1
00:00:00,670 --> 00:00:03,370
BARTON ZWIEBACH: Let's talk
now about entanglement.

2
00:00:11,330 --> 00:00:14,220
So we talk about
entanglement when we have

3
00:00:14,220 --> 00:00:17,620
two non-interacting particles.

4
00:00:17,620 --> 00:00:20,860
You don't need a strong
interaction between particles

5
00:00:20,860 --> 00:00:22,780
to produce entanglement,
the particles

6
00:00:22,780 --> 00:00:25,480
can be totally non-interacting.

7
00:00:25,480 --> 00:00:32,575
Suppose particle 1 can be
in any of these states--

8
00:00:35,500 --> 00:00:38,840
u 1, u 2.

9
00:00:38,840 --> 00:00:42,980
Let's assume just u 1 and u 2.

10
00:00:42,980 --> 00:00:49,130
And particle 2 can be
in states v 1 and v 2.

11
00:00:55,000 --> 00:00:57,700
And you have these two
particles flying around,

12
00:00:57,700 --> 00:01:00,410
these are possible
states of particle 1

13
00:01:00,410 --> 00:01:02,740
and possible states
of particle 2.

14
00:01:02,740 --> 00:01:06,010
Now you want to describe
the full system, the quantum

15
00:01:06,010 --> 00:01:09,680
state of the two particles.

16
00:01:09,680 --> 00:01:11,420
States of the two particles.

17
00:01:14,740 --> 00:01:16,480
Two particles.

18
00:01:19,800 --> 00:01:25,420
Well, it seems reasonable
that to describe

19
00:01:25,420 --> 00:01:28,420
the state of the two particles
that are not interacting,

20
00:01:28,420 --> 00:01:31,870
I should tell you what
particle 1 is doing

21
00:01:31,870 --> 00:01:34,910
and what particle 2 is doing.

22
00:01:34,910 --> 00:01:38,440
OK, so particle 1
could be doing this.

23
00:01:38,440 --> 00:01:41,360
Could be u 1.

24
00:01:41,360 --> 00:01:47,420
And particle 2
could be doing v 1.

25
00:01:51,600 --> 00:01:55,610
And in a sense, by
telling you that,

26
00:01:55,610 --> 00:01:57,860
we've said what
everything is doing.

27
00:01:57,860 --> 00:02:01,410
Particle 1 is doing u 1,
particle 2 is doing u 2.

28
00:02:01,410 --> 00:02:07,280
And mathematically, we like
to make this look like a state

29
00:02:07,280 --> 00:02:10,100
and we want to write
it in a coherent way.

30
00:02:10,100 --> 00:02:14,430
And we sort of multiply
these two things,

31
00:02:14,430 --> 00:02:18,750
but we must say
sort of multiply,

32
00:02:18,750 --> 00:02:21,110
because this strange
multiplication, this,

33
00:02:21,110 --> 00:02:24,860
you know, we think of
them as vectors or states,

34
00:02:24,860 --> 00:02:26,560
so how do you multiply states?

35
00:02:26,560 --> 00:02:29,150
So you put something
called the tensor

36
00:02:29,150 --> 00:02:32,405
product, a little
multiplication like this.

37
00:02:35,610 --> 00:02:42,720
So you could say, don't worry,
it's kind of like a product,

38
00:02:42,720 --> 00:02:44,400
and it's the way we do it.

39
00:02:44,400 --> 00:02:48,300
We don't move things across,
the first state here,

40
00:02:48,300 --> 00:02:50,700
the second state here, and
that's a possible state.

41
00:02:53,460 --> 00:02:57,060
Now, I could have
a different state.

42
00:02:57,060 --> 00:03:02,550
Because particle 1, in
fact, could be doing

43
00:03:02,550 --> 00:03:04,920
something a little different.

44
00:03:04,920 --> 00:03:12,660
Could be doing alpha 1
u 1 plus alpha 2 u 2,

45
00:03:12,660 --> 00:03:21,930
and maybe particle 2 is doing
beta 1 v 1 plus beta 2 v 2.

46
00:03:25,910 --> 00:03:28,520
And this would be all right.

47
00:03:28,520 --> 00:03:31,850
I'm telling you what particle
1 is doing and I'm telling you

48
00:03:31,850 --> 00:03:36,420
what particle 2 is
doing and the rules

49
00:03:36,420 --> 00:03:40,940
of tensor multiplication or
this kind of multiplication

50
00:03:40,940 --> 00:03:44,100
to combine those states
are just like a product,

51
00:03:44,100 --> 00:03:48,750
except that as I said, you
never move the states across.

52
00:03:48,750 --> 00:03:54,660
So you just distribute, so
you have alpha 2, beta 1,

53
00:03:54,660 --> 00:04:01,290
the number goes out, u 1 v 1--

54
00:04:01,290 --> 00:04:04,310
that's the first factor--

55
00:04:04,310 --> 00:04:23,000
plus alpha 1 beta 2 u 1 v 2
plus alpha 2 beta 1 u 2 v 1

56
00:04:23,000 --> 00:04:33,120
plus alpha 2 beta 2 u 2 v 2.

57
00:04:33,120 --> 00:04:35,640
I think I got it right.

58
00:04:35,640 --> 00:04:36,420
Let me know.

59
00:04:39,820 --> 00:04:43,400
I just multiplied and
got the numbers out.

60
00:04:43,400 --> 00:04:48,350
The numbers can be move
out across this product.

61
00:04:48,350 --> 00:04:53,390
OK, so that's a state and that's
a superposition of states,

62
00:04:53,390 --> 00:05:03,410
so actually, I could try to
write a different state now.

63
00:05:03,410 --> 00:05:06,550
You see, we're just
experimenting, but here

64
00:05:06,550 --> 00:05:09,430
is another state.

65
00:05:09,430 --> 00:05:19,240
u 1 v 1 plus u 2 v 2.

66
00:05:25,090 --> 00:05:32,700
Now this is a state that
actually seems different.

67
00:05:32,700 --> 00:05:34,940
Quite different.

68
00:05:34,940 --> 00:05:38,030
Because I don't seem
to be able to say

69
00:05:38,030 --> 00:05:42,080
that what particle
1 is doing and what

70
00:05:42,080 --> 00:05:45,645
particle 2 is doing separately.

71
00:05:45,645 --> 00:05:49,820
You see, I can say when
particle 1 is doing u 1,

72
00:05:49,820 --> 00:05:52,180
particle 2 is doing v 1.

73
00:05:52,180 --> 00:05:56,810
And if when particle 1 is
doing u 2, this is v 2.

74
00:05:56,810 --> 00:06:00,470
But can I write
this as some state

75
00:06:00,470 --> 00:06:05,660
of the first particle times some
state of the second particle?

76
00:06:10,190 --> 00:06:14,490
Well, let's see.

77
00:06:14,490 --> 00:06:17,800
Maybe I can and can
write it in this form.

78
00:06:17,800 --> 00:06:22,050
This is the most general
state that you can say,

79
00:06:22,050 --> 00:06:26,940
particle 1 is doing this,
and particle 2 is doing that.

80
00:06:26,940 --> 00:06:28,360
So can they do that?

81
00:06:28,360 --> 00:06:32,860
Well, I can compare these
two terms with those

82
00:06:32,860 --> 00:06:39,360
and they conclude that
alpha 1 beta 1 must be 1.

83
00:06:45,410 --> 00:06:51,870
Alpha 2 beta 2 must be also 1.

84
00:06:51,870 --> 00:07:00,880
But no cross products exist,
so alpha 1 beta 2 must be 0

85
00:07:00,880 --> 00:07:06,380
and alpha 2 beta 1 must be 0.

86
00:07:06,380 --> 00:07:10,130
And that's a problem
because either alpha 1

87
00:07:10,130 --> 00:07:13,330
is 0, which is
inconsistent, or beta 2

88
00:07:13,330 --> 00:07:15,830
is 0, which is
inconsistent with that,

89
00:07:15,830 --> 00:07:20,240
so now this state
is un-factorizable.

90
00:07:22,840 --> 00:07:28,510
It's a funny state in which you
cannot say that this quantum

91
00:07:28,510 --> 00:07:31,930
state can be described by
telling what the first particle

92
00:07:31,930 --> 00:07:34,810
is doing and what the
second particle is doing.

93
00:07:34,810 --> 00:07:37,510
What the first particle is
doing depends on the second

94
00:07:37,510 --> 00:07:40,540
and what the second is
doing depends on the first.

95
00:07:40,540 --> 00:07:43,391
This is an entangled state.

96
00:07:57,650 --> 00:08:03,800
And then we can build entangled
states and our very strange

97
00:08:03,800 --> 00:08:05,240
states.

98
00:08:05,240 --> 00:08:08,210
So with two
particles with spins,

99
00:08:08,210 --> 00:08:14,150
for example, we can build
an entangled states of 2

100
00:08:14,150 --> 00:08:18,160
spin 1/2 particles.

101
00:08:23,820 --> 00:08:26,580
And this state could
look like this--

102
00:08:26,580 --> 00:08:32,370
the first particle is up along
z and the second particle is

103
00:08:32,370 --> 00:08:37,940
down along z, plus
a particle that

104
00:08:37,940 --> 00:08:43,900
is down along z for
the first particle,

105
00:08:43,900 --> 00:08:46,480
but the second is up along z.

106
00:08:51,200 --> 00:08:53,410
And these are 2
spin 1/2 particles

107
00:08:53,410 --> 00:08:57,920
and in the usual notation,
these experiments in quantum

108
00:08:57,920 --> 00:09:00,470
mechanics and
black hole physics,

109
00:09:00,470 --> 00:09:03,490
people speak of Alice and Bob.

110
00:09:03,490 --> 00:09:07,690
Alice has one particle,
Bob has the other particle.

111
00:09:07,690 --> 00:09:12,120
Maybe Alice is in the
moon and has her electron

112
00:09:12,120 --> 00:09:15,950
and Bob is on earth
and has his electron,

113
00:09:15,950 --> 00:09:20,450
and the two electrons, one
on the earth and in the moon,

114
00:09:20,450 --> 00:09:23,490
are in this state.

115
00:09:23,490 --> 00:09:30,240
So then we say that Alice and
Bob share an entangled pair.

116
00:09:30,240 --> 00:09:33,790
And all kinds of
strange things happen.

117
00:09:33,790 --> 00:09:37,350
People can do those things
in the lab-- not quite one

118
00:09:37,350 --> 00:09:39,330
in the earth and
one in the moon,

119
00:09:39,330 --> 00:09:43,500
but one photon at one
place and another photon

120
00:09:43,500 --> 00:09:46,950
entangled with it at
100 kilometers away,

121
00:09:46,950 --> 00:09:50,220
that's pretty doable.

122
00:09:50,220 --> 00:09:56,160
And they are in this funny state
in which their properties are

123
00:09:56,160 --> 00:10:01,160
currently that in
surprising ways.

124
00:10:01,160 --> 00:10:03,220
So what happens here?

125
00:10:03,220 --> 00:10:06,790
Suppose Alice
goes-- or let's say

126
00:10:06,790 --> 00:10:11,320
Bob goes along and
measures his spin

127
00:10:11,320 --> 00:10:13,900
and he finds his spin down.

128
00:10:13,900 --> 00:10:18,520
So-- oh, you look here,
oh, here is down for Bob.

129
00:10:18,520 --> 00:10:22,990
So at this moment, the whole
state collapses into this.

130
00:10:22,990 --> 00:10:26,320
Because up with Bob
didn't get realized.

131
00:10:26,320 --> 00:10:30,880
So once Bob measures
and he finds down,

132
00:10:30,880 --> 00:10:32,540
the whole state goes into this.

133
00:10:32,540 --> 00:10:37,360
So if Alice-- on the moon
or in another galaxy--

134
00:10:37,360 --> 00:10:40,780
at that instant
looks at her spin,

135
00:10:40,780 --> 00:10:45,250
she will find it's
up before light

136
00:10:45,250 --> 00:10:47,940
has had time to get there.

137
00:10:47,940 --> 00:10:50,150
Instantaneously.

138
00:10:50,150 --> 00:10:53,260
It will go into this state.

139
00:10:53,260 --> 00:10:58,670
People were sure somehow this
violates special relativity.

140
00:10:58,670 --> 00:11:00,210
It doesn't.

141
00:11:00,210 --> 00:11:02,555
You somehow when you think
about this carefully,

142
00:11:02,555 --> 00:11:04,720
you can't quite
send information,

143
00:11:04,720 --> 00:11:07,210
but the collapse
is instantaneous

144
00:11:07,210 --> 00:11:09,090
in quantum mechanics.

145
00:11:09,090 --> 00:11:12,910
Somehow, Bob and Alice cannot
communicate information

146
00:11:12,910 --> 00:11:16,990
by sharing this entangled pair,
but it's an interesting thing

147
00:11:16,990 --> 00:11:19,860
why it cannot happen.

148
00:11:19,860 --> 00:11:22,920
Einstein again objected to this.

149
00:11:22,920 --> 00:11:26,180
And he said, this
is a fake thing.

150
00:11:29,430 --> 00:11:30,980
You guys are going to share--

151
00:11:30,980 --> 00:11:34,590
and now, of course, they have
to share many entangled pairs

152
00:11:34,590 --> 00:11:38,300
to do experiments, so maybe
1,000 entangled pairs.

153
00:11:38,300 --> 00:11:44,170
And Einstein would say, no,
that's not what's happening.

154
00:11:44,170 --> 00:11:49,400
What's happening is that some of
your entangled pairs are this.

155
00:11:49,400 --> 00:11:55,580
That is, Bob is down, Alice
is up, some of them are this--

156
00:11:55,580 --> 00:11:58,560
and there's no such thing
as this entanglement

157
00:11:58,560 --> 00:12:01,970
and indeed, if you
measure and you find down,

158
00:12:01,970 --> 00:12:04,830
she will find up, and if
you measure and you find up,

159
00:12:04,830 --> 00:12:10,750
she will find down, and there's
nothing too mysterious here.

160
00:12:10,750 --> 00:12:17,050
But then came John Bell in
1964 and discovered his Bell

161
00:12:17,050 --> 00:12:22,210
inequalities that demonstrated
that if Alice and Bob can

162
00:12:22,210 --> 00:12:25,720
measure in three
different directions,

163
00:12:25,720 --> 00:12:30,550
they will find correlations
that are impossible to explain

164
00:12:30,550 --> 00:12:32,680
with classical physics.

165
00:12:32,680 --> 00:12:36,640
It took a lot of originally
for Bell to discover this,

166
00:12:36,640 --> 00:12:39,040
that you have to measure
in three directions,

167
00:12:39,040 --> 00:12:42,280
and therefore, the
kind of correlations

168
00:12:42,280 --> 00:12:48,130
that appear in entangled states
are very subtle and pretty

169
00:12:48,130 --> 00:12:51,060
difficult to disentangle.

170
00:12:51,060 --> 00:12:57,160
So that's why entanglement
is a very peculiar subject.

171
00:12:57,160 --> 00:13:01,250
People think about it a lot
because it's very mysterious.

172
00:13:01,250 --> 00:13:04,720
It somehow violates
classical notions,

173
00:13:04,720 --> 00:13:07,890
but in a very subtle way.