms

892
01:02:04,992 --> 01:02:05,492
of entropy.

893
01:02:08,920 --> 01:02:11,800
This is one thing,
this area law,

894
01:02:11,800 --> 01:02:13,135
because the area law is nice.

895
01:02:19,050 --> 01:02:21,440
Then this can be used as
a distinguishing property

896
01:02:21,440 --> 01:02:23,381
of a ground state.

897
01:02:23,381 --> 01:02:24,880
If you go to a
highly excited state,

898
01:02:24,880 --> 01:02:27,730
then you find the generic
distribution inside

899
01:02:27,730 --> 01:02:29,639
and A and B will be entangled.

900
01:02:29,639 --> 01:02:30,930
And [INAUDIBLE] highly excited.

901
01:02:30,930 --> 01:02:33,890
But only in the ground state,
somehow only near the boundary

902
01:02:33,890 --> 01:02:35,290
they are entangled.

903
01:02:35,290 --> 01:02:38,020
This is also makes sense,
because if it were high energy

904
01:02:38,020 --> 01:02:40,270
then you can excite it,
because from the very far away

905
01:02:40,270 --> 01:02:45,123
from H AB, then of course,
everything would be entangled.

906
01:02:45,123 --> 01:02:48,460
AUDIENCE: So just
one other question.

907
01:02:48,460 --> 01:02:50,930
So why are you
dividing by epsilon?

908
01:02:50,930 --> 01:02:54,220
HONG LIU: No, this
estimation is a number.

909
01:02:54,220 --> 01:02:55,840
Making the dimension
is numbered.

910
01:02:55,840 --> 01:02:58,810
This is only a short-distance
information here.

911
01:02:58,810 --> 01:03:01,154
So that is spatial
short-distance cut off.

912
01:03:01,154 --> 01:03:02,883
AUDIENCE: Right,
the only reason I

913
01:03:02,883 --> 01:03:06,320
ask is that intuitively it
seems that the entropy should

914
01:03:06,320 --> 01:03:12,741
be proportional to this thin
volume around the boundary.

915
01:03:12,741 --> 01:03:14,240
But that doesn't--
OK, I don't know.

916
01:03:14,240 --> 01:03:16,760
HONG LIU: Yeah, you have to make
it into a dimensionless number.

917
01:03:16,760 --> 01:03:18,426
You have to be
proportional to the area.

918
01:03:21,260 --> 01:03:25,740
So there's a very simple way
to understand that thing.

919
01:03:25,740 --> 01:03:33,420
So the area divided by lattice
spacing-- so what is this guy?

920
01:03:33,420 --> 01:03:35,270
This is wonderfully
a lattice volume.

921
01:03:37,850 --> 01:03:42,520
That is the lattice
area on the surface.

922
01:03:42,520 --> 01:03:45,150
So essentially this tells you
how many degrees of freedom

923
01:03:45,150 --> 01:03:48,320
are lying on that boundary.

924
01:03:48,320 --> 01:03:49,460
AUDIENCE: OK, interesting.

925
01:03:49,460 --> 01:03:53,380
HONG LIU: Yeah, OK?

926
01:03:53,380 --> 01:03:54,680
So is this clear?

927
01:03:54,680 --> 01:03:58,950
OK, I think I'm spending
too much time on this.

928
01:03:58,950 --> 01:04:06,620
Anyway, so excited
in the last decade--

929
01:04:06,620 --> 01:04:08,550
the main excitement
about entangled

930
01:04:08,550 --> 01:04:09,920
entropy in the last decade.

931
01:04:09,920 --> 01:04:11,719
The one thing is
about this behavior,

932
01:04:11,719 --> 01:04:14,010
then you can use it as a
characterization of the ground

933
01:04:14,010 --> 01:04:15,270
state.

934
01:04:15,270 --> 01:04:18,230
But a lot of the
excitement about this thing

935
01:04:18,230 --> 01:04:21,050
is on those dot, dot, dots.

936
01:04:21,050 --> 01:04:27,570
So people actually find
that the subleading term--

937
01:04:27,570 --> 01:04:30,450
some excitement.

938
01:04:30,450 --> 01:04:38,550
Yeah, I'm just saying--
from last decade people

939
01:04:38,550 --> 01:04:48,820
have discover that the
subleading terms in entangled

940
01:04:48,820 --> 01:04:54,520
entropy other than
this area term which

941
01:04:54,520 --> 01:04:59,338
come in from long-range
entanglements.

942
01:05:02,330 --> 01:05:06,470
So this captures the
short-range entanglement

943
01:05:06,470 --> 01:05:11,800
from the long-range
entanglement can actually

944
01:05:11,800 --> 01:05:17,365
provide important
characterizations of a system.

945
01:05:35,180 --> 01:05:38,560
So let me just
mention two examples.

946
01:05:38,560 --> 01:05:40,770
So it will be little
bit fast because we

947
01:05:40,770 --> 01:05:48,753
are a little bit-- do you
want a break for the last day?

948
01:05:51,900 --> 01:05:53,840
AUDIENCE: Well, when
you say it like that--

949
01:05:53,840 --> 01:05:54,676
HONG LIU: OK, good.

950
01:05:59,210 --> 01:06:00,030
OK, thanks.

951
01:06:15,130 --> 01:06:17,540
So let me just quickly
talk about two examples

952
01:06:17,540 --> 01:06:19,735
because I want to talk
about the holographic case.

953
01:06:22,590 --> 01:06:25,550
So the first thing is something
called the topological can

954
01:06:25,550 --> 01:06:41,670
be used to characterize
so-called topological order

955
01:06:41,670 --> 01:06:43,083
in 2 plus 1 dimensions.

956
01:06:53,360 --> 01:06:58,310
So since the '90s, mid
'80s, and the '980s,

957
01:06:58,310 --> 01:07:02,420
people discovered
that if you look

958
01:07:02,420 --> 01:07:11,490
at the typical gapped
system-- so when

959
01:07:11,490 --> 01:07:15,330
we say a gapped system means
that system between the ground

960
01:07:15,330 --> 01:07:20,130
state and the first excited
state have a finite energy gap.

961
01:07:22,800 --> 01:07:27,460
So our general understanding
of a gap system is that

962
01:07:27,460 --> 01:07:31,930
in the ground state because
you have a finite gap,

963
01:07:31,930 --> 01:07:33,430
and there's sufficient
low energies,

964
01:07:33,430 --> 01:07:35,730
you simply cannot
excite anything.

965
01:07:35,730 --> 01:07:38,430
You have low energy because you
have a finite gap to the first

966
01:07:38,430 --> 01:07:39,070
excited state.

967
01:07:39,070 --> 01:07:40,462
You cannot excite anything.

968
01:07:40,462 --> 01:07:42,920
So essentially, when you look
at the correlation functions,

969
01:07:42,920 --> 01:07:45,110
we can look at
all the variables.

970
01:07:45,110 --> 01:07:47,566
They all only have
short-range correlations.

971
01:07:47,566 --> 01:07:49,190
The cannot have
long-range correlation,

972
01:07:49,190 --> 01:07:52,534
because there's nothing to
propagate you to long distance.

973
01:07:52,534 --> 01:07:53,950
And because long
distance requires

974
01:07:53,950 --> 01:07:55,430
massive degrees of freedom.

975
01:07:55,430 --> 01:07:58,055
And because you have a gap, you
don't have such massive degrees

976
01:07:58,055 --> 01:07:58,890
of freedom.

977
01:07:58,890 --> 01:08:03,610
So the typical conventional
idea about the gap system

978
01:08:03,610 --> 01:08:05,840
is that all correlations
should be short-ranged.

979
01:08:08,680 --> 01:08:12,500
And so this is a
condition that meets them.

980
01:08:12,500 --> 01:08:14,730
But in the late
'80s and early '90s,

981
01:08:14,730 --> 01:08:17,930
our colleague, Xiao-Gang
Wen, he introduced

982
01:08:17,930 --> 01:08:24,870
this notion of a
topological ordered system

983
01:08:24,870 --> 01:08:27,375
based on fractional
quantum Hall effect.

984
01:08:29,910 --> 01:08:37,439
So he actually reasoned that
there can be gap systems even

985
01:08:37,439 --> 01:08:39,979
though I have a finite gap,
but in the ground state

986
01:08:39,979 --> 01:08:42,850
actually contain
long-range correlations--

987
01:08:42,850 --> 01:08:45,520
nontrivial, long-range
correlations.

988
01:08:45,520 --> 01:08:49,080
And those long-range
correlations cannot be seen

989
01:08:49,080 --> 01:08:50,547
using the standard observables.

990
01:08:50,547 --> 01:08:52,630
So if you are using the
standard observables-- say

991
01:08:52,630 --> 01:08:56,100
correlation functions of local
operators, you don't see them.

992
01:08:56,100 --> 01:08:59,060
You only see the
short-range correlations.

993
01:08:59,060 --> 01:09:00,560
And those long-range
correlations

994
01:09:00,560 --> 01:09:03,100
are topological in nature.

995
01:09:03,100 --> 01:09:04,880
You have to see it
in some subtle ways.

996
01:09:04,880 --> 01:09:07,770
And then in the '80s, except
in the late '80s, early '90s,

997
01:09:07,770 --> 01:09:11,010
he was trying to argue
in some [INAUDIBLE] way

998
01:09:11,010 --> 01:09:14,550
to argue there is such kind
of subtle correlations.

999
01:09:14,550 --> 01:09:19,670
But then around 2000,
maybe 2005, then

1000
01:09:19,670 --> 01:09:22,760
he realized actually you can
see it very easily directly

1001
01:09:22,760 --> 01:09:24,852
from the ground
state wave function.

1002
01:09:24,852 --> 01:09:26,560
You just calculate
the entangled entropy.

1003
01:09:29,750 --> 01:09:31,840
You just calculate
entangled entropy.

1004
01:09:31,840 --> 01:09:34,479
So in entangled entropy,
you always have this--

1005
01:09:34,479 --> 01:09:37,930
so this is a 2 plus 1
system, and then the boundary

1006
01:09:37,930 --> 01:09:39,702
is just a line.

1007
01:09:39,702 --> 01:09:41,410
So essentially, you
just have a boundary.

1008
01:09:41,410 --> 01:09:45,500
It's a number-- the
length of the boundary

1009
01:09:45,500 --> 01:09:48,929
divided by your
cutoff, but then it

1010
01:09:48,929 --> 01:09:51,970
turns out for these kind of
topological ordered system,

1011
01:09:51,970 --> 01:09:55,310
there is a finite constant
as a subleading term.

1012
01:09:58,190 --> 01:10:04,170
And this finite constant is
independent of the shape of A

1013
01:10:04,170 --> 01:10:06,740
and independent of length
of A and independent

1014
01:10:06,740 --> 01:10:09,970
of whatever A is purely
topological in nature.

1015
01:10:09,970 --> 01:10:15,080
And so if you have
achieved a gap system then

1016
01:10:15,080 --> 01:10:16,500
this gamma will be 0.

1017
01:10:16,500 --> 01:10:20,210
So you look at a free
massive scale of field--

1018
01:10:20,210 --> 01:10:21,740
then this gamma will be 0.

1019
01:10:21,740 --> 01:10:23,840
But for those topological
ordered systems,

1020
01:10:23,840 --> 01:10:25,440
this gamma would be non-zero.

1021
01:10:25,440 --> 01:10:30,020
And it tells you actually there
is long-range correlations

1022
01:10:30,020 --> 01:10:35,070
which are encoded in the
entanglement entropy,

1023
01:10:35,070 --> 01:10:37,910
because this cannot come from
the local thing because this

1024
01:10:37,910 --> 01:10:41,250
does not depend on the shape of
A, does not depend on N sub A,

1025
01:10:41,250 --> 01:10:43,600
does not depend
on anything of A.

1026
01:10:43,600 --> 01:10:45,400
So this is really
topological in nature.

1027
01:10:45,400 --> 01:10:48,320
It tells you this can only
be some long-range, subtle

1028
01:10:48,320 --> 01:10:50,390
correlation, which
is only capture

1029
01:10:50,390 --> 01:10:51,765
by these entangled
entropy rather

1030
01:10:51,765 --> 01:10:53,556
than by the standard
correlation functions.

1031
01:10:53,556 --> 01:10:54,180
OK

1032
01:10:54,180 --> 01:11:00,270
So this is one discovery which
is generated a lot of interest

1033
01:11:00,270 --> 01:11:02,670
in the entangled entropy
because now this can

1034
01:11:02,670 --> 01:11:05,720
be used to define the phases.

1035
01:11:05,720 --> 01:11:08,010
OK, you can actually
define non-trivial phases

1036
01:11:08,010 --> 01:11:11,179
using this number.

1037
01:11:11,179 --> 01:11:12,470
And so this is the first thing.

1038
01:11:15,280 --> 01:11:18,860
So this is about in
condensed matter systems.

1039
01:11:18,860 --> 01:11:23,330
In quantum field theories ,
this can be used to characterize

1040
01:11:23,330 --> 01:11:33,670
the number of degrees
of freedom of a QFT.

1041
01:11:38,180 --> 01:11:40,950
Again, this is a long story.

1042
01:11:40,950 --> 01:11:43,670
But I don't have time.

1043
01:11:43,670 --> 01:11:45,990
So let me just tell
you a short story.

1044
01:11:45,990 --> 01:11:50,906
So first let me look at 1
plus 1 dimensions, a CFT.

1045
01:11:56,170 --> 01:12:00,640
So for 1 plus 1 dimension
of CFT or important

1046
01:12:00,640 --> 01:12:04,470
the quantity is so-called
essential charge.

1047
01:12:04,470 --> 01:12:08,940
So have you all heard of
essential charge or not really?

1048
01:12:08,940 --> 01:12:12,500
It just says, imagine you have
a 1 plus 1 dimensional CFT.

1049
01:12:12,500 --> 01:12:16,450
And for each CFT, you can
define a single number, which

1050
01:12:16,450 --> 01:12:18,240
is called the center charge.

1051
01:12:18,240 --> 01:12:20,720
And this center charge
is very important

1052
01:12:20,720 --> 01:12:23,680
because it controls
the asymptotic density

1053
01:12:23,680 --> 01:12:25,320
of state of the system.

1054
01:12:25,320 --> 01:12:27,611
So if you look at the system
go to very high energies,

1055
01:12:27,611 --> 01:12:29,110
and the density of
state essentially

1056
01:12:29,110 --> 01:12:31,440
is controlled by this number C.

1057
01:12:31,440 --> 01:12:33,950
So essentially C can
be used to characterize

1058
01:12:33,950 --> 01:12:35,770
a number of degrees of freedom.

1059
01:12:35,770 --> 01:12:38,400
And the C will appear
in many other places.

1060
01:12:38,400 --> 01:12:40,800
But I don't have
time to explain.

1061
01:12:40,800 --> 01:12:43,480
But anyway, there's something
called the central charge,

1062
01:12:43,480 --> 01:12:46,222
which can be used to
characterize numbers of degrees

1063
01:12:46,222 --> 01:12:47,820
of freedom of the system.

1064
01:12:47,820 --> 01:12:51,040
And thus, if you look at the
entanglement entropy for 1

1065
01:12:51,040 --> 01:12:54,239
plus 1 dimensional system for
1 plus 1 dimension of CFT.

1066
01:12:54,239 --> 01:12:55,655
So in 1 plus 1
dimension, you only

1067
01:12:55,655 --> 01:12:57,630
have a line in the
spatial direction.

1068
01:12:57,630 --> 01:13:03,480
So then this takes A just to
be a sacrament of length l.

1069
01:13:03,480 --> 01:13:05,250
Then you find when
you integrate out

1070
01:13:05,250 --> 01:13:07,440
everything else, the
entangled entropy

1071
01:13:07,440 --> 01:13:13,210
for the A given by C divided
by 3 log L divided by epsilon.

1072
01:13:13,210 --> 01:13:16,040
Again, epsilon is a
short distance cutoff.

1073
01:13:16,040 --> 01:13:18,440
And the C is your center charge.

1074
01:13:18,440 --> 01:13:24,520
So you see that the
entanglement entropy-- first you

1075
01:13:24,520 --> 01:13:27,650
notice that formula
becomes degenerate when

1076
01:13:27,650 --> 01:13:30,780
you go to the two dimensions.

1077
01:13:30,780 --> 01:13:32,224
It's equal to 1
plus 1 dimensions.

1078
01:13:32,224 --> 01:13:34,640
1 plus 1 dimension the were
played by something like this.

1079
01:13:34,640 --> 01:13:37,750
You have log epsilon.

1080
01:13:37,750 --> 01:13:41,140
So the key important
thing is not the prefactor

1081
01:13:41,140 --> 01:13:41,990
because of the log.

1082
01:13:41,990 --> 01:13:43,850
The prefactor is
actually reversal.

1083
01:13:43,850 --> 01:13:46,620
And it's actually controlled
by the central charge.

1084
01:13:46,620 --> 01:13:48,750
So that saves you from
entangled entropy.

1085
01:13:48,750 --> 01:13:50,840
You can actually read
the central charge

1086
01:13:50,840 --> 01:13:52,306
and actually read what
are the numbers of degrees

1087
01:13:52,306 --> 01:13:53,514
of was freedom of the system.

1088
01:13:57,240 --> 01:14:00,110
So the fact that the central
charge in the 1 plus 1

1089
01:14:00,110 --> 01:14:03,720
dimensional CFT captures the
number of degrees of freedom

1090
01:14:03,720 --> 01:14:06,730
was known-- known
before people thought

1091
01:14:06,730 --> 01:14:10,070
about entangled entropy.

1092
01:14:10,070 --> 01:14:15,014
But it turned out-- and
people spent many years

1093
01:14:15,014 --> 01:14:17,180
trying too hard to generalize
this concept to higher

1094
01:14:17,180 --> 01:14:18,017
dimensional series.

1095
01:14:20,485 --> 01:14:22,610
And it turned out they are
very hard to generalize.

1096
01:14:22,610 --> 01:14:24,890
People didn't really
know how to do it.

1097
01:14:24,890 --> 01:14:27,170
But now we know
because you can just

1098
01:14:27,170 --> 01:14:30,090
generalize the entangled
entropy to higher dimensions.

1099
01:14:30,090 --> 01:14:33,640
And then you find the same
kind of central charge

1100
01:14:33,640 --> 01:14:35,720
which appeared.

1101
01:14:35,720 --> 01:14:38,780
You find again
things can be defined

1102
01:14:38,780 --> 01:14:41,180
as analog of the single
charge in 1 plus 1 dimension

1103
01:14:41,180 --> 01:14:43,800
appears in the entangled entropy
in the higher dimensions.

1104
01:14:43,800 --> 01:14:45,970
And then they can be
used to characterize

1105
01:14:45,970 --> 01:14:49,110
the number degrees of freedom.

1106
01:14:49,110 --> 01:14:51,610
And so in some sense, the
entangled entropy really

1107
01:14:51,610 --> 01:14:55,354
provides a unified way to
think about this central charge

1108
01:14:55,354 --> 01:14:57,520
and to characterize the
number of degrees of freedom

1109
01:14:57,520 --> 01:15:01,960
across all dimensions--
across all dimensions.

1110
01:15:01,960 --> 01:15:03,575
And so this is a key formula.

1111
01:15:06,790 --> 01:15:11,160
Actually, I first realized by
our old friend Frank Wilczek

1112
01:15:11,160 --> 01:15:14,650
when they studied
free field theory.

1113
01:15:14,650 --> 01:15:17,190
And people later generalized
to show this actually

1114
01:15:17,190 --> 01:15:17,940
works for any CFT.

1115
01:15:22,700 --> 01:15:24,009
Good.

1116
01:15:24,009 --> 01:15:26,300
So now finally, we can talk
about holographic entangled

1117
01:15:26,300 --> 01:15:30,120
entropy with lots
of preparations.

1118
01:15:30,120 --> 01:15:33,700
So this just gives
you some taste

1119
01:15:33,700 --> 01:15:38,690
of actually why we are actually
interested in entangled entropy

1120
01:15:38,690 --> 01:15:43,230
and why this actually not only
people in quantum computing

1121
01:15:43,230 --> 01:15:47,920
or quantum information
people are interested in it.

1122
01:15:47,920 --> 01:15:49,670
People are doing
condensed matter-- people

1123
01:15:49,670 --> 01:15:53,620
doing quantum field theory
are also interested in it.

1124
01:15:56,980 --> 01:15:58,730
And now people who are
doing string theory

1125
01:15:58,730 --> 01:16:01,040
are also interested
in it because

1126
01:16:01,040 --> 01:16:02,830
of this holographic
entanglement entropy.

1127
01:16:16,070 --> 01:16:18,070
So suppose we have our CFT.

1128
01:16:21,150 --> 01:16:22,840
So a two-dimensional
CFT with a gravity

1129
01:16:22,840 --> 01:16:32,350
dual-- so a dual to some
theory in d plus 1 dimensional

1130
01:16:32,350 --> 01:16:35,330
[INAUDIBLE] space time.

1131
01:16:35,330 --> 01:16:41,540
So let me give you origin A. How
do I find the entangled-- what

1132
01:16:41,540 --> 01:16:43,050
is the counterpart
of this entangled

1133
01:16:43,050 --> 01:16:46,555
entropy on the gravity side?

1134
01:16:46,555 --> 01:16:47,554
So this is the question.

1135
01:16:50,420 --> 01:16:52,460
So let me just draw this figure.

1136
01:16:52,460 --> 01:16:53,920
Suppose this is
the box again that

1137
01:16:53,920 --> 01:16:56,340
represents your total system.

1138
01:16:56,340 --> 01:16:59,260
And then this carves
out of region A.

1139
01:16:59,260 --> 01:17:01,846
Then the question what
does this translate into?

1140
01:17:05,500 --> 01:17:06,612
And the graph decides.

1141
01:17:06,612 --> 01:17:08,070
OK, so now in the
boundary, we have

1142
01:17:08,070 --> 01:17:12,080
some region A is equal to 0.

1143
01:17:21,060 --> 01:17:31,980
So for this question, in
some sense it's difficult

1144
01:17:31,980 --> 01:17:33,770
because it's not
like other questions

1145
01:17:33,770 --> 01:17:35,090
we've talked about before.

1146
01:17:35,090 --> 01:17:37,890
So we also know
partition function.

1147
01:17:37,890 --> 01:17:40,580
You can say the partition
function cannot be the same.

1148
01:17:40,580 --> 01:17:42,850
Those things appear
to be natural.

1149
01:17:42,850 --> 01:17:46,300
Here, there is no natural
thing you can think about,

1150
01:17:46,300 --> 01:17:49,730
because you cannot
define entangled entropy

1151
01:17:49,730 --> 01:17:52,313
as a partition function.

1152
01:17:52,313 --> 01:17:55,460
Entangled entropy have to
involve in so-called trace

1153
01:17:55,460 --> 01:17:56,750
out the degrees of freedom.

1154
01:17:56,750 --> 01:17:59,610
Then take the logarithm.

1155
01:17:59,610 --> 01:18:03,040
It's highly non-local and
a complicated procedure.

1156
01:18:03,040 --> 01:18:06,980
And it's obvious how you would
define on the gravity side--

1157
01:18:06,980 --> 01:18:10,080
how you would guess
on the gravity side.

1158
01:18:10,080 --> 01:18:13,710
So it's quite remarkable
and the Ryu-Takayanagi,

1159
01:18:13,710 --> 01:18:15,990
they just made the guess.

1160
01:18:15,990 --> 01:18:18,570
And then worked.

1161
01:18:18,570 --> 01:18:34,250
So the proposal is just find
the minimal area surface,

1162
01:18:34,250 --> 01:18:50,070
let me call gamma A,
which extends in the back

1163
01:18:50,070 --> 01:18:52,660
and with the boundary
of A as the boundary.

1164
01:18:57,050 --> 01:18:58,910
You first find the surface.

1165
01:18:58,910 --> 01:19:01,600
And then you say the
entangled entropy for A

1166
01:19:01,600 --> 01:19:07,070
is just the area of this
gamma A divided by 40 Newton.

1167
01:19:11,450 --> 01:19:12,850
And this is the Ryu-Takayanagi.

1168
01:19:22,720 --> 01:19:27,730
So the idea is that you find
a surface of a minimal area

1169
01:19:27,730 --> 01:19:31,120
is going into the back, but
ending on the boundary of A.

1170
01:19:31,120 --> 01:19:35,740
So this is your gamma A. And
you find such a minimal surface.

1171
01:19:35,740 --> 01:19:38,756
Then you divide by 40 Newton.

1172
01:19:38,756 --> 01:19:39,255
OK?

1173
01:19:45,280 --> 01:19:48,182
So let me just
mention one thing.

1174
01:19:58,540 --> 01:20:03,390
Let me just note a simple
scene, because again,

1175
01:20:03,390 --> 01:20:08,390
the A of the boundary, this
is d minus 2 dimensional.

1176
01:20:11,680 --> 01:20:15,540
And A and the gamma A are
all d minus 1 dimensional.

1177
01:20:19,650 --> 01:20:21,450
So this is d minus
2 dimension is

1178
01:20:21,450 --> 01:20:23,270
what we said before
because A is just

1179
01:20:23,270 --> 01:20:25,430
a region in your
spatial section.

1180
01:20:25,430 --> 01:20:27,430
So A is a d minus 1 dimensional.

1181
01:20:27,430 --> 01:20:29,990
And the boundary A will
d minus 2 dimensional.

1182
01:20:29,990 --> 01:20:31,740
And the gamma A
would be a surface

1183
01:20:31,740 --> 01:20:37,280
which is ending on the boundary
of A. This ends on that thing.

1184
01:20:37,280 --> 01:20:40,050
So this is also the d
minus 1 dimensional.

1185
01:20:40,050 --> 01:20:41,660
And this is d minus
1 dimensional.

1186
01:20:41,660 --> 01:20:44,070
So this area have
dimension d minus 1.

1187
01:20:44,070 --> 01:20:45,940
And that's precisely
the dimension

1188
01:20:45,940 --> 01:20:49,760
of G Newton in the d plus
1 dimensional [INAUDIBLE]

1189
01:20:49,760 --> 01:20:50,770
of space time.

1190
01:20:50,770 --> 01:20:52,850
And then this is a
dimensionless number.

1191
01:20:52,850 --> 01:20:54,930
OK, because we did it
the SA with dimensions.

1192
01:20:59,350 --> 01:21:01,640
So you can say this is
a very difficult guess.

1193
01:21:01,640 --> 01:21:07,950
You can also say this is a very
simple guess, because clearly

1194
01:21:07,950 --> 01:21:10,560
this formula is motivated by
the black hole entropy formula.

1195
01:21:13,930 --> 01:21:16,110
You may say, ah,
black hole entropy

1196
01:21:16,110 --> 01:21:18,570
is something divided 4GN.

1197
01:21:18,570 --> 01:21:21,260
If it is entangled
entropy, it's also entropy.

1198
01:21:21,260 --> 01:21:24,250
Maybe it's also
something divided by 4GN.

1199
01:21:24,250 --> 01:21:26,710
And maybe it's some
surface divided by 4GN.

1200
01:21:26,710 --> 01:21:30,300
And then the special surface
would be the minimal surface.

1201
01:21:30,300 --> 01:21:33,630
So in some sense, it's
a very naive guess.

1202
01:21:33,630 --> 01:21:36,610
And you could say it's
a very simple guess.

1203
01:21:36,610 --> 01:21:39,330
But as I said, it's also
a very difficult guess,

1204
01:21:39,330 --> 01:21:42,030
because essentially,
other than that, you don't

1205
01:21:42,030 --> 01:21:44,230
have other starting points.

1206
01:21:44,230 --> 01:21:44,960
Yes?

1207
01:21:44,960 --> 01:21:46,840
AUDIENCE: So [INAUDIBLE].

1208
01:21:46,840 --> 01:21:49,570
So we have A in the CFT.

1209
01:21:49,570 --> 01:21:53,515
Where do we know exactly
where to place A in the AdS?

1210
01:21:53,515 --> 01:21:55,790
Because it's not--

1211
01:21:55,790 --> 01:21:58,076
HONG LIU: This is
a good question.

1212
01:21:58,076 --> 01:22:00,010
This is what I'm
going to say next.

1213
01:22:00,010 --> 01:22:02,370
Yeah, but let me
just emphasize that.

1214
01:22:02,370 --> 01:22:04,657
So when we talk
about A, we're always

1215
01:22:04,657 --> 01:22:06,240
talking about the
constant time slice,

1216
01:22:06,240 --> 01:22:10,230
because you have to
specify some time.

1217
01:22:10,230 --> 01:22:12,240
And actually, when we
consider the ground state

1218
01:22:12,240 --> 01:22:14,480
or typical state does
not depend on time.

1219
01:22:14,480 --> 01:22:17,180
And it actually does not
matter which slice we choose.

1220
01:22:17,180 --> 01:22:18,860
You just choose a time slice.

1221
01:22:18,860 --> 01:22:21,064
And then you specify
region A, because to talk

1222
01:22:21,064 --> 01:22:22,480
about degrees of
freedom, you only

1223
01:22:22,480 --> 01:22:24,380
talk about it in
the spatial section.

1224
01:22:29,720 --> 01:22:33,040
So if you have a ground state
or have any state which does not

1225
01:22:33,040 --> 01:22:35,140
depend on time, then
the gravity side

1226
01:22:35,140 --> 01:22:36,400
also does not depend on time.

1227
01:22:36,400 --> 01:22:38,440
It's a time-dependent geometry.

1228
01:22:38,440 --> 01:22:41,190
So time slicing in the
field series naturally

1229
01:22:41,190 --> 01:22:43,310
extends in the gravity side.

1230
01:22:43,310 --> 01:22:45,480
So in the gravity
side, this surface

1231
01:22:45,480 --> 01:22:48,252
would be in the constant
time slicing, which

1232
01:22:48,252 --> 01:22:49,460
would go to the gravity side.

1233
01:22:52,080 --> 01:22:54,320
AUDIENCE: Professor,
one more thing.

1234
01:22:54,320 --> 01:22:57,980
With the CFT and
AdS-- I know we like

1235
01:22:57,980 --> 01:23:00,840
to think one existing on
the boundary of the other.

1236
01:23:00,840 --> 01:23:04,765
But how does-- that's sort
of an abstraction, right?

1237
01:23:04,765 --> 01:23:06,640
HONG LIU: No, that you
should really consider

1238
01:23:06,640 --> 01:23:09,300
as a real thing-- real stuff.

1239
01:23:09,300 --> 01:23:14,050
Yeah, if you just think
of that as an abstraction,

1240
01:23:14,050 --> 01:23:16,770
then you miss a
lot of intuition,

1241
01:23:16,770 --> 01:23:19,170
because you use that
as a genuine boundary,

1242
01:23:19,170 --> 01:23:20,870
will give you a
lot of intuition.

1243
01:23:20,870 --> 01:23:24,997
And many things
become very natural.

1244
01:23:24,997 --> 01:23:26,330
Many things become very natural.

1245
01:23:26,330 --> 01:23:27,775
Yes?

1246
01:23:27,775 --> 01:23:29,775
AUDIENCE: So comparing
to the other formula that

1247
01:23:29,775 --> 01:23:33,160
also has an area
of the boundary,

1248
01:23:33,160 --> 01:23:35,877
all this tells us
is just it helps

1249
01:23:35,877 --> 01:23:38,130
us count the number of
degrees of freedom in the CFT.

1250
01:23:38,130 --> 01:23:39,546
So it all it tells
us is basically

1251
01:23:39,546 --> 01:23:40,837
the epsilon in the denominator?

1252
01:23:40,837 --> 01:23:44,990
HONG LIU: No, no, they have a
complete different dimension.

1253
01:23:44,990 --> 01:23:47,476
No, this one higher dimension.

1254
01:23:47,476 --> 01:23:49,542
AUDIENCE: So that's
another strange--

1255
01:23:49,542 --> 01:23:51,250
HONG LIU: No, this is
not strange at all,

1256
01:23:51,250 --> 01:23:53,200
because they're not
supposed to be the same.

1257
01:23:53,200 --> 01:23:56,050
No, this is the
way it should be.

1258
01:23:56,050 --> 01:23:57,690
So this is d minus
1 dimensional.

1259
01:23:57,690 --> 01:24:00,540
And that is the d
minus 2 dimensional.

1260
01:24:00,540 --> 01:24:02,785
It's just completely different.

1261
01:24:02,785 --> 01:24:05,130
AUDIENCE: Well, right,
but they are not-- there

1262
01:24:05,130 --> 01:24:07,972
is something the
same about them.

1263
01:24:07,972 --> 01:24:10,250
HONG LIU: No, there is
something to same about them.

1264
01:24:10,250 --> 01:24:12,910
But these two have
completely different physics.

1265
01:24:12,910 --> 01:24:15,310
So this has a close analog
with black hole physics.

1266
01:24:18,884 --> 01:24:20,800
Don't think they're
motivated by that formula.

1267
01:24:20,800 --> 01:24:23,250
I think they're motivated
by the black hole formula.

1268
01:24:23,250 --> 01:24:25,420
And people also try to connect
that formula to the black hole

1269
01:24:25,420 --> 01:24:25,730
formula.

1270
01:24:25,730 --> 01:24:26,659
That's another story.

1271
01:24:26,659 --> 01:24:27,742
Anyway, it doesn't matter.

1272
01:24:30,290 --> 01:24:31,140
OK, good.

1273
01:24:31,140 --> 01:24:32,050
So this is a guess.

1274
01:24:32,050 --> 01:24:33,300
Yes?

1275
01:24:33,300 --> 01:24:36,830
AUDIENCE: So is this result for
the ground state on the CFT?

1276
01:24:36,830 --> 01:24:38,500
HONG LIU: No, this is valid.

1277
01:24:38,500 --> 01:24:42,570
Yeah, for the
time-independent state.

1278
01:24:42,570 --> 01:24:45,533
The eigenstate is not
invoked with time.

1279
01:24:45,533 --> 01:24:47,116
AUDIENCE: The thing
I don't understand

1280
01:24:47,116 --> 01:24:50,374
is on the site of the
CFT, there is this state

1281
01:24:50,374 --> 01:24:51,915
that you did as an
input to calculate

1282
01:24:51,915 --> 01:24:53,890
the entanglement entropy.

1283
01:24:53,890 --> 01:24:56,850
And what's the corresponding
thing on the AdS side?

1284
01:24:56,850 --> 01:24:59,120
HONG LIU: The AdS side is
the corresponding geometry.

1285
01:24:59,120 --> 01:25:02,710
So it's as said before, each
state on the field theory

1286
01:25:02,710 --> 01:25:04,440
should correspond
to some geometry

1287
01:25:04,440 --> 01:25:07,050
with normalizable modes.

1288
01:25:07,050 --> 01:25:09,510
Yeah, so it's the
corresponding geometry.

1289
01:25:09,510 --> 01:25:13,810
And the ground state, then
this will be just pure ideas.

1290
01:25:13,810 --> 01:25:16,130
And if you look at the
final temperature state,

1291
01:25:16,130 --> 01:25:18,190
then this will be
the black hole.

1292
01:25:18,190 --> 01:25:20,960
And then if you are able to
construct some other geometry

1293
01:25:20,960 --> 01:25:24,640
corresponding to some other
state, you can use that.

1294
01:25:24,640 --> 01:25:29,170
AUDIENCE: How about this entropy
on independent on which state

1295
01:25:29,170 --> 01:25:30,950
we choose for CFT?

1296
01:25:30,950 --> 01:25:33,400
HONG LIU: No, of
course it depends.

1297
01:25:33,400 --> 01:25:35,700
No this formula does not.

1298
01:25:35,700 --> 01:25:37,500
But the geometry does.

1299
01:25:37,500 --> 01:25:39,380
The geometry does.

1300
01:25:39,380 --> 01:25:44,020
This is a minimal surface in
whatever [INAUDIBLE] geometry.

1301
01:25:44,020 --> 01:25:47,040
AUDIENCE: Does the region--
after we choose the region,

1302
01:25:47,040 --> 01:25:48,350
we can choose any state--

1303
01:25:48,350 --> 01:25:49,532
HONG LIU: Yes, yes.

1304
01:25:49,532 --> 01:25:50,990
AUDIENCE: And as
a minimal surface,

1305
01:25:50,990 --> 01:25:52,965
is it independent on
which state we choose?

1306
01:25:52,965 --> 01:25:55,090
HONG LIU: Of course, it
would depend on which state

1307
01:25:55,090 --> 01:25:58,050
you choose, because
each state corresponds

1308
01:25:58,050 --> 01:25:59,480
to a different geometry.

1309
01:25:59,480 --> 01:26:03,370
And the minimal surface
depends on your geometry.

1310
01:26:03,370 --> 01:26:06,540
So each state-- so
remember previously,

1311
01:26:06,540 --> 01:26:10,480
the state on the
CFT side is mapped

1312
01:26:10,480 --> 01:26:15,110
to the geometry with
normalizable boundary

1313
01:26:15,110 --> 01:26:17,740
conditions with
normalizable modes.

1314
01:26:17,740 --> 01:26:20,120
Yeah, anyway, just let
me say just geometry.

1315
01:26:20,120 --> 01:26:21,500
So a state with the geometry.

1316
01:26:21,500 --> 01:26:24,060
If you have a different state,
you have a different geometry.

1317
01:26:24,060 --> 01:26:26,670
But by definition,
all this geometry

1318
01:26:26,670 --> 01:26:28,430
would be asymptotic AdS.

1319
01:26:28,430 --> 01:26:32,450
So in a boundary, they
should all be asymptotic AdS.

1320
01:26:32,450 --> 01:26:38,180
And so that what means to
have [INAUDIBLE] mode excited.

1321
01:26:38,180 --> 01:26:39,270
OK?

1322
01:26:39,270 --> 01:26:39,770
Yes?

1323
01:26:39,770 --> 01:26:41,700
AUDIENCE: So this
entropy formula

1324
01:26:41,700 --> 01:26:45,540
takes into account the entropy
of gravitational excitations

1325
01:26:45,540 --> 01:26:47,470
in AdS, if you can
put it that way.

1326
01:26:47,470 --> 01:26:48,760
But black hole entropy--

1327
01:26:48,760 --> 01:26:55,480
HONG LIU: No, let's try
not to jump too fast.

1328
01:26:55,480 --> 01:27:00,230
And I think you are trying
to extrapolate too fast.

1329
01:27:00,230 --> 01:27:03,290
This formula is
supposed to calculate

1330
01:27:03,290 --> 01:27:06,330
the entangled entropy
in the field series

1331
01:27:06,330 --> 01:27:09,900
side in the [INAUDIBLE]
limit and the notch

1332
01:27:09,900 --> 01:27:12,330
lambda toward coupling limit.

1333
01:27:12,330 --> 01:27:15,160
And we can talk about the
gravitational interpretation

1334
01:27:15,160 --> 01:27:15,770
later.

1335
01:27:15,770 --> 01:27:18,830
But this formula is about
the field series entanglement

1336
01:27:18,830 --> 01:27:19,330
entropy.

1337
01:27:19,330 --> 01:27:19,829
OK?

1338
01:27:22,280 --> 01:27:22,780
Good.

1339
01:27:22,780 --> 01:27:24,350
Any other questions?

1340
01:27:27,250 --> 01:27:30,190
OK, good.

1341
01:27:30,190 --> 01:27:33,160
So there are many support of it.

1342
01:27:33,160 --> 01:27:34,900
So this is a rule of the game.

1343
01:27:34,900 --> 01:27:36,510
You make a guess.

1344
01:27:36,510 --> 01:27:39,390
Then you do a check.

1345
01:27:39,390 --> 01:27:44,480
If the check worked, then
that gives you confidence,

1346
01:27:44,480 --> 01:27:46,910
and then you do another check.

1347
01:27:46,910 --> 01:27:49,727
And here you find it will fail.

1348
01:27:49,727 --> 01:27:51,560
And if you do a sufficient
number of checks,

1349
01:27:51,560 --> 01:27:52,852
then people believed you.

1350
01:27:52,852 --> 01:27:54,060
Then people will believe you.

1351
01:27:54,060 --> 01:27:56,020
And everybody
started checking it.

1352
01:27:56,020 --> 01:27:59,730
And then sooner or
later we will see

1353
01:27:59,730 --> 01:28:03,450
whether this fails or works.

1354
01:28:03,450 --> 01:28:07,882
Anyway, so this is nice because
this is simple to compute,

1355
01:28:07,882 --> 01:28:10,090
because finding a minimal
surface on the gravity side

1356
01:28:10,090 --> 01:28:18,220
is in some sense, it's a
straightforward a mathematical

1357
01:28:18,220 --> 01:28:20,655
problem conceptually,
even though technically it

1358
01:28:20,655 --> 01:28:21,800
may not be simple.

1359
01:28:21,800 --> 01:28:25,210
But it's straightforward
conceptually.

1360
01:28:25,210 --> 01:28:29,747
Anyway, so you can use these
to calculate many quantities.

1361
01:28:32,300 --> 01:28:42,180
Let me just say in
support of this--

1362
01:28:42,180 --> 01:28:45,540
let me just say it in
words just to save time.

1363
01:28:45,540 --> 01:28:47,430
So the first thing
you should check

1364
01:28:47,430 --> 01:28:51,710
is that this satisfies the
subadditive condition which

1365
01:28:51,710 --> 01:28:53,440
we wrote down earlier,
because that's

1366
01:28:53,440 --> 01:28:55,360
a very non-trivial condition.

1367
01:28:55,360 --> 01:28:57,370
It should be satisfied
by the entropy.

1368
01:28:57,370 --> 01:29:01,500
So first, you should check that,
because if you violate that,

1369
01:29:01,500 --> 01:29:04,840
than this proposal is gone.

1370
01:29:04,840 --> 01:29:06,840
And then there are
other things that you

1371
01:29:06,840 --> 01:29:10,680
can try to use to reproduce
all the known behavior we know

1372
01:29:10,680 --> 01:29:12,920
about entangled
entropy, for example,

1373
01:29:12,920 --> 01:29:16,890
this behavior and this
behavior-- this area along.

1374
01:29:21,470 --> 01:29:24,120
And then we actually
build up confidence

1375
01:29:24,120 --> 01:29:26,250
by checking all
those known results.

1376
01:29:26,250 --> 01:29:28,950
And then you can try
to derive [INAUDIBLE]

1377
01:29:28,950 --> 01:29:32,300
and to see whether [INAUDIBLE]
makes physical sense.

1378
01:29:32,300 --> 01:29:36,500
So essentially,
that's how it works.

1379
01:29:36,500 --> 01:29:39,620
And this is nice also
at the tactical level,

1380
01:29:39,620 --> 01:29:43,530
because entangled entropy is
something very hard to compute.

1381
01:29:43,530 --> 01:29:44,990
I don't know.

1382
01:29:44,990 --> 01:29:48,110
You guys may not have experience
of calculating entangled

1383
01:29:48,110 --> 01:29:48,930
entropy.

1384
01:29:48,930 --> 01:29:50,650
But ask Frank Wilczek.

1385
01:29:50,650 --> 01:29:52,500
He was one of the
first few people

1386
01:29:52,500 --> 01:29:54,530
to do it in some
free field theory.

1387
01:29:54,530 --> 01:29:58,260
Even for free field
theory, people could mostly

1388
01:29:58,260 --> 01:30:00,330
do it in 1 plus 1 dimension.

1389
01:30:00,330 --> 01:30:04,540
Going beyond 1 plus 1 dimension
free scalar field theory in 2

1390
01:30:04,540 --> 01:30:07,540
plus 1 dimension
3 plus 1 dimension

1391
01:30:07,540 --> 01:30:10,480
for some simple region
like a circle or sphere

1392
01:30:10,480 --> 01:30:12,810
you have to do a
numerical calculation.

1393
01:30:12,810 --> 01:30:15,310
You have to discretize
the field theory

1394
01:30:15,310 --> 01:30:17,500
to do very massive
numerical calculation.

1395
01:30:17,500 --> 01:30:19,890
It's not easy to compute.

1396
01:30:19,890 --> 01:30:21,220
It's not easy to compute.

1397
01:30:21,220 --> 01:30:25,210
But this guy is actually easy
to compute in comparison.

1398
01:30:25,210 --> 01:30:27,940
So this guy at the
technical level

1399
01:30:27,940 --> 01:30:32,245
provides a huge convenience
to calculate many things.

1400
01:30:34,950 --> 01:30:36,740
Yeah, so this is a side remark.

1401
01:30:36,740 --> 01:30:42,540
So now let's just do
some calculations.

1402
01:30:42,540 --> 01:30:44,290
So first, let me show
that this actually

1403
01:30:44,290 --> 01:30:45,723
satisfies strong subaddivity.

1404
01:30:56,850 --> 01:31:03,380
So remember, the formula
we had before is S(AC),

1405
01:31:03,380 --> 01:31:04,880
unfortunately I erased it.

1406
01:31:13,710 --> 01:31:16,430
OK, so this is one
of the inequalities.

1407
01:31:16,430 --> 01:31:18,910
So we are just drawing
it 1 plus 1 dimension

1408
01:31:18,910 --> 01:31:20,950
because it's easy to draw.

1409
01:31:20,950 --> 01:31:23,590
But a similar thing can
be easily generalized

1410
01:31:23,590 --> 01:31:25,180
in a higher dimension.

1411
01:31:25,180 --> 01:31:28,330
So let's look at just the
line in 1 plus 1 dimension.

1412
01:31:28,330 --> 01:31:31,335
So this can call
this region A. This

1413
01:31:31,335 --> 01:31:34,460
is region C. This is region B.

1414
01:31:34,460 --> 01:31:36,335
You can also make them separate.

1415
01:31:36,335 --> 01:31:37,820
It doesn't matter.

1416
01:31:37,820 --> 01:31:39,500
You can also make them separate.

1417
01:31:39,500 --> 01:31:41,530
It's easy to work out.

1418
01:31:41,530 --> 01:31:45,060
So I'm just giving
you one cases.

1419
01:31:45,060 --> 01:31:49,270
So suppose a minimal service
for the AC reading is like this.

1420
01:31:49,270 --> 01:31:50,960
And the minimal
surface the BC reading

1421
01:31:50,960 --> 01:31:54,370
would be like this-- so
minimal surface like that.

1422
01:31:57,950 --> 01:32:03,980
So let me call this curve gamma
AC and this curve gamma BC.

1423
01:32:03,980 --> 01:32:10,290
OK, I could not find
the colored chalk today.

1424
01:32:10,290 --> 01:32:14,070
So I did not have
the colored chalk.

1425
01:32:14,070 --> 01:32:17,447
Let me see whether
this chalk works.

1426
01:32:17,447 --> 01:32:18,530
Is this the colored chalk?

1427
01:32:18,530 --> 01:32:19,030
Yes.

1428
01:32:21,740 --> 01:32:24,440
So now, let me define
two other surfaces.

1429
01:32:24,440 --> 01:32:36,436
Let me call this one gamma
1 and this one gamma 2.

1430
01:32:36,436 --> 01:32:38,810
In principle, I should use
two different chalks for that.

1431
01:32:43,720 --> 01:32:47,240
You think it looks
like something?

1432
01:32:47,240 --> 01:32:52,850
OK, anyway, so we have
gamma AC plus gamma

1433
01:32:52,850 --> 01:32:57,240
BC equals to gamma
1 plus gamma 2

1434
01:32:57,240 --> 01:33:02,730
because-- gamma AC-- this
is length-- gamma BC is

1435
01:33:02,730 --> 01:33:03,980
equal to gamma 1 plus gamma 2.

1436
01:33:12,820 --> 01:33:17,000
And then just from
the definition

1437
01:33:17,000 --> 01:33:18,840
of the minimal
surface, then the gamma

1438
01:33:18,840 --> 01:33:23,700
1 must be greater than gamma
C in terms of the length

1439
01:33:23,700 --> 01:33:27,360
because the minimal
surface associated with C

1440
01:33:27,360 --> 01:33:30,740
must be the minimal area.

1441
01:33:30,740 --> 01:33:33,320
It must be smaller than gamma 1.

1442
01:33:33,320 --> 01:33:39,410
Under the area of gamma 2 must
be greater than gamma ABC,

1443
01:33:39,410 --> 01:33:44,850
because the gamma 2 is
the boundary is the ABC.

1444
01:33:44,850 --> 01:33:48,060
And the gamma 2 must be
greater than gamma BC

1445
01:33:48,060 --> 01:33:50,310
because this is supposed
to be the minimal surface.

1446
01:33:54,200 --> 01:33:59,310
So now we conclude that
gamma AC plus gamma

1447
01:33:59,310 --> 01:34:05,730
BC is greater than
gamma ABC plus gamma C.

1448
01:34:05,730 --> 01:34:08,070
And then this translates
into that formula.

1449
01:34:08,070 --> 01:34:08,830
OK?

1450
01:34:08,830 --> 01:34:15,362
Very simple, and elegant proof--
very simple and elegant proof.

1451
01:34:15,362 --> 01:34:17,070
And now you can prove
another inequality.

1452
01:34:20,546 --> 01:34:21,920
You can prove
another inequality,

1453
01:34:21,920 --> 01:34:29,675
which is S(AC) plus S(BC)
greater than S(A) and S(B).

1454
01:34:32,410 --> 01:34:37,700
So now, I will define
another surface.

1455
01:34:37,700 --> 01:34:39,640
OK?

1456
01:34:39,640 --> 01:34:47,860
So now, let me just
redraw it-- A, C, B.

1457
01:34:47,860 --> 01:34:50,000
So I have AC like this.

1458
01:34:50,000 --> 01:34:51,980
I have BC like this.

1459
01:34:51,980 --> 01:35:02,216
And now, let me call this one
gamma 1 tilde-- this one gamma

1460
01:35:02,216 --> 01:35:02,893
2 tilde.

1461
01:35:12,410 --> 01:35:14,590
This is annoying.

1462
01:35:14,590 --> 01:35:15,460
OK.

1463
01:35:15,460 --> 01:35:20,420
And again, gamma AB plus
gamma AC plus gamma BC

1464
01:35:20,420 --> 01:35:26,670
equals to gamma 1 tilde
plus gamma 2 tilde.

1465
01:35:26,670 --> 01:35:30,790
So the gamma 1 tilde
ends in A. Gamma

1466
01:35:30,790 --> 01:35:32,950
2 tilde ends in B.
That means gamma

1467
01:35:32,950 --> 01:35:38,150
1 tilde must be greater than
gamma A and gamma 2 tilde

1468
01:35:38,150 --> 01:35:41,310
must be greater
than gamma B. OK?

1469
01:35:41,310 --> 01:35:43,500
So now you show that
gamma AC plus gamma

1470
01:35:43,500 --> 01:35:49,630
BC is greater than gamma A plus
gamma B. So this is very easy.

1471
01:35:49,630 --> 01:35:52,020
So this is a very
simple and elegant.

1472
01:35:52,020 --> 01:35:58,250
So if you want to really
to be impressed by this,

1473
01:35:58,250 --> 01:36:03,640
I urge you to look at the
proof of the strong subaddivity

1474
01:36:03,640 --> 01:36:10,210
itself in say in some textbook
or in the original papers.

1475
01:36:10,210 --> 01:36:12,220
It's actually
highly non-trivial.

1476
01:36:12,220 --> 01:36:15,350
You need some double,
double concave functions.

1477
01:36:15,350 --> 01:36:19,900
It's quite a height involved.

1478
01:36:19,900 --> 01:36:27,550
OK, so this is a
great confidence boost

1479
01:36:27,550 --> 01:36:30,345
that satisfies the strong
subadditivity condition.

1480
01:36:36,372 --> 01:36:37,830
So now, let's look
at the last one.

1481
01:36:37,830 --> 01:36:39,620
Let's try to reproduce
this formula.

1482
01:36:39,620 --> 01:36:41,190
Let's try to do a
simple calculation

1483
01:36:41,190 --> 01:36:42,680
to reproduce this formula.

1484
01:36:42,680 --> 01:36:46,880
So this formula is supposed
to be true for any CFT.

1485
01:36:46,880 --> 01:36:54,020
This should be also applied
for the holographic CFT

1486
01:36:54,020 --> 01:36:56,185
and to check whether
this formula works.

1487
01:36:59,310 --> 01:37:05,940
The reason you would like to
choose an example which is new.

1488
01:37:05,940 --> 01:37:07,639
But this example
is the simplest.

1489
01:37:07,639 --> 01:37:09,430
So that's the reason
I choose this example,

1490
01:37:09,430 --> 01:37:13,590
even though it's just
reproduced all the results.

1491
01:37:13,590 --> 01:37:18,680
So let's now look at the
entangled entropy of 1

1492
01:37:18,680 --> 01:37:19,982
plus 1 dimensional CFT.

1493
01:37:24,390 --> 01:37:27,850
So here, now you
have a CFT 1 plus 1

1494
01:37:27,850 --> 01:37:29,150
should be dual to the AdS3.

1495
01:37:33,660 --> 01:37:36,330
You will do some theory in AdS3.

1496
01:37:36,330 --> 01:37:41,460
So the S squared-- so let's
only look at the vacuum

1497
01:37:41,460 --> 01:37:44,650
then we just work with AdS3.

1498
01:37:44,650 --> 01:37:46,299
So let me write down
the AdS3 metric.

1499
01:37:51,950 --> 01:37:53,803
So dx is the
boundary directions.

1500
01:37:56,760 --> 01:37:59,140
So as I said, each
CFT is characterized

1501
01:37:59,140 --> 01:38:00,090
by a central charge.

1502
01:38:04,230 --> 01:38:05,690
And you can obtain
a central charge

1503
01:38:05,690 --> 01:38:07,720
from various different ways.

1504
01:38:07,720 --> 01:38:11,170
And again, from the
holographic, you

1505
01:38:11,170 --> 01:38:12,710
can try to express
the single charge

1506
01:38:12,710 --> 01:38:14,506
in terms of gravity quantities.

1507
01:38:14,506 --> 01:38:17,130
And there is many ways to derive
it that we will not go through

1508
01:38:17,130 --> 01:38:18,570
with that calculation.

1509
01:38:18,570 --> 01:38:21,330
So let me just write
down the results.

1510
01:38:21,330 --> 01:38:28,330
So far the holographic
CFTs, the single charge

1511
01:38:28,330 --> 01:38:31,000
is related to the
gravity quantity

1512
01:38:31,000 --> 01:38:34,160
by 3R divided by 2 G Newton.

1513
01:38:36,740 --> 01:38:38,860
So this is a result
which I just caught.

1514
01:38:38,860 --> 01:38:42,330
I will not derive it for you.

1515
01:38:42,330 --> 01:38:45,090
But you can derive it
in many different ways.

1516
01:38:45,090 --> 01:38:50,150
So for CFT, to do
a gravity idea 3,

1517
01:38:50,150 --> 01:38:52,360
its central charge
is given by 3R,

1518
01:38:52,360 --> 01:38:56,820
which is the AdS radius divided
by 2 times the Newton constant.

1519
01:38:56,820 --> 01:38:59,760
And the Newton constant
in three dimensions

1520
01:38:59,760 --> 01:39:00,980
has mass dimension 1.

1521
01:39:00,980 --> 01:39:04,110
So this is a
dimensionless number--

1522
01:39:04,110 --> 01:39:06,850
a dimensionless number.

1523
01:39:06,850 --> 01:39:10,694
So now let's calculate
the entanglement entropy

1524
01:39:10,694 --> 01:39:12,116
using that formula.

1525
01:39:15,750 --> 01:39:18,260
So this now reduced
your calculation

1526
01:39:18,260 --> 01:39:23,770
very similar to our
Wilson loop calculation.

1527
01:39:23,770 --> 01:39:25,340
So let's call this
L divided by 2.

1528
01:39:25,340 --> 01:39:27,320
So lets call this X direction.

1529
01:39:27,320 --> 01:39:29,040
So this is a V direction.

1530
01:39:29,040 --> 01:39:30,760
So this is L divided by 2.

1531
01:39:30,760 --> 01:39:32,050
This is minus L divided by 2.

1532
01:39:32,050 --> 01:39:34,780
This region is A.
Then we need to find

1533
01:39:34,780 --> 01:39:40,470
a curve, because this now 1 plus
1 dimensions-- one dimensions.

1534
01:39:40,470 --> 01:39:44,130
So we need to find the curve
which end on this segment of A.

1535
01:39:44,130 --> 01:39:46,340
OK?

1536
01:39:46,340 --> 01:39:47,090
So is this clear?

1537
01:39:49,650 --> 01:39:51,380
So this is the x direction.

1538
01:39:51,380 --> 01:39:55,530
And so this is 0, x equals to 0.

1539
01:39:55,530 --> 01:39:57,350
Yes?

1540
01:39:57,350 --> 01:39:58,690
OK.

1541
01:39:58,690 --> 01:40:04,150
So what we need to find is find
the minimal surface connecting

1542
01:40:04,150 --> 01:40:07,270
these two points-- a minimal
curve-- a minimal length

1543
01:40:07,270 --> 01:40:08,862
curve connecting
these two points.

1544
01:40:11,960 --> 01:40:15,180
So it's as we said
before, we should

1545
01:40:15,180 --> 01:40:18,780
look at the constant time slice
because this is a vacuum which

1546
01:40:18,780 --> 01:40:19,720
is time independent.

1547
01:40:19,720 --> 01:40:21,810
Let's look at the
constant time slice.

1548
01:40:21,810 --> 01:40:24,240
The constant time
slice, the metric

1549
01:40:24,240 --> 01:40:27,440
becomes-- so in this
space, the metric becomes r

1550
01:40:27,440 --> 01:40:30,414
square divided by z square
because the time does not

1551
01:40:30,414 --> 01:40:30,913
change.

1552
01:40:34,490 --> 01:40:40,850
So now if I treat the x as a
function of z, and then these

1553
01:40:40,850 --> 01:40:41,630
will become.

1554
01:40:41,630 --> 01:40:43,200
So this curve lets
me parametrize

1555
01:40:43,200 --> 01:40:44,760
x as a function of z.

1556
01:40:44,760 --> 01:40:48,140
Then this just becomes r
square divided by z square 1

1557
01:40:48,140 --> 01:40:52,110
plus x prime square dz square.

1558
01:40:56,400 --> 01:41:01,040
And so this is dl square.

1559
01:41:01,040 --> 01:41:02,910
So the length of
the curve square

1560
01:41:02,910 --> 01:41:05,150
would be just
parametrized by this guy.

1561
01:41:05,150 --> 01:41:05,650
OK?

1562
01:41:10,630 --> 01:41:13,310
And we have to satisfy
the boundary condition--

1563
01:41:13,310 --> 01:41:17,340
by symmetry, we only need to
consider the right stuff OK,

1564
01:41:17,340 --> 01:41:18,990
because it's symmetric.

1565
01:41:18,990 --> 01:41:21,820
So we impose the boundary
condition at the x

1566
01:41:21,820 --> 01:41:23,420
z equal to 0.

1567
01:41:23,420 --> 01:41:24,990
It's L divided by 2.

1568
01:41:27,885 --> 01:41:29,220
We can see the right half.

1569
01:41:42,655 --> 01:41:45,155
So now you can just write down
the entanglement entropy now.

1570
01:41:52,110 --> 01:41:55,000
So S(A) will be 1 over 4GN.

1571
01:41:58,250 --> 01:42:01,270
So I integrate
half of the curve.

1572
01:42:01,270 --> 01:42:03,537
Let me just call
here z is equal to 0.

1573
01:42:03,537 --> 01:42:04,620
Let me call this point z0.

1574
01:42:07,610 --> 01:42:12,790
So I just integrate
from 0 to z0.

1575
01:42:12,790 --> 01:42:17,300
Z0 we will find out when you
find out this minimal length

1576
01:42:17,300 --> 01:42:17,800
curve.

1577
01:42:17,800 --> 01:42:22,480
So this one will be r/z dz.

1578
01:42:22,480 --> 01:42:25,320
Then just taking the
square root of this guy--

1579
01:42:25,320 --> 01:42:27,680
the square of 1
over x prime square.

1580
01:42:27,680 --> 01:42:31,500
So this is the length
and the times 2,

1581
01:42:31,500 --> 01:42:34,530
because we only can see
that this is half of it.

1582
01:42:38,520 --> 01:42:41,110
And now you extremize it.

1583
01:42:41,110 --> 01:42:44,010
Then you find the
minimal length curve.

1584
01:42:44,010 --> 01:42:45,510
And then you plug
the solution here.

1585
01:42:45,510 --> 01:42:47,096
Then you find the action.

1586
01:42:47,096 --> 01:42:48,720
Actually we don't
need to extremize it,

1587
01:42:48,720 --> 01:42:50,920
because this is a
well-known problem.

1588
01:42:50,920 --> 01:42:53,450
And this is called
the Poincare disk.

1589
01:42:53,450 --> 01:42:57,160
So this is a prototype of a
two-dimensional hyperbolic

1590
01:42:57,160 --> 01:42:58,390
space.

1591
01:42:58,390 --> 01:43:03,003
And the minimal length curve
inside your hyperbolic space

1592
01:43:03,003 --> 01:43:07,949
is maybe a 16th century problem.

1593
01:43:07,949 --> 01:43:08,740
Yeah, I don't know.

1594
01:43:08,740 --> 01:43:10,800
Actually, maybe 17th century.

1595
01:43:10,800 --> 01:43:13,990
Anyway, so the answer
is actually very long.

1596
01:43:13,990 --> 01:43:19,150
But you can also easily
work out yourself.

1597
01:43:19,150 --> 01:43:22,880
So the result is actually
this is just exactly a circle.

1598
01:43:22,880 --> 01:43:24,300
OK?

1599
01:43:24,300 --> 01:43:25,650
Exactly a half circle.

1600
01:43:31,120 --> 01:43:32,800
So that's actually
a half circle.

1601
01:43:32,800 --> 01:43:35,260
That means that x equals 2.

1602
01:43:35,260 --> 01:43:37,712
The half circle with a
radius L divided by 2.

1603
01:43:37,712 --> 01:43:39,520
So this has just
become L squared

1604
01:43:39,520 --> 01:43:41,280
divided by 4 minus d square.

1605
01:43:44,590 --> 01:43:48,030
In particular, z0 corresponding
to the point which

1606
01:43:48,030 --> 01:43:51,360
x equal to 0 just L divided 2.

1607
01:43:54,740 --> 01:43:59,000
So now you just
plug this into here.

1608
01:43:59,000 --> 01:44:00,900
You just plug this here.

1609
01:44:00,900 --> 01:44:05,250
Let me just write it a
little bit fast, because we

1610
01:44:05,250 --> 01:44:08,660
are a little bit out of time.

1611
01:44:08,660 --> 01:44:12,090
So you plug this in.

1612
01:44:12,090 --> 01:44:14,080
Then you can scale the z out.

1613
01:44:14,080 --> 01:44:18,619
You can scale z to be scaled
L divided by 2 factors out.

1614
01:44:18,619 --> 01:44:19,660
And let me just scale it.

1615
01:44:19,660 --> 01:44:21,730
You can rewrite the
integral as follows.

1616
01:44:21,730 --> 01:44:25,820
So after the scaling
it's say 2z scale

1617
01:44:25,820 --> 01:44:30,020
with L divided by 2z
to do the scaling.

1618
01:44:30,020 --> 01:44:33,380
Then you find this
expression becomes 2r/4GN.

1619
01:44:37,420 --> 01:44:40,840
And the upper limit becomes 1.

1620
01:44:40,840 --> 01:44:43,032
And the lower limit you see.

1621
01:44:43,032 --> 01:44:44,990
When you plug this in,
actually the lower limit

1622
01:44:44,990 --> 01:44:47,970
is divergent because
there's 1 over z here.

1623
01:44:47,970 --> 01:44:49,710
As always, we need
to put the cutoff.

1624
01:44:49,710 --> 01:44:53,300
And also, this divergence
is expected because of below

1625
01:44:53,300 --> 01:44:55,700
that here it depends
on short distance

1626
01:44:55,700 --> 01:44:57,900
cutoff it would be divergent.

1627
01:44:57,900 --> 01:45:00,110
So we need to for the
short-distance cutoff here.

1628
01:45:00,110 --> 01:45:04,430
So let me put it here epsilon.

1629
01:45:04,430 --> 01:45:07,895
And this will be 2
epsilon divided by L

1630
01:45:07,895 --> 01:45:10,360
when I do this scaling.

1631
01:45:10,360 --> 01:45:15,880
And then you have
dz/z 1 minus d square.

1632
01:45:15,880 --> 01:45:18,420
So the integral reduces
to something like this.

1633
01:45:18,420 --> 01:45:21,900
And now you can evaluate
this integral trivially.

1634
01:45:21,900 --> 01:45:28,040
Then you find the epsilon
goes to the limit.

1635
01:45:28,040 --> 01:45:35,280
Then you find the one term
survives 2r divided by 4GN log

1636
01:45:35,280 --> 01:45:36,500
L divided by epsilon.

1637
01:45:39,360 --> 01:45:41,420
So now let's try
to rewrite in terms

1638
01:45:41,420 --> 01:45:46,290
of this C-- we
rewrite in terms of C.

1639
01:45:46,290 --> 01:45:55,920
So this is 1/3 3r divided by
2GN log L divided by epsilon.

1640
01:45:55,920 --> 01:45:59,924
And so this is precisely
that formula C divided by 3

1641
01:45:59,924 --> 01:46:03,830
log L of epsilon.

1642
01:46:03,830 --> 01:46:07,030
And with that, you just count
from the minimal surface

1643
01:46:07,030 --> 01:46:08,565
in the hyperbolic
space is a circle.

1644
01:46:12,050 --> 01:46:15,320
And it takes a lot of effort
to calculate this thing,

1645
01:46:15,320 --> 01:46:19,622
even to do a free field
series calculation.

1646
01:46:19,622 --> 01:46:21,330
But in gravity you
can do it very easily.

1647
01:46:24,490 --> 01:46:27,310
So now let me just
quickly mention

1648
01:46:27,310 --> 01:46:31,430
what happens if you do
at a finite temperature.

1649
01:46:31,430 --> 01:46:34,610
So you can also do this
at a finite temperature.

1650
01:46:38,924 --> 01:46:41,340
So since we're out of time,
we'll be doing it a little bit

1651
01:46:41,340 --> 01:46:42,950
fast.

1652
01:46:42,950 --> 01:46:47,230
So first doing it at a
finite temperature is easier.

1653
01:46:47,230 --> 01:46:53,510
Let me just say finite T.
And then I will connect

1654
01:46:53,510 --> 01:46:55,600
to the black hole entropy.

1655
01:46:55,600 --> 01:46:59,189
I will show that
this formula actually

1656
01:46:59,189 --> 01:47:01,230
reduces to the black hole
entropy in some limits.

1657
01:47:15,709 --> 01:47:17,250
So for these problems,
actually, it's

1658
01:47:17,250 --> 01:47:19,306
easier to consider
CFT on the circle.

1659
01:47:26,700 --> 01:47:27,906
OK.

1660
01:47:27,906 --> 01:47:29,280
So when the CFT
is on the circle,

1661
01:47:29,280 --> 01:47:31,030
the boundary will be
a circle because this

1662
01:47:31,030 --> 01:47:34,070
is 1 plus 1 dimension.

1663
01:47:34,070 --> 01:47:37,320
So remember, in
the global ideas,

1664
01:47:37,320 --> 01:47:40,750
it's like sitting
there in the boundary.

1665
01:47:40,750 --> 01:47:43,360
And in the 1 plus 1 dimensional
case and the boundary

1666
01:47:43,360 --> 01:47:44,390
is just really a circle.

1667
01:47:44,390 --> 01:47:46,860
And this is the bark

1668
01:47:46,860 --> 01:47:50,560
And a finite temperature,
you put the black hole there.

1669
01:47:50,560 --> 01:47:52,670
Let me just put the
black hole here.

1670
01:47:52,670 --> 01:47:53,572
OK?

1671
01:47:53,572 --> 01:47:54,925
Just put the black hole there.

1672
01:47:57,820 --> 01:48:01,390
So now you can ask the
following question.

1673
01:48:01,390 --> 01:48:04,990
So now let's consider
some reaching A. So

1674
01:48:04,990 --> 01:48:08,200
let's consider A is very small.

1675
01:48:08,200 --> 01:48:10,160
Yo can work out the
minimal surface.

1676
01:48:10,160 --> 01:48:13,250
So it acts the same as
our Wilson loop story.

1677
01:48:13,250 --> 01:48:14,880
So you find a minimal surface.

1678
01:48:14,880 --> 01:48:17,660
But if A is sufficiently
small, then the geometry around

1679
01:48:17,660 --> 01:48:21,130
here is still AdS.

1680
01:48:21,130 --> 01:48:23,370
And then you just
get a minimal surface

1681
01:48:23,370 --> 01:48:26,030
around here-- so a
small deformation

1682
01:48:26,030 --> 01:48:28,810
from the vacuum behavior.

1683
01:48:28,810 --> 01:48:31,320
So we'll make A
larger and larger.

1684
01:48:34,140 --> 01:48:40,750
So this z0 depends on L. If
we make L larger and larger,

1685
01:48:40,750 --> 01:48:46,070
this is z0 will be
deeper in the bark.

1686
01:48:46,070 --> 01:48:49,752
So if you make A larger, so
this will go more to the bark

1687
01:48:49,752 --> 01:48:51,460
and then will be
deformed, and then we'll

1688
01:48:51,460 --> 01:48:54,574
see the black hole geometry.

1689
01:48:54,574 --> 01:48:55,990
But now to answer
the question how

1690
01:48:55,990 --> 01:49:02,410
about if I make A to be
larger-- this region to be

1691
01:49:02,410 --> 01:49:08,110
A to be larger than the half
of the size of the circle.

1692
01:49:08,110 --> 01:49:11,960
So the minimal surface
should be able to be

1693
01:49:11,960 --> 01:49:14,050
to be smoothly
deformed back into A.

1694
01:49:14,050 --> 01:49:15,800
So now you have to do
something like this.

1695
01:49:15,800 --> 01:49:19,338
It turns out the minimal surface
want to go around the horizon

1696
01:49:19,338 --> 01:49:20,796
and then doing
something like this.

1697
01:49:24,940 --> 01:49:27,065
AUDIENCE: Why don't you do
it the other way around?

1698
01:49:27,065 --> 01:49:29,116
HONG LIU: No, it's
because the other won't.

1699
01:49:29,116 --> 01:49:31,750
It's because A is like this.

1700
01:49:31,750 --> 01:49:34,580
And if you do
something like this,

1701
01:49:34,580 --> 01:49:37,110
then it can be
deformed back into A.

1702
01:49:37,110 --> 01:49:39,160
It's mostly without
crossing the black hole.

1703
01:49:39,160 --> 01:49:42,200
AUDIENCE: Yes, or it could be
like a phase jump or something

1704
01:49:42,200 --> 01:49:44,854
at some point when one
surface becomes longer than

1705
01:49:44,854 --> 01:49:45,710
[INAUDIBLE].

1706
01:49:45,710 --> 01:49:49,750
HONG LIU: No, in this
situation, it's not that thing.

1707
01:49:49,750 --> 01:49:51,090
It's something like this.

1708
01:49:51,090 --> 01:49:57,040
Now, if you make A even
longer, eventually,

1709
01:49:57,040 --> 01:49:58,997
let's take the A even longer.

1710
01:49:58,997 --> 01:50:00,580
And that would be
something like this.

1711
01:50:03,730 --> 01:50:06,270
Yeah, because you always
want to hug near the horizon.

1712
01:50:06,270 --> 01:50:09,284
And eventually, we'll
make A to be tiny,

1713
01:50:09,284 --> 01:50:10,950
then it will become
something like this.

1714
01:50:10,950 --> 01:50:13,260
You have something
close on the horizon.

1715
01:50:13,260 --> 01:50:14,996
Then you have
something like this.

1716
01:50:14,996 --> 01:50:16,370
It's roughly
something like this.

1717
01:50:21,530 --> 01:50:25,421
Yeah, is it clear to you?

1718
01:50:25,421 --> 01:50:26,170
It doesn't matter.

1719
01:50:26,170 --> 01:50:28,142
[LAUGHTER]

1720
01:50:34,560 --> 01:50:38,810
So now let's consider--
anyway, yeah,

1721
01:50:38,810 --> 01:50:40,100
let me just draw this again.

1722
01:50:40,100 --> 01:50:41,929
It actually does matter.

1723
01:50:41,929 --> 01:50:42,428
Sorry.

1724
01:50:47,930 --> 01:50:50,870
So let's take A
to be very small.

1725
01:50:50,870 --> 01:50:52,150
Let's take A to be a very big.

1726
01:50:52,150 --> 01:50:54,890
This part will be very small.

1727
01:50:54,890 --> 01:50:59,070
So we are finally we
will go like this.

1728
01:50:59,070 --> 01:51:03,080
Anyway, something like this
when you hug the black hole.

1729
01:51:03,080 --> 01:51:05,360
And eventually, when
these two points shrink,

1730
01:51:05,360 --> 01:51:08,370
so take A with the total region.

1731
01:51:08,370 --> 01:51:10,880
Take A to be the total space.

1732
01:51:10,880 --> 01:51:12,460
What do you get?

1733
01:51:12,460 --> 01:51:16,710
Then what you see that the
minimal surface essentially

1734
01:51:16,710 --> 01:51:19,490
is just essentially a surface
hugger on the horizon.

1735
01:51:27,100 --> 01:51:29,350
And this is a precise
black hole formula,

1736
01:51:29,350 --> 01:51:31,640
because then this is a
imprecise a black hole formula.

1737
01:51:31,640 --> 01:51:34,400
And this is a situation you
expect it to be black hole

1738
01:51:34,400 --> 01:51:46,540
formula, because if you take
the A to be the host space,

1739
01:51:46,540 --> 01:51:50,776
then rho A is just equal to
the rho of the whole system.

1740
01:51:53,580 --> 01:51:58,020
And the S(A) by definition
is just equal to the S

1741
01:51:58,020 --> 01:52:00,310
of the whole system.

1742
01:52:00,310 --> 01:52:04,280
And this is just a thermal
density, a thermal entropy.

1743
01:52:04,280 --> 01:52:07,500
And this is given by
the black hole formula.

1744
01:52:07,500 --> 01:52:10,080
It's given by the
black hole formula.

1745
01:52:10,080 --> 01:52:12,470
And this is given
by [INAUDIBLE].

1746
01:52:12,470 --> 01:52:14,382
So you find in these
special cases, actually,

1747
01:52:14,382 --> 01:52:15,840
you recover the
black hole formula.

1748
01:52:19,480 --> 01:52:21,467
So I have some small things.

1749
01:52:23,580 --> 01:52:24,830
We can talk a little bit more.

1750
01:52:24,830 --> 01:52:26,610
But I think we'll skip it.

1751
01:52:26,610 --> 01:52:30,840
You can also show that actually
for general dimensions,

1752
01:52:30,840 --> 01:52:33,180
you always recover the area
more without the details

1753
01:52:33,180 --> 01:52:38,380
of the geometry, because no
matter what kind of dimension

1754
01:52:38,380 --> 01:52:41,000
you look at, the
typical state which will

1755
01:52:41,000 --> 01:52:42,430
be normalizable to geometry.

1756
01:52:42,430 --> 01:52:45,360
When you approach the AdS--
when you approach the boundary,

1757
01:52:45,360 --> 01:52:48,330
it's always like AdS.

1758
01:52:48,330 --> 01:52:50,010
And then you find in
a higher dimension,

1759
01:52:50,010 --> 01:52:52,394
it's always like this.

1760
01:52:52,394 --> 01:52:53,810
It says, when this
minimal surface

1761
01:52:53,810 --> 01:52:56,500
is close to the boundary,
it becomes perpendicular

1762
01:52:56,500 --> 01:52:58,406
to the boundary.

1763
01:52:58,406 --> 01:52:59,780
Here is a circle,
then of course,

1764
01:52:59,780 --> 01:53:01,238
it's perpendicular
to the boundary.

1765
01:53:01,238 --> 01:53:02,870
But you see this
feature is actually

1766
01:53:02,870 --> 01:53:06,930
generalized to higher
dimensions-- to any dimensions.

1767
01:53:06,930 --> 01:53:12,210
So what you see is that
in the general dimension,

1768
01:53:12,210 --> 01:53:14,751
a minimal surface will be
just near the boundary will

1769
01:53:14,751 --> 01:53:16,917
be just perpendicular to
going down from everywhere.

1770
01:53:19,306 --> 01:53:20,680
And then from
there, you can show

1771
01:53:20,680 --> 01:53:25,640
that you always have the
area law from this behavior.

1772
01:53:28,190 --> 01:53:30,540
You always have the area law.

1773
01:53:30,540 --> 01:53:37,010
So this can give you an
exercise for yourself.

1774
01:53:37,010 --> 01:53:38,980
Yeah, so let me just
finish by saying

1775
01:53:38,980 --> 01:53:42,490
a couple of philosophical
remarks, and then we'll finish.

1776
01:53:48,680 --> 01:53:54,620
So this RT formula
is really remarkable,

1777
01:53:54,620 --> 01:53:58,110
because it's actually,
as I said, technically

1778
01:53:58,110 --> 01:54:02,390
provides a very simple way to
calculate entangled entropy.

1779
01:54:02,390 --> 01:54:09,670
But actually, conceptually,
it's even more profound

1780
01:54:09,670 --> 01:54:15,830
because it tells you that
the space time is related

1781
01:54:15,830 --> 01:54:21,950
to the entanglement, because
it simulates something which

1782
01:54:21,950 --> 01:54:28,000
is very subtle from the field
theory side, because this S(A),

1783
01:54:28,000 --> 01:54:30,480
as we said, you have to do
some complicated procedure

1784
01:54:30,480 --> 01:54:33,400
to trace out degrees of
freedom-- take the log.

1785
01:54:33,400 --> 01:54:36,060
And that turns out to be
related to the geometry

1786
01:54:36,060 --> 01:54:37,292
in a very simple way.

1787
01:54:37,292 --> 01:54:39,500
So essentially, you see that
entanglement-- it's just

1788
01:54:39,500 --> 01:54:41,110
captured by the space time.

1789
01:54:41,110 --> 01:54:43,130
And in particular,
it's something

1790
01:54:43,130 --> 01:54:45,660
you can say the
geometry is essentially

1791
01:54:45,660 --> 01:54:47,035
equal to the
quantum information.

1792
01:54:56,020 --> 01:54:59,130
So this is a very exciting
also from the gravity

1793
01:54:59,130 --> 01:55:02,470
perspective, because if they
ask you if you really understand

1794
01:55:02,470 --> 01:55:05,930
this duality, then you can
actually using the quantum

1795
01:55:05,930 --> 01:55:10,610
information of the field theory
to understand the subtle things

1796
01:55:10,610 --> 01:55:13,050
about the geometry.

1797
01:55:13,050 --> 01:55:14,906
It just tells you the
geometry is actually

1798
01:55:14,906 --> 01:55:16,280
equal to the
quantum information.

1799
01:55:19,660 --> 01:55:27,570
So for many years, actually not
that many years, so previously,

1800
01:55:27,570 --> 01:55:32,160
we have people doing
quantum information.

1801
01:55:32,160 --> 01:55:36,250
And we have people doing
this matter, which is

1802
01:55:36,250 --> 01:55:41,390
the quantum many-body system.

1803
01:55:41,390 --> 01:55:45,870
And this matter, or QCD, so this
is kind of quantum many-body.

1804
01:55:49,470 --> 01:55:55,462
And then we have people doing
black hole, string theory,

1805
01:55:55,462 --> 01:55:56,128
and the gravity.

1806
01:56:00,170 --> 01:56:04,490
So they are all very
different fields.

1807
01:56:04,490 --> 01:56:08,454
But now, they're all connected
by this holographic duality.

1808
01:56:12,050 --> 01:56:14,500
Somehow essentially,
they connect all of them.

1809
01:56:14,500 --> 01:56:17,440
So now, we essentially
just see a unified picture,

1810
01:56:17,440 --> 01:56:23,570
a unified paradigm, to
understand all quantum systems,

1811
01:56:23,570 --> 01:56:27,710
including quantum systems
without gravity, with gravity.

1812
01:56:27,710 --> 01:56:29,480
So I will stop here.

1813
01:56:29,480 --> 01:56:31,930
[APPLAUSE]