qm6.eng.srt

1
00:00:03,352 --> 00:00:07,091
Stanford University

2
00:00:11,384 --> 00:00:13,850
There was an issue that came up last time

3
00:00:14,104 --> 00:00:15,351
I didn't understand the question

4
00:00:15,560 --> 00:00:17,066
There was a question that kept coming up

5
00:00:17,682 --> 00:00:21,326
And I didn't understand what it was until the very end of the class

6
00:00:23,149 --> 00:00:24,701
It was an interesting and a good question

7
00:00:25,153 --> 00:00:27,801
So I wanna go over it because it is important

8
00:00:28,866 --> 00:00:33,153
It had to do with a spin in a magnetic field emitting a photon

9
00:00:36,274 --> 00:00:39,180
And I maintained that if it emits a photon

10
00:00:39,600 --> 00:00:42,704
Remember the, the way we set things up

11
00:00:43,089 --> 00:00:48,261
We set let's take the Hamiltonian to be omega over two

12
00:00:48,687 --> 00:00:49,952
Let's say sigma z

13
00:00:51,365 --> 00:00:55,527
That has two eigenvalues, plus omega over two and minus omega over two

14
00:00:55,888 --> 00:00:59,837
So the two eigenvalues let's call it E_1 the energy

15
00:01:01,157 --> 00:01:05,917
The lower of the two eigenvalues is minus omega over two 

16
00:01:06,185 --> 00:01:13,537
And the upper on is, let's call it E minus and E plus 

17
00:01:14,869 --> 00:01:16,276
This is plus omega over two 

18
00:01:17,484 --> 00:01:20,632
So there was two energy levels, if we draw an energy level diagram

19
00:01:21,320 --> 00:01:24,903
Energy, we will have two eigenvalues, it would be zero over here

20
00:01:25,086 --> 00:01:26,442
But that would not be one of them

21
00:01:26,982 --> 00:01:29,222
But then we would have one over here, omega over two

22
00:01:30,182 --> 00:01:31,253
And minus omega over two 

23
00:01:33,634 --> 00:01:34,964
Alright what could happen under such 

24
00:01:35,429 --> 00:01:38,089
We have not studied this process here yet 

25
00:01:38,774 --> 00:01:40,237
But let's talk about it for a moment 

26
00:01:41,067 --> 00:01:44,408
Given two states like this, it is possible for an atom or a spin

27
00:01:44,877 --> 00:01:48,436
Or a system with energy levels like this to emit a photon

28
00:01:49,585 --> 00:01:52,323
Emit a photon and go from the upper one to the lower one 

29
00:01:52,953 --> 00:01:57,254
If it's in the upper, if it's in the lower state obviously it cannot emit a photon

30
00:01:58,567 --> 00:01:59,891
Doesn't have the energy to do it

31
00:02:00,852 --> 00:02:02,980
But if it's in an upper state it can emit a photon

32
00:02:04,026 --> 00:02:07,102
And that photon will have the energy difference 

33
00:02:07,577 --> 00:02:08,797
between the upper and the lower state 

34
00:02:09,019 --> 00:02:12,342
So we will have an energy, of frequency, of energy omega 

35
00:02:13,686 --> 00:02:15,613
Sorry not frequency, it will be frequency 

36
00:02:16,011 --> 00:02:18,752
But more important, it will have energy of omega

37
00:02:19,534 --> 00:02:22,632
There is an h bar missing, I have been setting h bar equals to zero

38
00:02:23,015 --> 00:02:26,119
No, h bar equals one 

39
00:02:30,257 --> 00:02:32,127
I've been setting h bar equals to one 

40
00:02:33,608 --> 00:02:37,257
It's really h bar omega, but I will continue to set h bar equals to one

41
00:02:37,457 --> 00:02:42,918
Because we all grown up, we are not afraid to set h bar equals to one right

42
00:02:44,817 --> 00:02:45,476
What?

43
00:02:45,825 --> 00:02:48,139
>> It was zero last quarter

44
00:02:49,352 --> 00:02:52,675
it was zero last quarter, yeah, it was zero last quarter that's correct 

45
00:02:54,052 --> 00:02:59,316
Okay so we draw a picture of the photon transition and a photon being emitted

46
00:03:00,989 --> 00:03:07,787
Now somebody said what if the spin was not pointed up or down

47
00:03:08,638 --> 00:03:10,592
What if it pointed, here is the z axis 

48
00:03:10,840 --> 00:03:15,359
What if it pointed, let's say in this way along the x axis

49
00:03:16,500 --> 00:03:18,115
And then it emitted a photon

50
00:03:18,565 --> 00:03:23,159
Would it not emit the photon of half the energy namely omega over two

51
00:03:23,438 --> 00:03:27,348
The answer is no if would not emit a photon of a energy omega over two

52
00:03:28,699 --> 00:03:31,429
Let's discuss first what we are having classically

53
00:03:32,053 --> 00:03:36,845
Classically if we have a roller, a spinning roller in a magnetic field 

54
00:03:37,745 --> 00:03:42,575
That had an energy, which it was the typical energy would be the co..

55
00:03:43,070 --> 00:03:46,811
The cosine of the angle between the axis of the spin

56
00:03:48,269 --> 00:03:51,349
Here's the axis of the spin, here's the magnetic field 

57
00:03:53,178 --> 00:03:58,546
The energy of that rotating object would depend on the orientation of the spin

58
00:03:58,993 --> 00:04:00,073
Relative to the magnetic field

59
00:04:00,748 --> 00:04:02,067
The energy would be minimum down

60
00:04:02,732 --> 00:04:05,787
And it will gradually increase until it was up

61
00:04:06,540 --> 00:04:10,878
That's number one, number two classically and quantum mechanically 

62
00:04:11,753 --> 00:04:13,032
Classically the actual value, 

63
00:04:14,133 --> 00:04:16,542
and quantum mechanically the expectation value of the spin

64
00:04:17,375 --> 00:04:21,382
processes around the B axis with frequency omega 

65
00:04:22,728 --> 00:04:24,530
That also happens classically 

66
00:04:25,850 --> 00:04:33,625
It's in the classical mechanics notes in the Poisson brackets section

67
00:04:35,079 --> 00:04:38,129
Where we study exactly that question

68
00:04:39,061 --> 00:04:42,237
It wasn't called spin, it was called angular momentum, the spinning top

69
00:04:43,915 --> 00:04:44,837
Does exactly the same thing 

70
00:04:45,678 --> 00:04:49,210
And that spinning top if it really is a little electro magnet

71
00:04:50,302 --> 00:04:56,727
A little magnetic system rotating, it will radiate 

72
00:04:58,106 --> 00:05:00,838
In the process of going around with frequency omega 

73
00:05:01,690 --> 00:05:07,960
It will emit radiation, at what frequency? Omega 

74
00:05:09,290 --> 00:05:12,277
It will emit, radiation at the

75
00:05:12,911 --> 00:05:14,618
And how much radiation would it emit?

76
00:05:14,942 --> 00:05:16,582
Now it's classical it's not emitting photon 

77
00:05:16,953 --> 00:05:21,152
Emitting classical radiation, and it will gradually, as it emit radiation

78
00:05:22,338 --> 00:05:25,881
Fall over until it gets down and then no more radiation

79
00:05:26,266 --> 00:05:28,365
How much energy would it emit all together?

80
00:05:30,025 --> 00:05:35,091
And the answer depend on the angle that it started at

81
00:05:35,612 --> 00:05:41,097
If it starts up it would emit all together an amount of energy that is proportional 

82
00:05:41,637 --> 00:05:46,674
essentially omega, if it's pointing down it will emit no energy

83
00:05:47,511 --> 00:05:50,699
If it's pointing horizontally, it would emit, let's say omega over two

84
00:05:52,021 --> 00:05:58,106
So it would be a continuously possible value of emission of radiation at the same frequency

85
00:05:59,101 --> 00:06:04,006
Quantum mechanically it can only emit a photon with 

86
00:06:04,506 --> 00:06:06,812
whatever that energy the photon has

87
00:06:07,787 --> 00:06:12,042
And the answer is, it's related but different answer 

88
00:06:12,919 --> 00:06:19,008
If it emits, it will emit a photon of energy omega

89
00:06:21,028 --> 00:06:24,808
On the other hand the probability of whatever it emits is not one necessarily

90
00:06:25,390 --> 00:06:33,740
And in fact let's suppose the initial state is some alpha up plus beta down

91
00:06:35,912 --> 00:06:41,326
If alpha and beta are equal, that corresponds to oriented along the x axis

92
00:06:43,016 --> 00:06:47,013
If alpha is one and beta is zero, that's pointing up

93
00:06:48,146 --> 00:06:51,444
If alpha is zero and beta is one, that's pointing down

94
00:06:54,230 --> 00:07:01,561
And up electron, or an up spin, will emit a photon of frequency omega

95
00:07:02,104 --> 00:07:06,121
Of energy omega, so if was just the up state, yes indeed

96
00:07:06,461 --> 00:07:08,051
It will emit omega

97
00:07:08,999 --> 00:07:11,216
If it was a down state, it emits nothing 

98
00:07:12,445 --> 00:07:14,733
If it starts at the superposition of the states 

99
00:07:15,450 --> 00:07:18,604
Then with probability alpha star alpha it emits

100
00:07:19,679 --> 00:07:22,677
And with probability beta star beta it doesn't emit 

101
00:07:24,114 --> 00:07:29,515
What is the average energy, given that it's a probability alpha star alpha 

102
00:07:29,738 --> 00:07:33,389
that emits with a photon of frequency omega, energy omega 

103
00:07:33,615 --> 00:07:36,938
And beta star beta that it emits nothing 

104
00:07:37,799 --> 00:07:40,621
What is the average energy, and the average energy 

105
00:07:40,912 --> 00:07:42,511
is the same as the classical energy

106
00:07:44,122 --> 00:07:46,642
The average energy is the same as the classical energy 

107
00:07:47,261 --> 00:07:52,324
But each individual event emits a photon of either 

108
00:07:53,274 --> 00:07:57,518
Either it emits no photon or it emits a photon of energy omega 

109
00:07:58,369 --> 00:08:00,085
So that was, who asked the question?

110
00:08:00,323 --> 00:08:01,169
Somebody here asked a question 

111
00:08:01,466 --> 00:08:06,014
So that's the upshot

112
00:08:08,932 --> 00:08:11,690
And this is sort of the internal consistency of quantum mechanics

113
00:08:11,913 --> 00:08:13,086
And of course to really, ya?

114
00:08:15,713 --> 00:08:22,346
>> If the electron will emit a photon when it's in the down energy

115
00:08:22,646 --> 00:08:24,711
>> presumably could avert the photon?

116
00:08:24,945 --> 00:08:28,065
Yes, that's correct

117
00:08:28,676 --> 00:08:31,892
>> How precise of the photon after B to omega 

118
00:08:33,908 --> 00:08:36,128
That depends on the life time of the state 

119
00:08:37,017 --> 00:08:40,222
And which in terms depend on how strongly coupled 

120
00:08:41,019 --> 00:08:47,783
the depends on the dipole moment, the stronger the dipole moment

121
00:08:48,246 --> 00:08:50,889
The magnetic moment, the faster it would emit

122
00:08:52,049 --> 00:08:54,547
The faster it would emit, it's another way the sort of life time

123
00:08:54,972 --> 00:08:56,013
the excited state will be 

124
00:08:57,095 --> 00:09:02,822
It's the inversed life time which set this scale of exactly what you are asking

125
00:09:03,047 --> 00:09:06,702
The precision with which you will have to be exactly 

126
00:09:09,599 --> 00:09:11,859
There is a spread in the energy level

127
00:09:12,266 --> 00:09:14,990
The spread in the energy level are uncertainty in the energy level

128
00:09:15,805 --> 00:09:22,002
And the uncertainty of the energy level is the inversed to the life time of the state

129
00:09:22,686 --> 00:09:26,553
So right, so you are absolutely right, it was a spread

130
00:09:27,254 --> 00:09:32,629
It's governed by the energy-time uncertainty principle

131
00:09:33,033 --> 00:09:36,637
But we are not there yet, whether we will get there I don't know

132
00:09:36,935 --> 00:09:37,492
Okay 

133
00:09:37,983 --> 00:09:41,588
>> Is this phenomenon of photon emission which use 

134
00:09:41,788 --> 00:09:43,714
>> to create an apparatus to detect it?

135
00:09:44,046 --> 00:09:44,676
What apparatus

136
00:09:45,123 --> 00:09:48,749
>> to create an apparatus that detects the spin of an electron 

137
00:09:49,162 --> 00:09:54,566
You could, that's one way to do it, that's one way to do it

138
00:09:55,162 --> 00:09:58,351
I decided to not think about that as an apparatus

139
00:09:58,708 --> 00:10:01,883
I didn't like it, because it didn't satisfied the rule that 

140
00:10:02,344 --> 00:10:07,310
If you measure a component of a spin you leave the system with exactly that 

141
00:10:08,605 --> 00:10:17,707
After thinking about it, I decided not qualified as a genuine measuring apparatus

142
00:10:19,617 --> 00:10:22,331
So we are gonna come to measurement 

143
00:10:22,659 --> 00:10:24,045
and the meaning of measurement in a little while

144
00:10:24,600 --> 00:10:27,254
But we will soon move, half an hour gone already

145
00:10:31,501 --> 00:10:33,048
#Okay if there are no more questions, 

146
00:10:33,491 --> 00:10:36,613
I'm gonna move on to the subject of entanglement 

147
00:10:39,056 --> 00:10:42,716
I have a few words to say about the collapse of a wave function

148
00:10:42,943 --> 00:10:47,488
Let's just talk about for a moment the idea of collapse of the wave function

149
00:10:51,745 --> 00:10:53,947
We've talked about how the quantum states evolve 

150
00:10:54,634 --> 00:10:58,381
According to the Schrodinger equation and that is the way they evolve

151
00:10:58,762 --> 00:11:02,689
But that's the way they evolve between events 

152
00:11:03,620 --> 00:11:06,482
where somebody or something are detectors or 

153
00:11:06,707 --> 00:11:10,260
an apparatus or just some other systems, an external system

154
00:11:10,744 --> 00:11:13,982
interact with the original quantum system

155
00:11:14,800 --> 00:11:18,757
So we have a quantum system, we have other thing in the world

156
00:11:19,339 --> 00:11:24,192
Including observers, including apparatuses, and just including other things

157
00:11:25,508 --> 00:11:28,807
If we take away all the other things 

158
00:11:29,991 --> 00:11:34,466
The system might be left in some state, and then the system will evolve 

159
00:11:34,746 --> 00:11:36,224
according to the Schrodinger equation 

160
00:11:37,090 --> 00:11:39,240
And then we maybe bring back the other things

161
00:11:39,711 --> 00:11:41,546
The other things might be an apparatus 

162
00:11:41,842 --> 00:11:43,971
Where did the apparatus go? The apparatus is hiding 

163
00:11:44,182 --> 00:11:48,594
But it could be an apparatus or something else 

164
00:11:49,865 --> 00:11:53,836
But let's say it's an apparatus, and the apparatus measures some specific thing 

165
00:11:55,197 --> 00:11:59,553
That were happened during the course of the interaction with the apparatus

166
00:12:00,741 --> 00:12:05,397
If a measurement takes place, and the system is left in an eigen state 

167
00:12:07,373 --> 00:12:09,662
corresponding to the particular outcome 

168
00:12:11,105 --> 00:12:15,561
So for example we might start, let's suppose we want to measure L

169
00:12:16,744 --> 00:12:19,547
And our system after a while might be in a state

170
00:12:20,348 --> 00:12:25,884
Some summation over i, a bunch of complex number alpha sub i

171
00:12:26,783 --> 00:12:35,051
Times eigenvectors of L, eigenvectors of L with eigenvalues lambda sub i

172
00:12:35,641 --> 00:12:38,279
Okay what's the probability for the outcome of the experiment?

173
00:12:39,125 --> 00:12:43,445
The system got here by evolving using the Schrodinger equation 

174
00:12:44,499 --> 00:12:47,632
We started it out, we evolve it according to Schrodinger equation

175
00:12:47,826 --> 00:12:49,571
Here is what we got, at the end of the day

176
00:12:50,405 --> 00:12:52,849
And now we make a measurement of L

177
00:12:53,849 --> 00:12:56,435
And we get some specific value of lambda

178
00:12:56,843 --> 00:12:58,220
Random, we don't know what it's gonna be?

179
00:12:59,262 --> 00:13:00,214
But it's gonna be something 

180
00:13:01,399 --> 00:13:03,693
Let's say it's lambda seven, we measured lambda seven

181
00:13:04,861 --> 00:13:11,126
There after, or right after that, the system is in state lambda seven

182
00:13:14,844 --> 00:13:17,178
The course of the operation of measurement 

183
00:13:17,825 --> 00:13:21,922
is not governed by the Schrodinger equation for the system

184
00:13:22,323 --> 00:13:26,133
It's governed by the nature of the measuring process

185
00:13:27,124 --> 00:13:31,738
And it simply tells you that you come out with one of the eigenvalues 

186
00:13:33,178 --> 00:13:37,942
as the measured value, and the state vector will be just one of these

187
00:13:38,884 --> 00:13:40,558
Which one? Random 

188
00:13:40,998 --> 00:13:43,310
Random depending on the measure, now that's funny 

189
00:13:43,632 --> 00:13:47,143
because you might think, look, well first of all


190
00:13:47,466 --> 00:13:48,766
let's give it a terminology, 

191
00:13:49,638 --> 00:13:53,499
the terminology is the way the function collapse around one point

192
00:13:54,203 --> 00:13:56,544
It is collapse around one single eigenvalue

193
00:13:57,846 --> 00:13:58,953
You can't predict which one 

194
00:13:59,996 --> 00:14:04,401
But all of the structure here with all the other one becomes irrelevant 

195
00:14:05,095 --> 00:14:09,165
And only one state pops into existence so to speak

196
00:14:10,116 --> 00:14:11,354
Alright now what's wrong with this?

197
00:14:11,713 --> 00:14:16,930
What's wrong with this? There is absolutely nothing wrong with it

198
00:14:17,109 --> 00:14:19,185
It's correct description of the measurement process 

199
00:14:19,541 --> 00:14:23,114
But what is very unsatisfying about this


200
00:14:23,640 --> 00:14:26,804
is after all, apparatus is just quantum systems 

201
00:14:29,188 --> 00:14:33,489
Why do we have two rules for the wave systems evolve?

202
00:14:35,301 --> 00:14:40,119
One is Schrodinger equation, and one is arbitrary collapsed wave function

203
00:14:41,055 --> 00:14:44,096
The answer is that we, 

204
00:14:45,763 --> 00:14:49,503
if we want to think quantum mechanically about the whole system

205
00:14:50,339 --> 00:14:53,305
A system plus apparatus, that's fine we can do it, 

206
00:14:53,505 --> 00:14:54,526
and I hope we could get to it tonight

207
00:14:55,789 --> 00:14:57,421
And then it's pure Schrodinger equation 

208
00:14:57,933 --> 00:15:02,771
but we cannot separate the system into just system and apparatus

209
00:15:03,067 --> 00:15:06,146
We have to think about the whole quantum system as being

210
00:15:06,346 --> 00:15:10,452
system plus apparatus, that raise the question immediately right now

211
00:15:11,397 --> 00:15:13,492
How do you combine systems together?

212
00:15:14,434 --> 00:15:20,295
If you have more than one system, it could be an electron spin and the apparatus

213
00:15:21,207 --> 00:15:23,148
We have two systems now, 

214
00:15:23,660 --> 00:15:26,575
imagine we had a good quantum theory of the apparatus

215
00:15:26,807 --> 00:15:30,309
The apparatus is a physical system quantum theories governs it

216
00:15:31,486 --> 00:15:36,220
How do we describe the combined system where we take two components

217
00:15:37,571 --> 00:15:39,700
One spin, one apparatus, put them together

218
00:15:39,943 --> 00:15:41,740
How do we make it into a single quantum system?

219
00:15:42,464 --> 00:15:44,636
And then how with that single quantum system 

220
00:15:44,873 --> 00:15:46,569
do we describe this process of measurement?

221
00:15:47,944 --> 00:15:52,502
That is a very, confused a generation of physicists

222
00:15:54,512 --> 00:15:57,728
Partly because Bohr was very stubborn about this


223
00:15:58,197 --> 00:15:59,517
He didn't quite had it right

224
00:16:00,747 --> 00:16:06,928
And he was a nice man, a very nice man, very glad, I never met him

225
00:16:10,275 --> 00:16:13,262
Though he was a nice man, but thing is he is also a very stubborn man

226
00:16:15,140 --> 00:16:18,447
And this particular thing, he was mostly right

227
00:16:19,216 --> 00:16:20,703
But here and there, he was wrong, 

228
00:16:22,457 --> 00:16:25,579
It was apparently very tough to be around him where he was 

229
00:16:27,142 --> 00:16:29,077
People tried to explained to him over and over again 

230
00:16:29,523 --> 00:16:31,431
You have to think of the combined system 

231
00:16:31,650 --> 00:16:33,718
and you have to think of the combined quantum system 

232
00:16:33,979 --> 00:16:36,005
If you want to understand the quantum mechanics of measurement 

233
00:16:36,211 --> 00:16:37,900
And he said no, he didn't believed that, anyway

234
00:16:38,714 --> 00:16:43,331
Combined systems, we have two systems now

235
00:16:43,558 --> 00:16:46,563
Let's just not worry about what they are 

236
00:16:47,138 --> 00:16:53,029
They are two systems, we will get more specific in the moment 

237
00:16:53,324 --> 00:16:57,330
But let's give them names, let's call one of the systems A 

238
00:16:57,531 --> 00:16:59,941
and A does not stand, it could stand for apparatus 

239
00:17:00,557 --> 00:17:02,915
But it doesn't have to, it's just A

240
00:17:03,678 --> 00:17:08,697
System A, I will call the space of states of system A, S sub A

241
00:17:09,068 --> 00:17:10,347
S sub A could stand for system A 

242
00:17:10,683 --> 00:17:14,889
but now I had it to mean the space of quantum state of the system A

243
00:17:15,748 --> 00:17:18,032
And we have another system, I'm gonna call I

244
00:17:23,320 --> 00:17:26,157
It's another system with it's own space of state I

245
00:17:26,935 --> 00:17:35,874
This system A has state which are labeled by a basis of states

246
00:17:37,312 --> 00:17:39,710
You chose a basis of states in the system A

247
00:17:40,485 --> 00:17:41,786
And you can label them |a>, 

248
00:17:42,109 --> 00:17:46,893
|a> can go from one to whatever the number of state of the system A is

249
00:17:48,118 --> 00:17:50,712
Whatever the dimensions of the space of state A is 

250
00:17:51,407 --> 00:17:53,771
And system I has state |i>

251
00:17:57,521 --> 00:18:04,483
Any state of system A, if you didn't have I, 

252
00:18:05,340 --> 00:18:08,714
would be a superposition of this basis states

253
00:18:09,854 --> 00:18:13,731
Any state of system I would be a superposition of these space of states

254
00:18:16,278 --> 00:18:18,932
How do we put the systems together and make a single quantum system?

255
00:18:19,820 --> 00:18:21,332
Well, I'm gonna tell you what the rule is

256
00:18:21,541 --> 00:18:24,729
I will give you a quick book rule but it's pretty compelling

257
00:18:25,701 --> 00:18:28,139
And it is the way real things work, okay

258
00:18:28,919 --> 00:18:34,228
You built a space of state of the system A I

259
00:18:35,481 --> 00:18:38,616
Which means A and I together, both of them

260
00:18:39,745 --> 00:18:45,741
That space of state is called mathematically, 

261
00:18:46,272 --> 00:18:48,429
it is what is called the tensor product 

262
00:18:48,695 --> 00:18:50,476
I will tell you what it means in detail

263
00:18:50,874 --> 00:18:54,959
but it's the tensor product of the two space of states S A

264
00:18:55,709 --> 00:18:58,718
And this is the symbol of the tensor product S I

265
00:18:59,991 --> 00:19:02,144
Now what does a tensor product mean?

266
00:19:02,748 --> 00:19:05,735
Well, it mean something very simple 

267
00:19:06,034 --> 00:19:09,371
It means you construct a new space 

268
00:19:10,289 --> 00:19:14,311
New mathematical space whose vectors are labeled 

269
00:19:14,956 --> 00:19:17,225
by a value of |a> and value of |i>

270
00:19:18,190 --> 00:19:21,867
Another words, you pair off the basis vectors and all possible pairings

271
00:19:22,861 --> 00:19:31,537
One from A, one from I, and you construct a basis which is labeled |a i>

272
00:19:34,885 --> 00:19:40,166
If this, let's suppose the dimensionality of this space is D sub a

273
00:19:41,196 --> 00:19:45,986
And the dimensionality of this space is D sub i, or D sub

274
00:19:46,236 --> 00:19:53,573
I guess it's capital A I want here, D sub capital A and D sub capital I

275
00:19:55,878 --> 00:19:59,464
Another words the number of orthogonal states of each one of these separately 

276
00:19:59,713 --> 00:20:04,630
Then what's the, what's the dimensionality of the tensor product?

277
00:20:07,699 --> 00:20:11,418
D_A times D_I, you get to pick one from A and one from I

278
00:20:11,627 --> 00:20:14,430
Like in a Chinese restaurant, one from A and one from I, Okay?

279
00:20:15,105 --> 00:20:16,539
And how many are there all together?

280
00:20:16,916 --> 00:20:19,554
D_A times D_I, that's why it's called a product, 

281
00:20:19,903 --> 00:20:22,745
because the dimensionality is a product

282
00:20:23,881 --> 00:20:29,751
So this is D_A times D_I in dimensionality

283
00:20:30,851 --> 00:20:34,297
These are the base of states for the tensor product, okay

284
00:20:34,665 --> 00:20:36,863
They are labeled with two indices 

285
00:20:37,249 --> 00:20:40,370
Now you think of them as orthonormal basis 

286
00:20:41,255 --> 00:20:42,879
Orthonormal in the following sense 

287
00:20:43,399 --> 00:20:46,961
If you take two such vectors |a i> and let's say |b j>

288
00:20:49,437 --> 00:20:54,130
Where b is another one of the basis vector of A

289
00:20:54,859 --> 00:20:58,236
And j is another one of the this

290
00:20:58,623 --> 00:21:01,115
What's the inner product between these

291
00:21:01,351 --> 00:21:05,472
Let's take these to be, let's take these to be normalized vectors

292
00:21:06,384 --> 00:21:08,416
What is the inner product between them?

293
00:21:08,804 --> 00:21:12,976
And the rule for a tensor product is the inner product between this and this

294
00:21:13,194 --> 00:21:15,119
Of course we have to make a bra vector out of it 

295
00:21:15,339 --> 00:21:17,785
In order to, in order to take it's inner product 

296
00:21:18,420 --> 00:21:21,892
But if we make a bra vector out of this until it's inner product with this one

297
00:21:22,408 --> 00:21:28,994
The rule is that the inner product is zero unless a is equal to b

298
00:21:29,238 --> 00:21:33,316
And i is equals to j, another words, unless they matched

299
00:21:35,161 --> 00:21:39,622
If they don't match, these might be observably different 

300
00:21:40,330 --> 00:21:43,119
For example if this b is the same as this a

301
00:21:44,021 --> 00:21:45,501
But i was not the same as j

302
00:21:46,742 --> 00:21:53,027
Then one half of the system here would be, would be observable difference

303
00:21:53,884 --> 00:21:57,251
associated with the observable difference in the sub system I

304
00:21:58,801 --> 00:22:01,303
So in order for them to have an overlap

305
00:22:02,435 --> 00:22:05,639
b must equal a and j must equal i

306
00:22:06,096 --> 00:22:08,766
and then they are observably the same vector

307
00:22:09,620 --> 00:22:13,726
>> Excuse me, are these arbitrary vectors? a, and

308
00:22:14,010 --> 00:22:14,868
These are basis vectors 

309
00:22:15,168 --> 00:22:17,034
>> Part of the orthonormal basis?

310
00:22:17,440 --> 00:22:19,718
Yes you could pick any orthonormal basis from this one

311
00:22:20,090 --> 00:22:21,635
And any orthonormal basis from this one 

312
00:22:23,094 --> 00:22:27,234
Ya, but you could pick any orthonormal basis from this one 

313
00:22:29,593 --> 00:22:31,739
And put them together in this way, you will get the same result 

314
00:22:32,444 --> 00:22:36,205
Doesn't matter which basis you pick, but let's pick a basis to be concrete

315
00:22:37,043 --> 00:22:41,979
Let's pick a basis, if a and i are not the same as both b and j

316
00:22:42,667 --> 00:22:43,856
The inner product is zero

317
00:22:44,472 --> 00:22:45,826
I can write an equation for that 

318
00:22:50,261 --> 00:23:00,334
<b j|a i> is equal to delta, kronecker delta ab kronecker delta ij

319
00:23:01,811 --> 00:23:08,091
If i is not equal to j, or a is not equal to b, this is zero

320
00:23:08,825 --> 00:23:12,557
It's only none-zero if i is equal to j and a is equal to b

321
00:23:14,750 --> 00:23:20,168
So these vectors formally by construction, by construction

322
00:23:20,458 --> 00:23:22,830
Which means you simply assigning values to things now

323
00:23:23,418 --> 00:23:26,295
But in a way that it's useful

324
00:23:27,021 --> 00:23:32,200
These vectors form by definition or by construction, 

325
00:23:32,605 --> 00:23:35,818
an orthonormal basis for the combined system

326
00:23:38,147 --> 00:23:39,277
And this makes sense 

327
00:23:44,026 --> 00:23:47,312
>> And you've defined this depend upon the basis, but

328
00:23:47,551 --> 00:23:50,540
It does, but actually, defined it, let's put it this way

329
00:23:50,778 --> 00:23:52,313
It were depend on the basis 

330
00:23:52,948 --> 00:23:54,760
That construction in the end doesn't 

331
00:23:55,065 --> 00:23:57,516
The construction in the end doesn't, and that's not hard to show 

332
00:23:57,965 --> 00:24:00,557
This is, however, keep in mind that 

333
00:24:01,300 --> 00:24:05,528
these basis vector are not all of the vectors in a tensor products

334
00:24:05,958 --> 00:24:09,490
You can then take any linear combination of these basis vectors 

335
00:24:09,793 --> 00:24:14,651
And they are also a vector in the tensor product, so

336
00:24:14,903 --> 00:24:19,046
So the general vector in the tensor product is of the form 

337
00:24:20,234 --> 00:24:29,197
Sum over i and a, alpha, let's call it sum i a, |i a>

338
00:24:30,875 --> 00:24:33,617
A general linear combination of the coefficient 

339
00:24:33,876 --> 00:24:37,270
that depend on what's inside here times the vector

340
00:24:37,546 --> 00:24:38,582
times the basis vectors

341
00:24:39,003 --> 00:24:42,950
Now the answer is, if you have chosen another basis 

342
00:24:43,952 --> 00:24:47,752
Another set of basis, you could have gotten exactly the same vectors

343
00:24:48,257 --> 00:24:50,228
Except they would have been written differently

344
00:24:50,461 --> 00:24:53,844
They would have just been written in a different basis of a combined system

345
00:24:54,556 --> 00:24:56,477
Changing basis for the individual systems 

346
00:24:56,693 --> 00:24:59,451
corresponds to the basis changes for combined systems

347
00:24:59,649 --> 00:25:04,120
Okay, so that's good, we now have the idea of a tensor product

348
00:25:04,911 --> 00:25:06,714
And I'm gonna give you a concrete example now

349
00:25:07,571 --> 00:25:11,697
The concrete example is the next most complicated quantum system

350
00:25:13,144 --> 00:25:14,081
Namely two spins

351
00:25:15,452 --> 00:25:18,225
Well it's debatable whether it's the next most complicated 

352
00:25:18,409 --> 00:25:19,701
or the next to next most complicated

353
00:25:20,451 --> 00:25:25,578
And the reason I say that, is because two spins have four basis vectors

354
00:25:26,119 --> 00:25:28,540
You could imagine a system with only three basis vectors

355
00:25:29,073 --> 00:25:30,350
And such system exist 

356
00:25:31,058 --> 00:25:36,194
But since we are trying to learn how to combine systems

357
00:25:36,656 --> 00:25:38,867
Let's take two systems and combine them 

358
00:25:39,547 --> 00:25:42,316
Each of them being a spin

359
00:25:42,910 --> 00:25:47,561
So what do we have, we have an electron nail down at this position over here

360
00:25:48,219 --> 00:25:51,394
And it has a spin, which we are gonna call sigma 

361
00:25:51,861 --> 00:25:54,748
And it has three components, sigma x y and z

362
00:25:55,324 --> 00:26:04,421
So sigma x y and z, and we have another one over here 

363
00:26:06,001 --> 00:26:08,475
We need a name for it, now I've toy it for a while 

364
00:26:08,716 --> 00:26:10,728
We are calling this sigma one, this one sigma two

365
00:26:11,254 --> 00:26:13,267
But there are too many indices, 

366
00:26:13,623 --> 00:26:16,439
but then I have sigma one x and sigma one y, but they are bad

367
00:26:16,900 --> 00:26:18,677
So I decided to give this one another name 

368
00:26:19,325 --> 00:26:22,031
And the Greek letter that comes after sigma is tau

369
00:26:22,674 --> 00:26:25,653
I don't know if it's the next letter, the next one in the English alphabet

370
00:26:26,100 --> 00:26:28,594
That's t, ya, so the next one I think in the alphabet is tau

371
00:26:30,126 --> 00:26:31,549
And I simply call this one tau

372
00:26:32,811 --> 00:26:34,575
How many component does tau have?

373
00:26:35,146 --> 00:26:36,579
Oh tau is also a spin 

374
00:26:37,703 --> 00:26:39,971
So it also has three components x y and z 

375
00:26:44,116 --> 00:26:48,358
Supposing I chose for the basis vectors for each of these 

376
00:26:48,940 --> 00:26:51,050
The up and down basis vectors 

377
00:26:52,081 --> 00:26:58,348
Another words, the state in which sigma z is up or down

378
00:26:58,850 --> 00:27:03,314
And the state in which tau z are up or down

379
00:27:05,786 --> 00:27:12,944
Then our basis vector for sigma are up> and down>

380
00:27:14,527 --> 00:27:18,558
Our basis vectors for tau are exactly the same

381
00:27:20,768 --> 00:27:26,971
Up> and down>, and maybe in order to indicate that

382
00:27:27,544 --> 00:27:30,373
These are the vectors for the system tau

383
00:27:30,919 --> 00:27:32,777
And these are the vectors for system sigma 

384
00:27:33,418 --> 00:27:36,372
Let's use a slightly different kind of bracket notation here

385
00:27:36,614 --> 00:27:41,735
Let's use a curly bracket, I'm gonna give that very shortly 

386
00:27:42,188 --> 00:27:46,534
I just want to point out to you that these are different in meanings...and down

387
00:27:50,061 --> 00:27:55,058
And down}, so what is the basis vectors for the combined system

388
00:27:55,525 --> 00:27:56,854
If we put the system together, 

389
00:27:57,364 --> 00:27:59,735
what are the basis vectors for the combined system?

390
00:28:00,119 --> 00:28:02,014
There are four of them, two times two is four 

391
00:28:02,684 --> 00:28:06,536
And what are they? They are states in which both spins are up

392
00:28:08,661 --> 00:28:11,085
It's one of them, one spin 

393
00:28:11,394 --> 00:28:14,820
And I always take the first entry here correspond to sigma 

394
00:28:15,187 --> 00:28:17,470
And the second entry to correspond to tau

395
00:28:18,464 --> 00:28:25,747
Up down>, down up> and down down>

396
00:28:28,465 --> 00:28:34,105
That's the four basis vectors, each basis vectors carrying two labels now

397
00:28:35,361 --> 00:28:37,833
A sigma label and a tau label 

398
00:28:39,636 --> 00:28:42,721
That's the idea, that's you know, it's very reasonable idea 

399
00:28:43,674 --> 00:28:44,545
Yes?

400
00:28:45,013 --> 00:28:47,927
>> Supposedly these two component systems interact 

401
00:28:48,352 --> 00:28:50,049
>> Does that .. or not?

402
00:28:50,274 --> 00:28:54,510
No, the basis vectors stay the same, but the linear combination of basis vectors

403
00:28:54,747 --> 00:28:57,242
That they are describing the system would change, ya

404
00:28:58,400 --> 00:29:06,161
So any state in this space can be written as alpha up, times up up>

405
00:29:07,724 --> 00:29:17,617
Plus an alpha up down |up down> plus an alpha down up |down up>

406
00:29:18,927 --> 00:29:23,963
Plus an alpha down down |down down>

407
00:29:26,468 --> 00:29:28,652
That's the most general thing we can write 

408
00:29:30,138 --> 00:29:31,809
Let's call it |Psi>

409
00:29:32,044 --> 00:29:34,016
The answer to your question is 

410
00:29:34,229 --> 00:29:37,231
if you started out with a particular linear combination

411
00:29:38,417 --> 00:29:41,389
Let's say for example just alpha up up |up up>

412
00:29:41,613 --> 00:29:43,290
Both spin's up and you let them interact 

413
00:29:43,988 --> 00:29:46,918
Then in general what happen to these coefficients would change 

414
00:29:47,645 --> 00:29:51,628
In fact it could even happen that after a time alpha up up 

415
00:29:51,921 --> 00:29:53,333
might go to alpha down down

416
00:29:55,217 --> 00:29:59,166
In other words, two spin up might interact in such a way

417
00:29:59,409 --> 00:30:00,931
It's just wind up down, 

418
00:30:01,880 --> 00:30:04,559
that would not be a change in what the basis vectors are

419
00:30:05,678 --> 00:30:07,373
It would be a change in what the alphas are

420
00:30:21,726 --> 00:30:23,515
Alright another several questions we can ask

421
00:30:23,908 --> 00:30:29,075
Incidentally, one of the criteria for doing this correctly 

422
00:30:29,725 --> 00:30:34,821
is that if we take a combined system but then we only do experiments

423
00:30:35,370 --> 00:30:36,949
on one of the halves of it

424
00:30:38,403 --> 00:30:40,023
The other one doing whatever it's doing 

425
00:30:40,771 --> 00:30:42,484
And they are well separated from each other 

426
00:30:42,806 --> 00:30:44,689
Let's suppose they are well separated from each other 

427
00:30:45,028 --> 00:30:46,941
And we do an experiment on only one of them

428
00:30:47,434 --> 00:30:50,351
The answer that we get should be the same answer that we got

429
00:30:50,685 --> 00:30:53,009
When we even not think of it as a combined system

430
00:30:53,970 --> 00:30:56,476
If we have a theory of sigma

431
00:30:57,410 --> 00:31:00,120
Which we talked about now for five weeks I think 

432
00:31:01,118 --> 00:31:02,172
And we know all about it

433
00:31:02,413 --> 00:31:03,785
We know how to do experiments on it

434
00:31:04,105 --> 00:31:08,007
And now we think of it as half a system composed of sigma and tau

435
00:31:08,816 --> 00:31:10,954
But sigma and tau are far from each other let's say

436
00:31:11,211 --> 00:31:13,118
They are not interacting with each other substantially 

437
00:31:13,685 --> 00:31:17,077
And we do exactly the same experiment so we've been talking about up till now

438
00:31:17,290 --> 00:31:18,216
We should give the same answers

439
00:31:18,590 --> 00:31:20,811
So we should put these things together in a way 

440
00:31:21,439 --> 00:31:25,706
Which make sure that the earlier issues

441
00:31:26,043 --> 00:31:30,266
The earlier description of either sub system is still correct 

442
00:31:30,974 --> 00:31:32,864
Now I'm not gonna bother proving that 

443
00:31:33,099 --> 00:31:35,339
I think you will get the idea and you will see how it works 

444
00:31:36,310 --> 00:31:40,350
And the way we are putting things together does guarantee that 

445
00:31:42,620 --> 00:31:44,278
Okay now we can go into two directions from here 

446
00:31:45,003 --> 00:31:46,317
We can talk about the space of states

447
00:31:46,936 --> 00:31:48,099
Or we can talk about the observables 

448
00:31:48,370 --> 00:31:49,169
So we are gonna do both

449
00:31:49,480 --> 00:31:52,342
But I will take a vote, how many of you want to talk about the space of states?

450
00:31:52,856 --> 00:31:55,115
How many want to talk about the observables? 

451
00:31:56,059 --> 00:31:57,263
We will talk about the space of states first 

452
00:31:58,761 --> 00:32:00,351
You know I was gonna do that, you know, come on

453
00:32:03,600 --> 00:32:06,566
Alright , ya

454
00:32:08,526 --> 00:32:13,536
>> .. left and right for that again?

455
00:32:13,796 --> 00:32:14,494
What's that?

456
00:32:14,813 --> 00:32:16,587
Oh yeah yeah yeah, sure ,there are ya


457
00:32:17,327 --> 00:32:21,532
Supposing we wanted, supposing we wanted a state

458
00:32:22,400 --> 00:32:27,827
Which was left for sigma and up for tau, alright?

459
00:32:28,244 --> 00:32:31,496
So this would be a state which would be prepared 

460
00:32:31,960 --> 00:32:33,502
We have two apparatuses now, 

461
00:32:34,259 --> 00:32:36,073
I'm always uncomfortable with the word apparatuses

462
00:32:36,446 --> 00:32:38,561
I always thinking it would be apparati, I don't think that's right

463
00:32:39,404 --> 00:32:44,619
We have two apparatuses, one that we apply to sigma

464
00:32:44,937 --> 00:32:48,320
The other we apply to tau, and we now take our apparatus 

465
00:32:48,734 --> 00:32:53,512
One apparatus, sigma apparatus, we point it in the, let's say, the x y axis

466
00:32:54,422 --> 00:32:59,569
Not the x y, along the x axis, and the other one and we point along the z axis

467
00:33:00,310 --> 00:33:06,462
And we are wind up discovering that sigma was along the x axis with plus one

468
00:33:07,249 --> 00:33:11,235
And tau was along the z axis also with plus one

469
00:33:12,124 --> 00:33:19,161
So that means that sigma was up but tau was right

470
00:33:20,716 --> 00:33:22,531
How do we make a state which is

471
00:33:24,448 --> 00:33:27,799
So the answer is, I will tell you what to do, 

472
00:33:28,106 --> 00:33:30,858
but then we will eventually have a mathematic for this

473
00:33:33,713 --> 00:33:40,565
Sigma is definitely up, but tau is a linear superposition of up with down

474
00:33:42,033 --> 00:33:46,579
Easy, you just take up up plus> up down>

475
00:33:48,860 --> 00:33:51,105
Square root of two, one over square root of two 

476
00:33:54,683 --> 00:33:56,225
You can think of this another way

477
00:33:56,784 --> 00:33:59,324
You can say think of it as a product 

478
00:34:00,045 --> 00:34:11,555
A product of up> for sigma and right} for tau

479
00:34:13,509 --> 00:34:14,757
Now what is right}?

480
00:34:15,642 --> 00:34:18,393
Right} is up> plus down> right? Right

481
00:34:18,970 --> 00:34:20,142
Right is up> plus down>

482
00:34:20,841 --> 00:34:25,046
So this can also be written as up> times another, 

483
00:34:25,284 --> 00:34:26,819
I will put a big square bracket around it

484
00:34:29,593 --> 00:34:33,108
Up}, curly bracket, plus down}

485
00:34:35,233 --> 00:34:38,657
But if I want to normalize it I should put a square root of two downstairs

486
00:34:41,808 --> 00:34:49,363
So we just think of it as a product of sigma up and tau in the, 

487
00:34:50,221 --> 00:34:51,676
along the right direction

488
00:34:52,630 --> 00:34:53,959
But that's the same thing here

489
00:34:55,314 --> 00:34:58,230
We have one over square root of two up up> as this one

490
00:34:59,684 --> 00:35:02,995
And one over square root of two up down>, that's this one 

491
00:35:04,212 --> 00:35:05,716
I'm not sure this answer your questions or not 

492
00:35:06,236 --> 00:35:08,428
But I think it probably partly did, ya, okay 

493
00:35:12,118 --> 00:35:14,651
This kind of state is called the product state

494
00:35:15,606 --> 00:35:19,643
It's a product state in which you know exactly what's going on with sigma

495
00:35:20,147 --> 00:35:24,111
Namely it's in the up direction and you know exactly what's going along with tau

496
00:35:24,468 --> 00:35:27,327
It's in the y direction, it's product state 

497
00:35:29,084 --> 00:35:32,323
It's simply two separate things which are prepared independently 

498
00:35:33,173 --> 00:35:39,076
And here it is, a product state is by definition a product

499
00:35:40,489 --> 00:35:42,290
Let's write the most general product state 

500
00:35:44,562 --> 00:35:48,037
The most general, this is not the most general product state 

501
00:35:48,276 --> 00:35:51,038
The most general product state can be written as follows 

502
00:35:53,228 --> 00:35:59,374
Alpha up  up> plus alpha down  down>

503
00:36:01,442 --> 00:36:05,955
That's for sigma and then for tau let's call it beta

504
00:36:06,317 --> 00:36:09,017
Let's call the coefficient for tau, let's call it beta 

505
00:36:09,754 --> 00:36:18,436
Beta up up} squiggly plus beta down down}

506
00:36:20,728 --> 00:36:21,479
What does this mean? 

507
00:36:22,101 --> 00:36:27,444
This means that sigma was prepared in this particular linear combination here

508
00:36:28,455 --> 00:36:31,451
It might have been holding the detector in some funny angle 

509
00:36:32,411 --> 00:36:37,600
And this one is the respond to holding tau detector at some other angle 

510
00:36:39,127 --> 00:36:42,101
And you had only one of these not the other

511
00:36:42,329 --> 00:36:43,511
You would just have written this down 

512
00:36:43,832 --> 00:36:45,224
And the other one you would have written this down 

513
00:36:45,905 --> 00:36:48,211
Together you write down the product, that's the rule 

514
00:36:50,998 --> 00:36:53,500
Okay, of course you know, ultimately these rules are the rules 

515
00:36:53,707 --> 00:36:55,200
Because they agree with experiment but

516
00:36:55,474 --> 00:36:57,589
It's pretty plausible rule 

517
00:37:00,440 --> 00:37:06,797
This is a product, we can also write it alpha up beta up times up up>

518
00:37:08,643 --> 00:37:14,748
Plus alpha up beta down up down>, do I need to write the rest of it?

519
00:37:16,509 --> 00:37:23,725
Well I will, plus alpha down beta up down up>, 

520
00:37:24,916 --> 00:37:29,477
Plus alpha down beta down down down>

521
00:37:31,556 --> 00:37:35,913
Alright of course, this is of the form of the general vector 

522
00:37:36,308 --> 00:37:38,578
But there is something special about the coefficients here

523
00:37:38,783 --> 00:37:42,663
The coefficients factorized into 

524
00:37:43,054 --> 00:37:49,149
And therefore this is in a certain sense special to some extend

525
00:37:49,823 --> 00:37:51,880
Let's see if we can quantify how special 

526
00:37:54,818 --> 00:37:56,274
>> What kind of vector product is that?

527
00:37:56,961 --> 00:38:00,203
That's a tensor product, this is the element of a tensor product 

528
00:38:00,709 --> 00:38:04,986
And it's a tensor product of this vector with that vector

529
00:38:07,491 --> 00:38:13,606
It's a vector drawn in a composite system as a tensor product of 

530
00:38:13,880 --> 00:38:17,217
vector described in one system and a vector described in other system

531
00:38:18,552 --> 00:38:22,669
So strictly speaking the tensor product is usually used for the whole space

532
00:38:23,449 --> 00:38:27,074
But then you would say this is a product state within the tensor product space

533
00:38:27,431 --> 00:38:30,227
And I will just call it tensor product state or product state

534
00:38:32,001 --> 00:38:34,400
It's not general and to see how none general it is 

535
00:38:34,714 --> 00:38:43,083
We can count parameters, given a general system or a general state vector

536
00:38:43,740 --> 00:38:44,748
How many parameters? 

537
00:38:44,951 --> 00:38:47,704
How many independent parameters does it take to describe it?

538
00:38:48,844 --> 00:38:52,002
And the answer is four complex numbers 

539
00:38:52,523 --> 00:38:56,149
Alpha up up, alpha up down, alpha down up, alpha down down 

540
00:38:56,556 --> 00:38:59,180
I am not assuming that it has this special form 

541
00:39:00,492 --> 00:39:06,225
Just general four numbers, four complex numbers is eight real numbers

542
00:39:06,663 --> 00:39:08,272
But that's a little too much 

543
00:39:09,102 --> 00:39:11,693
Because we should assume the state is normalized 

544
00:39:12,879 --> 00:39:17,907
So that's one constrain, and we also know over all phase is not important

545
00:39:18,772 --> 00:39:22,723
So when you count parameters, you always subtracts two

546
00:39:23,159 --> 00:39:25,735
One for the normalization, one for the phase

547
00:39:26,317 --> 00:39:35,912
Eight minus two is six, six real parameters describing a general state of two spins

548
00:39:37,188 --> 00:39:43,575
How many here? Well the state describing the sigma particle 

549
00:39:44,409 --> 00:39:45,976
That has two real parameters 

550
00:39:46,477 --> 00:39:50,675
Four real parameters for alpha up and alpha down

551
00:39:51,497 --> 00:39:54,297
But then we subtract two, same thing here 

552
00:39:56,558 --> 00:39:59,822
This is a normalized state, and it's overall phase doesn't matter

553
00:40:00,565 --> 00:40:03,715
This is a normalized state, it's overall phase doesn't matter

554
00:40:04,403 --> 00:40:06,438
All together four real parameters

555
00:40:06,925 --> 00:40:09,597
It's the same four real parameters that you might have guess 

556
00:40:09,848 --> 00:40:11,081
if you set ahead just two spins 

557
00:40:11,871 --> 00:40:16,100
Each spin has two parameters necessary to write down it's state 

558
00:40:16,412 --> 00:40:17,358
two real parameters, 

559
00:40:18,074 --> 00:40:21,355
two real parameters could have been the polarization direction of the spin

560
00:40:23,024 --> 00:40:25,508
The axis along which the spin is oriented or 

561
00:40:25,954 --> 00:40:31,805
So there are four parameters but six parameters here

562
00:40:32,223 --> 00:40:36,527
That simply proves that not all states are product states

563
00:40:36,727 --> 00:40:38,728
In fact most states are not product states 

564
00:40:39,803 --> 00:40:42,085
We are gonna talk about some none product state soon enough

565
00:40:43,478 --> 00:40:45,350
But this is the notion of the product state 

566
00:40:46,992 --> 00:40:51,169
And the product state, we can really think of separate states 

567
00:40:52,002 --> 00:40:54,406
for each of the particles for each of the sub systems 

568
00:40:55,377 --> 00:40:59,365
But for more general state you can't really divide it 

569
00:41:00,115 --> 00:41:03,386
into a state for one subsystem and the state for the other sub system

570
00:41:04,018 --> 00:41:07,258
They are entangled, they are entangled and they are more elaborate

571
00:41:07,535 --> 00:41:14,500
They are more um, I don't know what the right word is

572
00:41:15,004 --> 00:41:17,476
Entangled, they are entangled in a more entangle way

573
00:41:18,873 --> 00:41:20,422
We are gonna discuss that, entanglement

574
00:41:20,806 --> 00:41:22,346
That's very important concept of course 

575
00:41:23,690 --> 00:41:27,435
Okay that's two spins, how many parameters six versus four

576
00:41:27,846 --> 00:41:30,391
They are entangled, alright, we talked about product state 

577
00:41:31,517 --> 00:41:35,132
And that's what we know about the space of state of the system 

578
00:41:35,431 --> 00:41:36,794
It takes six parameters to describe it

579
00:41:37,805 --> 00:41:39,704
While only four parameters of it's product state

580
00:41:40,116 --> 00:41:41,507
Product state is the one 

581
00:41:41,706 --> 00:41:44,234
which you can separately say there is the state for one system

582
00:41:44,989 --> 00:41:47,798
And the state for the other system, end of story

583
00:41:50,451 --> 00:41:53,945
Okay let's talk about observables, now we come to observables 

584
00:41:57,733 --> 00:42:00,572
What are the observables for this composite system?

585
00:42:03,530 --> 00:42:05,604
Well they must be the same observables 

586
00:42:07,115 --> 00:42:09,754
as for the separate systems by themselves

587
00:42:10,420 --> 00:42:13,748
Another words, sigma x y and z and tau x y and z

588
00:42:17,843 --> 00:42:24,162
They are the things you could measure with two apparatuses of the same kind

589
00:42:24,967 --> 00:42:28,814
As we used in our experiments on one spin

590
00:42:29,426 --> 00:42:33,658
We now have two of them, so we now have sigma x and sigma y and sigma z

591
00:42:34,164 --> 00:42:35,897
And tau x tau y and tau z

592
00:42:36,586 --> 00:42:42,199
Let's see if we can figure out how these operators what it operates on

593
00:42:42,571 --> 00:42:44,260
the state of the combined systems

594
00:42:45,330 --> 00:42:48,499
Alright you can sort of guess I will spell it out

595
00:42:48,994 --> 00:42:53,378
But here is the rule, let's take sigma x

596
00:42:55,641 --> 00:42:59,704
And let's apply it to the state of these sigma systems

597
00:42:59,984 --> 00:43:01,561
Obviously it belong to the sigma systems 

598
00:43:02,547 --> 00:43:05,280
If it applies to up>, it gives down>

599
00:43:06,590 --> 00:43:09,948
If it applies to down>, it gives up>

600
00:43:14,005 --> 00:43:20,472
You can check that, that's a fact with sigma x, it's equal to zero one one zero

601
00:43:21,348 --> 00:43:25,866
This off diagonal character means it takes up> to down> and down> to up>

602
00:43:27,031 --> 00:43:28,782
This is what sigma x does

603
00:43:29,247 --> 00:43:31,645
Ordinarily if we only have one sub system 

604
00:43:32,466 --> 00:43:35,267
Here is what it does on a combined system

605
00:43:35,674 --> 00:43:41,996
Sigma x on any vector which begins with up and has anything 

606
00:43:42,646 --> 00:43:47,838
Let's see, which one was a and which one was i?

607
00:43:50,148 --> 00:43:54,527
Second one was i? Tau went with i or Tau went with?

608
00:43:55,185 --> 00:43:56,056
Tau was i, okay

609
00:43:56,805 --> 00:44:05,123
So sigma x when it acts on up i, all it sees is the up>

610
00:44:05,523 --> 00:44:09,605
All it sees is the first thing here, and it's completely passive 

611
00:44:10,466 --> 00:44:12,182
with respect to the second index 

612
00:44:12,762 --> 00:44:15,262
Another words, it doesn't even notice the second index 

613
00:44:15,515 --> 00:44:17,759
And leaves it the same, it doesn't change it 

614
00:44:18,992 --> 00:44:24,082
But it does whatever it should do on the first entry

615
00:44:25,861 --> 00:44:33,520
Likewise, sigma x on down i> equals up i>

616
00:44:36,061 --> 00:44:41,737
You just take what it did for the single spin space

617
00:44:42,894 --> 00:44:46,288
And you just think of the other index as a kind of spectator

618
00:44:46,556 --> 00:44:50,089
It just goes along for the ride, doesn't do anything

619
00:44:50,868 --> 00:44:52,856
On the other hand, let's do one more sigma

620
00:44:53,117 --> 00:45:03,036
Sigma z on up i> now what does sigma z do when it acts on up>?

621
00:45:07,432 --> 00:45:17,562
Sigma z is one minus one zero zero, it just gives back up with value one

622
00:45:18,023 --> 00:45:24,662
So that's just up i>, and what about sigma z on down i>?

623
00:45:27,185 --> 00:45:29,110
And what happens when sigma z hits down?

624
00:45:30,359 --> 00:45:34,704
Minus down, so this gives minus down i>

625
00:45:36,585 --> 00:45:39,077
How about sigma y? What does sigma y (do)?

626
00:45:39,417 --> 00:45:41,583
Sigma y does something very much the same as sigma x

627
00:45:41,820 --> 00:45:46,048
Except it throws an i, not this i>, but numerical complex number i

628
00:45:47,273 --> 00:45:50,413
It throws it around because what is it, sigma x?

629
00:45:50,750 --> 00:45:55,369
This is sigma z, sigma x was one minus one 

630
00:45:56,209 --> 00:46:00,539
And sigma y was equal to minus i, i, zero zero

631
00:46:01,407 --> 00:46:04,685
Sigma y also flips up> to down> and down> to up>

632
00:46:05,029 --> 00:46:07,492
But it throws in a .. i here

633
00:46:07,746 --> 00:46:10,843
Okay so that's what sigma does 

634
00:46:11,954 --> 00:46:12,911
Now how about tau?

635
00:46:13,702 --> 00:46:19,451
Well tau does exactly the same thing except that 

636
00:46:20,328 --> 00:46:23,914
the first entry becomes the spectator

637
00:46:24,528 --> 00:46:28,146
And only the second entry gets acted on 

638
00:46:29,044 --> 00:46:30,968
So let's, let's do this one 

639
00:46:32,195 --> 00:46:43,094
Tau let's say x on a up>, a could be anything, a could be up> or down>

640
00:46:43,614 --> 00:46:51,604
Doesn't matter, tau x on a up> is equal to a times down>

641
00:46:51,897 --> 00:47:03,860
And tau x on a down> is equal to a up>

642
00:47:08,184 --> 00:47:18,709
Alright let's do tau z on a up> equals a up>

643
00:47:21,454 --> 00:47:30,835
And tau z on a down> is equal to minus a down>

644
00:47:33,093 --> 00:47:37,427
And you could figure out the action of the y component of the spin

645
00:47:39,363 --> 00:47:40,588
It's straightforward

646
00:47:41,194 --> 00:47:46,555
With this definition, it's quite clear that if you are talking about an observable 

647
00:47:46,894 --> 00:47:53,721
associated with sigma, that the tau degrees of freedom are completely passive

648
00:47:54,510 --> 00:48:00,301
Nothing happens to them, the eigenvectors of the components of sigma 

649
00:48:02,945 --> 00:48:08,891
Are simply product state of the eigenvector of sigma times any vector for tau

650
00:48:10,456 --> 00:48:12,820
Another words for example the eigenvectors

651
00:48:19,530 --> 00:48:26,073
the eigenvectors, the eigenvectors of, let's take sigma y

652
00:48:28,108 --> 00:48:37,023
Ordinarily the eigenvector of sigma y are one i or one minus i

653
00:48:38,004 --> 00:48:41,069
I can't remember which one has eigenvalues plus one or minus one

654
00:48:41,561 --> 00:48:43,297
But one way or the other

655
00:48:43,997 --> 00:48:50,815
But that means, that means the linear combination up plus i down>

656
00:48:51,905 --> 00:48:54,763
That's an eigenvector of sigma y

657
00:48:55,167 --> 00:48:56,870
If you want the eigenvector of sigma y 

658
00:48:57,755 --> 00:49:00,011
when we are talking about the composite system

659
00:49:00,320 --> 00:49:04,206
It's exactly the same except you put an i in here I guess

660
00:49:05,032 --> 00:49:09,224
An i in here and an i in here, same i, same i

661
00:49:09,713 --> 00:49:13,218
So for example, it could be up up plus i down up

662
00:49:14,222 --> 00:49:16,632
Or up down> plus i down down>

663
00:49:17,647 --> 00:49:21,011
The i is a spectator, it just goes along for the ride

664
00:49:21,582 --> 00:49:25,131
Anything which was an eigenvectors for the single system

665
00:49:25,979 --> 00:49:27,858
Remains the eigenvectors for the single system 

666
00:49:28,110 --> 00:49:30,853
except you have to add in the index for the other particle, 

667
00:49:31,308 --> 00:49:34,030
for the other subsystem 

668
00:49:34,842 --> 00:49:37,697
And that's all there is to it, with this definition as I said

669
00:49:38,787 --> 00:49:41,735
It's pretty clear that the separate subsystem 

670
00:49:42,449 --> 00:49:45,850
Behaves independently, quite independently 

671
00:49:47,796 --> 00:49:51,269
As long as you don't let them interact and feel each other

672
00:49:52,574 --> 00:49:55,031
Or do something interesting with each other

673
00:49:55,283 --> 00:49:56,588
As long as they are kept separated 

674
00:49:58,512 --> 00:50:06,452
Okay, um, other questions?

675
00:50:08,498 --> 00:50:12,673
>> The equation sigma x acting on |u> and gives down> 

676
00:50:12,951 --> 00:50:19,255
>> Shouldn't it, you have sigma x acting on

677
00:50:19,872 --> 00:50:20,927
>> the superposition of state?

678
00:50:22,822 --> 00:50:24,807
No no, sigma x

679
00:50:26,044 --> 00:50:30,407
>> in terms of the eigen states of the sigma x

680
00:50:32,796 --> 00:50:36,160
Well, let's just do it by matrices 

681
00:50:37,864 --> 00:50:46,708
sigma x is one one zero zero, up is one zero, okay?

682
00:50:47,533 --> 00:50:51,884
up> is one zero, what happens? This times this is zero

683
00:50:53,648 --> 00:50:55,231
This times this is one

684
00:50:56,536 --> 00:51:00,431
So up> has gone to down>, okay?

685
00:51:00,952 --> 00:51:04,466
Likewise, if we put a zero here and a one here 

686
00:51:06,547 --> 00:51:08,633
This would become one zero 

687
00:51:09,743 --> 00:51:13,218
So it just takes one zero and flips it to zero one

688
00:51:13,885 --> 00:51:15,940
And back and forth like that

689
00:51:16,363 --> 00:51:17,742
In general of course you would be right

690
00:51:17,989 --> 00:51:22,122
But for this particular case, I just gives you back

691
00:51:23,177 --> 00:51:24,408
>> I see what the ones and zeros work out

692
00:51:24,664 --> 00:51:27,024
>> but in the physical situation where 

693
00:51:27,700 --> 00:51:30,747
>> you were making the measurement to get the state

694
00:51:34,020 --> 00:51:38,666
Well this doesn't say, this doesn't say, let's see

695
00:51:40,165 --> 00:51:43,956
This doesn't say one zero and zero one are eigenvectors of sigma x

696
00:51:44,323 --> 00:51:48,249
Far from it, it's as far from an eigenvector you can get 

697
00:51:48,992 --> 00:51:52,302
When you acted sigma x on this one you get a completely orthogonal state

698
00:51:53,682 --> 00:51:57,133
So this is not the rule that says, this is not the equation 

699
00:51:57,428 --> 00:51:59,937
that says zero one or one zero are eigenvectors

700
00:52:02,036 --> 00:52:06,233
An eigenvector is a vector which is kept in the same direction 

701
00:52:07,286 --> 00:52:08,541
by action of the operator

702
00:52:09,192 --> 00:52:11,537
This is as far as you can get from that 

703
00:52:12,302 --> 00:52:14,701
The operator acts and gives you something orthogonal 

704
00:52:17,233 --> 00:52:20,956
And that's okay? Alright you have to work that out

705
00:52:22,405 --> 00:52:26,008
>> up> is a linear superposition of left and right

706
00:52:26,661 --> 00:52:27,661
Indeed it is

707
00:52:32,928 --> 00:52:37,140
Right that's correct 

708
00:52:38,658 --> 00:52:41,639
>> my question before class is, 

709
00:52:42,204 --> 00:52:47,566
>> Is there a physical meaning to applying operator to a none-eigen vector

710
00:52:47,982 --> 00:52:50,973
>>What is that operation equates to?

711
00:52:55,777 --> 00:53:02,214
Most of the times, that's just auxiliary operation in some calculation or other

712
00:53:05,002 --> 00:53:11,726
Well, it doesn't have any particular special physical meaning 

713
00:53:13,340 --> 00:53:18,627
But in this particular case, it tells you is the zero one one zero 

714
00:53:18,998 --> 00:53:20,779
It's about as far from eigenvalues as you can get 

715
00:53:21,416 --> 00:53:25,692
And what it tells you is that if you measure sigma x

716
00:53:26,774 --> 00:53:31,723
In an eigen state of sigma z, it's as random as it can be 

717
00:53:34,383 --> 00:53:35,293
It's as random as it can be 

718
00:53:40,550 --> 00:53:41,788
Alright we are going along good 

719
00:53:43,878 --> 00:53:45,325
>> (inaudible)

720
00:53:49,578 --> 00:53:51,962
>> What happens if I have sigma x tau x

721
00:53:52,335 --> 00:53:54,120
say that again

722
00:53:54,628 --> 00:53:56,770
>> (inaudible)

723
00:53:59,653 --> 00:54:05,192
>> What happens if I have sigma x tau x operating on 

724
00:54:07,455 --> 00:54:10,941
Sigma x times tau x? Okay we could work that out

725
00:54:13,652 --> 00:54:16,732
It's a little bit later, five minutes in the note

726
00:54:17,033 --> 00:54:19,276
But let's do it right now since we are at it

727
00:54:31,914 --> 00:54:39,190
Let's take sigma x tau x and tell me what vector you would like to apply it to?

728
00:54:42,781 --> 00:54:47,451
How about up up>? Let's do up up>

729
00:54:53,507 --> 00:54:58,182
What does tau x do? I keep forgetting 

730
00:54:59,212 --> 00:55:04,864
The one in the left is sigma right? This one is sigma and this one is tau right?

731
00:55:05,785 --> 00:55:12,357
Okay, alright so we saw the tau and we hit this

732
00:55:13,833 --> 00:55:17,625
What does tau x do? It operates on this one and let this one alone 

733
00:55:18,948 --> 00:55:22,364
So this gives us, do I have it wrong? Ya 

734
00:55:23,071 --> 00:55:25,584
Ya it operates on this one and let this one alone, right

735
00:55:26,674 --> 00:55:30,489
So it gives sigma x, I have not acted on that yet

736
00:55:30,942 --> 00:55:34,122
Times, now tau x flips this one

737
00:55:34,494 --> 00:55:35,742
So it was up down>

738
00:55:39,363 --> 00:55:40,691
Okay now what does sigma x do?

739
00:55:42,671 --> 00:55:53,339
It flips this one, so it just gives down down>, kind of boring 

740
00:55:53,857 --> 00:55:55,479
But that's what it does

741
00:55:56,052 --> 00:55:57,422
Let's do it in an opposite order 

742
00:55:59,407 --> 00:56:06,707
Let's take tau x times sigma x on up up> what does it do?

743
00:56:07,597 --> 00:56:08,993
It does exactly the same thing 

744
00:56:09,875 --> 00:56:13,066
Sigma x hits one of these and flips them 

745
00:56:13,598 --> 00:56:16,796
And then tau x comes and hits the other one and flips them

746
00:56:17,863 --> 00:56:21,114
So when tau x times sigma x acts, 

747
00:56:21,369 --> 00:56:24,252
it gives exactly the same thing as sigma x times tau x 

748
00:56:26,838 --> 00:56:32,310
Commutator of sigma x with tau x equals zero

749
00:56:33,588 --> 00:56:35,997
No difference between the order of operations

750
00:56:36,700 --> 00:56:41,453
Let's try it for sigma x times tau z

751
00:56:43,085 --> 00:56:46,239
Okay let's do it for sigma x times tau z and see what we get

752
00:56:46,959 --> 00:56:52,730
Oh I have not proved this because I only exhibit for one particular state

753
00:56:53,830 --> 00:57:03,573
In order to argue that sigma x times tau x equals tau x times sigma x

754
00:57:04,288 --> 00:57:11,086
Oops, you would have to show that they give the same thing 

755
00:57:11,312 --> 00:57:13,194
when acting on any basis vectors

756
00:57:13,587 --> 00:57:16,203
It's enough to show that they gives the same thing on basis vectors 

757
00:57:17,095 --> 00:57:20,085
Alright but that's pretty easy, not hard to do

758
00:57:20,310 --> 00:57:21,706
You can do it on any basis vectors 

759
00:57:21,947 --> 00:57:24,795
And you will find out the order of operation doesn't matter 

760
00:57:25,649 --> 00:57:31,997
In fact the same is true if you take sigma x times, let's say, tau z

761
00:57:33,041 --> 00:57:35,154
Let's do it again, shall I take a different vector?

762
00:57:35,413 --> 00:57:40,912
Let's take some other vector, instead of up down>, let's take down up>

763
00:57:41,871 --> 00:57:44,466
And see what it does, and then we will do it in the opposite order

764
00:57:45,523 --> 00:57:48,252
Alright what does tau z do when it hits this vector here?

765
00:57:49,848 --> 00:57:52,113
It only acts on the second entry

766
00:57:52,874 --> 00:57:57,049
And it's the z component so it just gives plus one on up> right?

767
00:57:59,573 --> 00:58:07,777
So this just gives sigma x times down up>

768
00:58:09,133 --> 00:58:12,705
Tau z on up> gives up>

769
00:58:15,171 --> 00:58:16,730
So that just goes right through

770
00:58:17,138 --> 00:58:19,828
And what does sigma x do when it hits down up>?

771
00:58:21,130 --> 00:58:26,558
This gets a little bit gobbly look, at this point my tongue starts rebelling 

772
00:58:28,064 --> 00:58:30,787
What does sigma x do when it hits tau? When it hits this guy

773
00:58:32,102 --> 00:58:37,608
It just flips down> to up>, it gives up up> right

774
00:58:38,930 --> 00:58:44,484
Let's do it in the opposite order tau z sigma x on down up

775
00:58:45,712 --> 00:58:49,480
Alright first sigma x, that flips down to up

776
00:58:49,982 --> 00:58:53,871
So it gives us tau z times up up>

777
00:58:55,720 --> 00:58:59,952
And then tau z just gives us one when it gives it up here

778
00:59:00,187 --> 00:59:02,790
So it gives the same thing again, so you can sort of see why

779
00:59:03,019 --> 00:59:06,969
They are acting on separate indices here, 

780
00:59:07,227 --> 00:59:09,017
it doesn't matter which way you order them

781
00:59:09,453 --> 00:59:12,457
This is very general, it doesn't matter which state you take, 

782
00:59:12,719 --> 00:59:13,970
It doesn't matter which component you take


783
00:59:14,228 --> 00:59:19,249
Any component of sigma commutes with any components of tau

784
00:59:20,503 --> 00:59:23,169
You can check that, just by knowing how they act 

785
00:59:24,956 --> 00:59:30,554
Alright so we can say sigma anything, I'm not sure how to call the index here

786
00:59:30,818 --> 00:59:35,551
Sigma, I don't wanna call it i, I already used i for something 

787
00:59:40,013 --> 00:59:48,352
But sigma n and sigma tau n for any m and n they commute 

788
00:59:49,172 --> 00:59:53,940
The meaning of that, is that if you have two separate sub systems 

789
00:59:55,010 --> 01:00:01,101
Which are combined in this way, the observable for one and 

790
01:00:01,308 --> 01:00:04,569
the observable for the other can be simultaneously measured

791
01:00:05,524 --> 01:00:09,394
In particular if you took the system from far apart to each other 

792
01:00:09,740 --> 01:00:11,567
And you treat them as a single quantum system 

793
01:00:12,307 --> 01:00:14,449
You would expect that measuring one of them 

794
01:00:15,014 --> 01:00:17,215
doesn't interfere with any measurement with the other one 

795
01:00:18,618 --> 01:00:19,825
And that's what this says, 

796
01:00:20,292 --> 01:00:25,777
that measuring a sigma does not preclude measuring a tau

797
01:00:26,414 --> 01:00:28,880
And vise versa, for any components of them

798
01:00:29,424 --> 01:00:32,402
So in that sense, they behave like independent systems 

799
01:00:33,142 --> 01:00:34,646
You can make any measurement on sigma 

800
01:00:35,296 --> 01:00:37,141
And any measurement on tau, ya?

801
01:00:39,502 --> 01:00:41,726
>>

802
01:00:42,121 --> 01:00:44,572
>> So these observables are each independent parts

803
01:00:44,843 --> 01:00:46,285
>> you knows it fit into the system

804
01:00:46,788 --> 01:00:48,287
>> and you could think about that observables one 

805
01:00:48,848 --> 01:00:50,310
>>

806
01:00:50,922 --> 01:00:54,728
It's as happy as it was when you added the other sub system

807
01:00:56,168 --> 01:01:00,306
Yes, that's a good question, are there observables which are some how

808
01:01:00,968 --> 01:01:03,443
more intermittently connected with having both of them?

809
01:01:04,447 --> 01:01:07,174
The answer is yes, for example sigma times tau

810
01:01:09,304 --> 01:01:13,343
Sigma times tau, sigma x times tau x or sigma y times tau y

811
01:01:13,929 --> 01:01:18,064
That sort of things are observables that 

812
01:01:18,567 --> 01:01:22,927
somehow no more intermittently about the combination of both of them

813
01:01:24,013 --> 01:01:27,074
And they know in particular about entanglement 

814
01:01:28,687 --> 01:01:31,466
>> If you started with independent

815
01:01:32,053 --> 01:01:33,397
>> You just said ..

816
01:01:34,963 --> 01:01:40,631
>> Now how are you making them so they dependent on each other?

817
01:01:40,888 --> 01:01:43,888
We haven't yet, we haven't yet, what we've done up till now 

818
01:01:44,408 --> 01:01:47,348
is to all the ingredients which tell you that 

819
01:01:47,617 --> 01:01:52,932
as long as you do separate experiments, separate operations

820
01:01:53,625 --> 01:01:59,184
On the two sub systems, that they will behave exactly as they would have

821
01:01:59,400 --> 01:02:00,882
Had the other system not been there

822
01:02:01,834 --> 01:02:06,546
But now you can ask what about, are there more in particular

823
01:02:06,896 --> 01:02:11,689
In particular, if the system, if the combined system, if it's in a product state

824
01:02:12,583 --> 01:02:16,654
If it's in a product state so that each component has it's own state vector 

825
01:02:17,353 --> 01:02:21,393
Then the quantum mechanics of each one of them is separate from the other

826
01:02:21,818 --> 01:02:22,905
They don't interfere with each other

827
01:02:23,091 --> 01:02:24,620
They don't know about each other in any way

828
01:02:25,388 --> 01:02:27,068
That's what this prescription ensures 

829
01:02:28,257 --> 01:02:30,134
No, they are not entangled yet 

830
01:02:31,083 --> 01:02:32,428
Not if they are product states 

831
01:02:32,726 --> 01:02:37,553
Now what we found out is that the product states are four parameters set

832
01:02:38,968 --> 01:02:42,393
We found that also that the full space of state has six parameters

833
01:02:42,738 --> 01:02:47,340
So there are apparently state which are not product states 

834
01:02:47,974 --> 01:02:50,808
Those are the state which would have to varying degrees

835
01:02:51,095 --> 01:02:52,217
A degree of entanglement 

836
01:02:53,166 --> 01:02:55,644
#I'm gonna write down the most entangled state 

837
01:02:56,268 --> 01:02:58,019
and I'm gonna examine it a little bit

838
01:02:59,472 --> 01:03:01,014
and see in what way it's odd

839
01:03:08,897 --> 01:03:15,504
up down>, there are many highly entangled states, maximally entangled state 

840
01:03:16,101 --> 01:03:18,736
I'm writing one, one particular one, minus

841
01:03:19,014 --> 01:03:24,838
I'm putting a minus here just because I happen to know that it's a particular

842
01:03:25,256 --> 01:03:28,696
It has a particular property which is not so important at this stage 

843
01:03:29,069 --> 01:03:32,028
But let's put the minus state, minus down up>

844
01:03:35,175 --> 01:03:36,723
One over square root of two of course 

845
01:03:37,044 --> 01:03:38,719
One over square root of two times the whole thing 

846
01:03:43,594 --> 01:03:45,091
Now this state is interesting 

847
01:03:46,562 --> 01:03:55,416
It has the property that if you, well, it has a correlation 

848
01:03:57,525 --> 01:04:01,491
If you know that tau was down, then sigma must be up

849
01:04:02,921 --> 01:04:07,536
On the other hand if you know that tau is up, then sigma must be down 

850
01:04:07,772 --> 01:04:09,227
Now what does that mean mathematically? 

851
01:04:09,459 --> 01:04:11,671
We are going to find out what exactly that mean mathematically 

852
01:04:12,385 --> 01:04:13,565
This is not a product state

853
01:04:13,934 --> 01:04:18,262
There is no way that this can be written as a product of one state

854
01:04:18,803 --> 01:04:21,914
It's a sum or difference in this case for two product states 

855
01:04:22,158 --> 01:04:23,264
But it's not a product state 

856
01:04:23,947 --> 01:04:33,689
How can we tell, how can we tell, so let's consider the expectation value

857
01:04:34,735 --> 01:04:39,321
In a state of the components of sigma 

858
01:04:40,108 --> 01:04:45,237
Let's go back to the one subsystem, to the one system quantum mechanics 

859
01:04:45,685 --> 01:04:49,035
One spin, and let's say what do we know about that?

860
01:04:51,790 --> 01:04:58,750
In particular I'm interested in expectation values of, in any state 

861
01:04:59,584 --> 01:05:04,943
Take a state, and take the expectation value of all three components 

862
01:05:13,195 --> 01:05:16,608
This is the average value, I'm not multiplying anything

863
01:05:17,117 --> 01:05:18,361
I'm just considering them individually 

864
01:05:23,561 --> 01:05:24,752
Can they all be zero?

865
01:05:27,609 --> 01:05:32,306
Can they all simultaneously be zero for an ordinary state of a

866
01:05:32,622 --> 01:05:35,720
For the ordinary two parameters family of states for the single system?

867
01:05:39,323 --> 01:05:41,028
Should we take a vote or should I just tell you

868
01:05:43,472 --> 01:05:45,820
The answer is no, they can not all be zero

869
01:05:46,182 --> 01:05:53,771
Here's the reason, well, the reason is given any state 

870
01:05:54,434 --> 01:05:58,516
There was always a direction in space, in three-dimensional space

871
01:05:59,707 --> 01:06:02,820
A direction for which, sigma in that direction, 

872
01:06:03,014 --> 01:06:07,833
the component of sigma in that direction is definite and equals to plus one

873
01:06:08,425 --> 01:06:11,747
Given any Psi, this is what I told you in the very beginning of the lecture 

874
01:06:12,503 --> 01:06:13,914
Or maybe even before the lecture


875
01:06:14,236 --> 01:06:20,004
Given any state, there are some linear combination of sigmas 

876
01:06:21,018 --> 01:06:23,051
which is definitely equal to plus one

877
01:06:24,930 --> 01:06:29,064
In that case if I find that particular combination of sigmas 

878
01:06:30,087 --> 01:06:34,417
Sigma n x, you know, dot sigma into n, that particular linear combination

879
01:06:35,033 --> 01:06:37,223
It's expectation value can't be zero 

880
01:06:37,529 --> 01:06:40,324
Because it's definitely up along that direction

881
01:06:41,892 --> 01:06:47,484
So the answer is, given any state there was always some direction

882
01:06:48,352 --> 01:06:51,960
Let's call it sigma dot n, that depends on Psi

883
01:06:52,247 --> 01:06:58,576
That depend on Psi, sigma dot n, for which this is not only not zero

884
01:06:59,109 --> 01:07:00,384
But it's equal to plus one 

885
01:07:01,218 --> 01:07:06,824
The average of sigma dot n is of course 

886
01:07:07,443 --> 01:07:11,666
If sigma dot n is definite, if it's known and there is no ambiguity about it

887
01:07:12,197 --> 01:07:15,975
Then the average is just equal to the value that it's known to be

888
01:07:17,162 --> 01:07:22,508
So there is always some n, alright, 

889
01:07:23,712 --> 01:07:32,537
for any Psi, there exists an n, such that, this is true

890
01:07:34,438 --> 01:07:40,422
Therefore it can not be true that all of these can simultaneously be zero

891
01:07:40,625 --> 01:07:43,116
For a state of a single spin

892
01:07:47,744 --> 01:07:49,451
You can't measure them all simultaneously 

893
01:07:50,180 --> 01:07:52,495
But if I give you a state I tell you the state

894
01:07:53,205 --> 01:07:57,857
Then I can then, you can tell me what direction of space 

895
01:07:58,672 --> 01:08:00,342
should you orient your detector 

896
01:08:00,907 --> 01:08:04,403
so that the answer comes out plus one, okay?

897
01:08:04,889 --> 01:08:06,739
For example I mean, I can just give you up

898
01:08:07,151 --> 01:08:08,739
Okay so what direction if I gave you up?

899
01:08:09,386 --> 01:08:14,971
What direction should you orient your detector? along the z axis

900
01:08:15,686 --> 01:08:18,821
What if I gave you up plus down, with square root of two?

901
01:08:20,193 --> 01:08:24,608
What direction should you orient your detector to make sure

902
01:08:24,838 --> 01:08:28,839
you get plus one? Along the x axis right?

903
01:08:30,671 --> 01:08:33,165
What if I gave you up> plus i down> ?

904
01:08:36,695 --> 01:08:42,559
I know, the sum, it's either plus i or minus i, okay? I forget which one 

905
01:08:43,498 --> 01:08:48,355
Given any combination there are enough direction that you can find the direction

906
01:08:49,134 --> 01:08:52,933
For which sigma in that direction is definitely plus one 

907
01:08:53,710 --> 01:08:58,428
And therefore it can not be that all of these expectation values are all zero

908
01:08:59,066 --> 01:09:03,007
Why? Because you know that sigma dot n is equal to plus one

909
01:09:03,598 --> 01:09:06,006
And that of course is just a linear combination of these 

910
01:09:06,957 --> 01:09:13,746
Okay so as I said, it's impossible if we have just a state of one spin

911
01:09:14,575 --> 01:09:17,271
Isolated by itself, it's been prepared somehow

912
01:09:18,323 --> 01:09:23,214
However it's prepared, there exists a direction in which the spin is definite

913
01:09:23,447 --> 01:09:25,527
And not equal to zero, okay

914
01:09:26,798 --> 01:09:33,190
What about the state? What about the state now?

915
01:09:34,906 --> 01:09:42,558
Let's take sigma and ask whether any of the components of sigma

916
01:09:43,619 --> 01:09:47,038
have a non zero expectation value in this state?

917
01:09:48,411 --> 01:09:50,301
What I'm gonna show, I will tell you the answer right now

918
01:09:50,932 --> 01:09:53,804
We are gonna show that all three component of sigma 

919
01:09:55,259 --> 01:09:58,860
have zero expectation value in this state

920
01:10:00,835 --> 01:10:04,661
That means the state cannot possibly be a product state

921
01:10:05,564 --> 01:10:09,085
A product state is one in which each component behaves 

922
01:10:09,320 --> 01:10:13,516
as if they were just a state of subsystem by itself 

923
01:10:15,131 --> 01:10:17,091
This state is gonna behave in a way that 

924
01:10:17,385 --> 01:10:19,944
no simple product state could possibly behave

925
01:10:21,038 --> 01:10:23,337
We are going to find, so let's do it, let's do it

926
01:10:24,441 --> 01:10:28,186
Okay let's put, the square root of two here is not important

927
01:10:28,569 --> 01:10:30,014
That's not the important thing 

928
01:10:30,555 --> 01:10:34,438
We have up down> minus down up>

929
01:10:38,837 --> 01:10:40,848
We are gonna take the expectation value 

930
01:10:41,126 --> 01:10:42,670
So that mean we have to make a sandwich 

931
01:10:43,623 --> 01:10:47,817
The sandwich would have up down> minus down up>

932
01:10:49,321 --> 01:10:51,083
In the middle here we are gonna put a sigma 

933
01:10:52,321 --> 01:10:54,817
We could do with sigma tau, and get the same answer 

934
01:10:55,488 --> 01:10:58,837
So let's put sigma z first 

935
01:10:59,671 --> 01:11:02,108
Let's put sigma z here and see what we get 

936
01:11:03,698 --> 01:11:08,467
First we act with sigma z, and what does sigma z gives when it hits up down>

937
01:11:11,222 --> 01:11:12,614
It just gives up down> right?

938
01:11:13,297 --> 01:11:15,825
Alright? So this just gives up down>, 

939
01:11:16,649 --> 01:11:18,977
but what does it give when it hits this guy over here?

940
01:11:20,107 --> 01:11:22,730
Minus down up>, but there is already a minus down up>

941
01:11:23,043 --> 01:11:28,521
So there is a plus, already a minus, the minus gets killed

942
01:11:30,750 --> 01:11:33,117
It's over here, so that's the inner product 

943
01:11:33,719 --> 01:11:35,330
Let's put some brackets around it

944
01:11:35,938 --> 01:11:41,000
It's the inner product with <up down minu <down up

945
01:11:42,422 --> 01:11:43,698
Okay let's calculate it 

946
01:11:44,289 --> 01:11:47,266
What's <down up| with |up down>? Zero

947
01:11:49,170 --> 01:11:52,461
What's <down up| with |down up>? One

948
01:11:52,972 --> 01:11:55,759
So we get a one, what's <down up| with |up down>? 

949
01:11:59,995 --> 01:12:01,726
Wowo, I got something wrong here

950
01:12:02,947 --> 01:12:06,571
What did I do here? This should be <up down| right?

951
01:12:08,533 --> 01:12:09,384
Okay so let's go back 

952
01:12:09,839 --> 01:12:11,697
<up down| with |up down> is one 

953
01:12:12,210 --> 01:12:14,926
<up down| with |down up> is zero

954
01:12:16,123 --> 01:12:19,850
what about <down up| with |up down>? Zero 

955
01:12:20,219 --> 01:12:25,296
What about <down up| with |down down>? One, but there's a minus sign here

956
01:12:27,318 --> 01:12:31,964
So this is orthogonal to this, the inner product to zero 

957
01:12:32,685 --> 01:12:37,301
And it says that the expectation value of sigma z is equal to zero

958
01:12:37,867 --> 01:12:43,075
Sigma z in this state, not any state but in this state is zero

959
01:12:43,868 --> 01:12:50,078
Let's try sigma x, let's try sigma x next and see what we get 

960
01:12:50,307 --> 01:12:54,447
So we have sigma x here, alright, let's spill out sandwich again 

961
01:12:55,980 --> 01:13:01,126
And the minus sign here, and what happens when sigma x equals |up down>?

962
01:13:01,634 --> 01:13:02,497
It hits |up down>?

963
01:13:04,126 --> 01:13:08,976
It flips up to down, sigma x is the flipper 

964
01:13:09,772 --> 01:13:11,146
And it flips the first entry here 

965
01:13:11,363 --> 01:13:12,763
So it flips the first entry to down 

966
01:13:15,021 --> 01:13:16,634
And flips this entry to up

967
01:13:19,450 --> 01:13:23,638
We are taking the inner product with this vector over here

968
01:13:23,868 --> 01:13:26,242
So let's see is there anything in common over here 

969
01:13:26,425 --> 01:13:27,459
with something in common here?

970
01:13:28,011 --> 01:13:31,112
No, so there is no way to get an inner product 

971
01:13:31,609 --> 01:13:34,000
<down up| with |down down> is zero etc. etc.

972
01:13:34,597 --> 01:13:35,997
again this is zero 

973
01:13:38,199 --> 01:13:43,796
so sigma x is equal to zero, expectation value 

974
01:13:46,088 --> 01:13:50,086
okay what about sigma y, let's put sigma y there

975
01:13:51,345 --> 01:13:54,249
sigma y is also a flip flop operator 

976
01:13:54,877 --> 01:13:57,611
It flips up to down and down to up but with an i

977
01:13:58,186 --> 01:14:01,187
I don't remember which, where the i goes 

978
01:14:01,706 --> 01:14:07,695
It either goes here, here, something like that 

979
01:14:08,881 --> 01:14:11,580
But whatever it does, it flips this up to down

980
01:14:11,918 --> 01:14:14,535
So we get a |down down>, and it flips this down to up

981
01:14:14,827 --> 01:14:15,799
So we get a |up up>

982
01:14:16,226 --> 01:14:18,641
Again, no inner product is zero

983
01:14:19,927 --> 01:14:24,740
So there we are, all three components have zero expectation value 

984
01:14:26,363 --> 01:14:30,650
But that's funny because we proved, well we didn't prove

985
01:14:31,102 --> 01:14:36,517
But I told you, that if I take any state of the single spin

986
01:14:38,139 --> 01:14:44,828
It's never the case that all three sigma-s have zero expectation value 

987
01:14:45,987 --> 01:14:49,849
What's the lesson? The lesson is there is a state of the composite system 

988
01:14:50,306 --> 01:14:58,283
That in no way ever be thought of as separate states for the two subsystems

989
01:14:58,575 --> 01:14:59,726
That's what entanglement is 

990
01:15:03,962 --> 01:15:05,862
That's an example of entanglement

991
01:15:10,506 --> 01:15:15,310
>> So what .. obvious that if you do that with product state 

992
01:15:15,835 --> 01:15:20,372
>> then it will satisfy the same property that one of them will be ..

993
01:15:20,603 --> 01:15:23,749
Yes, that's correct, that's correct

994
01:15:24,231 --> 01:15:28,626
Well if you show it for the single spin by itself

995
01:15:29,881 --> 01:15:33,291
Remember the other index that just goes along for the ride

996
01:15:33,530 --> 01:15:36,193
So however you derive it for a one spin with itself 

997
01:15:36,490 --> 01:15:38,101
You will do the same derivation 

998
01:15:38,348 --> 01:15:42,608
except put a sort of spectator index to ride along with it

999
01:15:43,249 --> 01:15:49,534
So it's obvious that if it's true for a one subsystem by itself that

1000
01:15:49,770 --> 01:15:51,073
It will be true for the composite 

1001
01:15:51,429 --> 01:15:54,221
What's not obvious is that it's true for one subsystem 

1002
01:15:55,562 --> 01:15:59,848
So ya, so that's an exercise, that's an exercise to show that

1003
01:16:00,663 --> 01:16:04,584
all three components of the spin cannot, 

1004
01:16:04,975 --> 01:16:07,772
or the expectation value cannot be zero

1005
01:16:08,315 --> 01:16:10,600
There is always a direction in which it's maximum 

1006
01:16:12,304 --> 01:16:17,131
>> Question? So how do you put a system in this strange state?

1007
01:16:17,609 --> 01:16:19,753
Ya, it's very natural state to have 

1008
01:16:20,228 --> 01:16:25,423
I will tell you, if you have two spins 

1009
01:16:25,635 --> 01:16:27,568
and the two spins are interacting with each other

1010
01:16:28,950 --> 01:16:35,222
They will have a, in particular if there are little magnetic moments 

1011
01:16:35,930 --> 01:16:37,916
Then they will have an interaction with each other 

1012
01:16:38,155 --> 01:16:40,059
which will prefer to make them anti-align

1013
01:16:41,844 --> 01:16:44,666
Now this is anti-align, |up down>

1014
01:16:45,595 --> 01:16:47,772
This is anti-align, |down up>

1015
01:16:48,809 --> 01:16:52,824
And in fact this combination here has the lowest energy 

1016
01:16:53,653 --> 01:16:55,394
This combination has the lowest energy 

1017
01:16:55,895 --> 01:16:58,365
It has less energy than |up down> plus |down up>

1018
01:16:58,645 --> 01:17:01,236
So if you bring these two spins together and you wait awhile 

1019
01:17:02,932 --> 01:17:07,490
They will irradiate a photon then eventually come to the state 

1020
01:17:09,591 --> 01:17:12,415
So it's actually not at all hard to get this kind of state

1021
01:17:12,859 --> 01:17:14,175
It's kind of easy to get this kind of state 

1022
01:17:14,992 --> 01:17:19,044
And we will discuss that when we talk about interactions

1023
01:17:20,214 --> 01:17:24,417
But it's a typical situation that this is what result

1024
01:17:25,213 --> 01:17:27,381
If you take two spins put them together and 

1025
01:17:27,783 --> 01:17:29,486
let them come to equilibrium by radiating a photon

1026
01:17:33,251 --> 01:17:38,321
A long time, first of all I have to close them to bring them together 

1027
01:17:38,802 --> 01:17:43,715
Second of all, how big the magnetic dipole of the spins are 

1028
01:17:47,282 --> 01:17:50,189
And that has to do with the strength of the coupling to the magnetic field 

1029
01:17:52,245 --> 01:17:54,580
So in ordinary laboratory situation, 

1030
01:17:55,421 --> 01:17:58,784
in an atom the electrons are close enough together 

1031
01:17:59,621 --> 01:18:04,562
That they will radiate the photon at not too long a time to achieve this 

1032
01:18:05,598 --> 01:18:08,764
so called singlet state, this is called the singlet state, singlet 

1033
01:18:11,594 --> 01:18:13,020
I'm not gonna tell you why right now 

1034
01:18:18,850 --> 01:18:20,537
There are four basis vectors all together 

1035
01:18:21,646 --> 01:18:23,311
The reason this is called the singlet 

1036
01:18:23,522 --> 01:18:25,643
because there are three other which are called the triplet, okay?

1037
01:18:26,116 --> 01:18:30,236
>> Can you get the over all up down plus down up ..

1038
01:18:32,700 --> 01:18:35,437
Ya, well not at the moment by eventually 

1039
01:18:35,942 --> 01:18:37,833
I will tell you the answer though 

1040
01:18:38,095 --> 01:18:39,833
If you think of these as little angular momenta 

1041
01:18:40,515 --> 01:18:42,397
And you put them in |up down> minus |down up>

1042
01:18:43,173 --> 01:18:44,843
The total angular momentum is zero

1043
01:18:45,246 --> 01:18:47,278
If you put then |down up> plus |down up>

1044
01:18:47,669 --> 01:18:48,984
The total angular momentum is one 

1045
01:18:49,900 --> 01:18:53,566
And energy is often proportional to angular momentum 

1046
01:18:53,893 --> 01:18:58,734
So, but, we will come to that, eventually we will come to that 

1047
01:18:59,250 --> 01:19:03,637
It's a fair question but let's not try to answer it right this minute

1048
01:19:06,036 --> 01:19:10,577
>> If you take it in general state of two systems

1049
01:19:10,839 --> 01:19:15,048
>> Is it possible to change the basis 

1050
01:19:15,388 --> 01:19:19,301
>> such that one basis create a sub space of product space, of product state?

1051
01:19:19,564 --> 01:19:22,394
>> the other is entirely pure product space

1052
01:19:22,743 --> 01:19:27,535
You are asking whether the notion of entanglement is an invariant notion

1053
01:19:28,416 --> 01:19:29,508
The answer is yes

1054
01:19:30,099 --> 01:19:33,343
The answer is yes, there is very definite quantity that you can calculate

1055
01:19:33,571 --> 01:19:38,205
called the entanglement entropy which measures the strength of entanglement 

1056
01:19:38,476 --> 01:19:40,801
Which is not dependent on any particular basis 

1057
01:19:41,504 --> 01:19:45,089
Now I might tell you right now, this state okay, 

1058
01:19:45,360 --> 01:19:48,260
this state does have a special feature that 

1059
01:19:48,493 --> 01:19:50,120
up down> plus down up> wouldn't have 

1060
01:19:50,479 --> 01:19:54,048
Supposing you took this state, and this is good, homework assignment 

1061
01:19:54,539 --> 01:19:58,328
Up and down individually for both sigma and tau 

1062
01:19:58,754 --> 01:20:00,909
can be rewritten in terms of left and right

1063
01:20:02,220 --> 01:20:08,448
For example, up is left plus right and down is left minus right

1064
01:20:08,737 --> 01:20:11,998
Or whatever, there's a square root of two there but you know what I mean

1065
01:20:12,361 --> 01:20:14,636
Okay supposing I write this state 

1066
01:20:15,086 --> 01:20:18,481
I plug in every place I see down, left minus right

1067
01:20:18,809 --> 01:20:20,666
And every place I see up, left plus right 

1068
01:20:21,311 --> 01:20:23,390
What do I get out of this, the answer is very simple 

1069
01:20:24,610 --> 01:20:28,139
I get |left right> minus |right left>

1070
01:20:30,007 --> 01:20:34,085
Supposing I rewrite it in terms of in and out, what do I get?

1071
01:20:35,052 --> 01:20:39,031
|in out> minus |out in>

1072
01:20:40,779 --> 01:20:44,357
So the structure that the state has and the nature of it

1073
01:20:45,312 --> 01:20:47,147
is actually the same in every basis 

1074
01:20:47,497 --> 01:20:48,811
Every basis it looks the same 

1075
01:20:50,578 --> 01:20:53,863
And that would not be true for |up down> plus |down up>

1076
01:20:54,814 --> 01:20:57,210
|down up> and.. they would.. um

1077
01:21:05,038 --> 01:21:06,762
They would get mixed up with other states

1078
01:21:07,381 --> 01:21:10,561
There are three other states, this is one, there are three others 

1079
01:21:11,150 --> 01:21:14,526
Three others, obviously there are three others

1080
01:21:14,779 --> 01:21:17,034
The name mixed up with each other when you change the basis 

1081
01:21:18,767 --> 01:21:21,027
But this is not obvious, this is not obvious at all

1082
01:21:21,844 --> 01:21:25,189
Alright let's do one last thing, I'm not sure it's last thing or not

1083
01:21:25,408 --> 01:21:26,691
I think it's probably last thing for tonight

1084
01:21:28,303 --> 01:21:34,353
#Let's ask about one of these observables which is not an observable 

1085
01:21:34,706 --> 01:21:37,776
that can be identified with either sub system 

1086
01:21:38,380 --> 01:21:40,774
but which are some how got to do with both

1087
01:21:41,797 --> 01:21:45,811
And let's look at that expectation value, very simple example 

1088
01:21:52,923 --> 01:22:03,943
Let's take, let's take the expectation value of the product of sigma z times tau z

1089
01:22:04,577 --> 01:22:07,296
We will do that one, we will do sigma x times tau x

1090
01:22:09,479 --> 01:22:15,312
And maybe we will do sigma x times tau y, we will do, we will try some examples

1091
01:22:16,124 --> 01:22:17,987
Let's take the expectation value of this 

1092
01:22:18,266 --> 01:22:20,158
Now to do this we again make a sandwich 

1093
01:22:22,391 --> 01:22:24,753
And let's do it

1094
01:22:28,075 --> 01:22:29,831
|up down>, oops, sorry

1095
01:22:40,957 --> 01:22:43,058
Now I should put the square root of two in 

1096
01:22:43,784 --> 01:22:45,777
The reason I should put the square root of two in now

1097
01:22:46,060 --> 01:22:48,465
is I'm not gonna get zero, so if I really want to get the numerical number 

1098
01:22:49,414 --> 01:22:50,833
I wanna put the square root of two in

1099
01:22:51,666 --> 01:22:58,448
And here we have <up down| minus <down up

|1100
01:23:00,349 --> 01:23:05,595
Also over square root of two, so in fact I could take the square roots of twos

1101
01:23:06,401 --> 01:23:09,222
And put them all into the factor one half here

1102
01:23:10,292 --> 01:23:13,139
This is the expectation value in the singlet state 

1103
01:23:13,956 --> 01:23:17,736
So we take a singlet state, and we take sigma z times tau z

1104
01:23:18,085 --> 01:23:18,977
Let's see what we get 

1105
01:23:19,766 --> 01:23:25,213
First of all, when sigma z, when tau z acts what does it do?

1106
01:23:25,449 --> 01:23:27,945
Do you remember? I don't, oh yes

1107
01:23:28,948 --> 01:23:35,427
tau z doesn't flip, it just gives you a sign which is plus for this one

1108
01:23:35,931 --> 01:23:37,278
and minus for this one

1109
01:23:38,565 --> 01:23:44,720
So we can remove the tau z just by putting the minus sign here alright 

1110
01:23:46,385 --> 01:23:47,644
Did I do that right? 

1111
01:23:51,332 --> 01:23:58,933
Let's see tau z, tau z acts on down to get minus down

1112
01:24:00,908 --> 01:24:04,809
It acts on up to give up, so it has no effect on this term 

1113
01:24:05,792 --> 01:24:07,433
And it changes the sign of this term 

1114
01:24:07,930 --> 01:24:13,210
I think I get it right, No? Ya, minus 

1115
01:24:14,976 --> 01:24:17,056
Now what about sigma z what does it do?

1116
01:24:18,853 --> 01:24:25,544
It flips the sign, no it doesn't flip, it gives plus on this one and minus on this one 

1117
01:24:27,116 --> 01:24:29,917
So it changes the sign of this one 

1118
01:24:33,888 --> 01:24:35,497
Okay now we have to put it together 

1119
01:24:36,202 --> 01:24:43,023
We have <up down|, |up down> with a minus sign, it's a minus one 

1120
01:24:44,045 --> 01:24:47,799
We have <down up| with |down up>, minus one

1121
01:24:49,230 --> 01:24:52,786
Cross terms give nothing, <up down| with |down up> gives nothing 

1122
01:24:53,598 --> 01:24:55,316
<down up| with |up down> gives nothing 

1123
01:24:55,847 --> 01:24:58,692
There are two terms, you get the two over two 

1124
01:25:00,624 --> 01:25:05,218
Hum, minus two over two equals minus one

1125
01:25:06,302 --> 01:25:08,179
What does is say? This is kind of interesting 

1126
01:25:11,052 --> 01:25:14,604
The expectation value of sigma x by itself was zero 

1127
01:25:15,493 --> 01:25:18,832
The expectation value of tau x by itself was zero

1128
01:25:20,333 --> 01:25:25,523
The expectation value of sigma x times tau x is minus one 

1129
01:25:26,636 --> 01:25:27,792
Let's see if we can see why 

1130
01:25:28,031 --> 01:25:31,974
Look at the state here, in this component over here 

1131
01:25:32,530 --> 01:25:40,242
Obviously, sorry, this was sigma z, sigma z we were talking about 

1132
01:25:41,645 --> 01:25:46,447
Sigma z are opposite in both contributions here 

1133
01:25:47,337 --> 01:25:53,123
Sigma z time this z for this piece is minus one, why is that?

1134
01:25:53,675 --> 01:25:55,049
This one's up, this one's down

1135
01:25:55,478 --> 01:25:58,935
If sigma z is plus for this one, sigma z is minus for this one 

1136
01:25:59,325 --> 01:26:02,776
The product is negative, the product is negative one for this one also 

1137
01:26:04,226 --> 01:26:12,031
In fact if you think about it, this state is an eigenvector of sigma z times tau z

1138
01:26:15,307 --> 01:26:18,970
What's the eigenvalue? Minus one

1139
01:26:20,186 --> 01:26:24,263
So in fact this state is the state in which you measure the product sigma z tau z

1140
01:26:24,518 --> 01:26:26,758
We get minus one every single time

1141
01:26:29,545 --> 01:26:31,533
So the expectation value is minus one 

1142
01:26:33,549 --> 01:26:37,908
Nevertheless the expectation value of the individual sigma x, 

1143
01:26:38,434 --> 01:26:40,716
sorry sigma z and tau z is zero

1144
01:26:42,194 --> 01:26:47,865
Alright so this is, this is beginning to get some of the flavor 

1145
01:26:48,087 --> 01:26:49,436
of what entangled state is

1146
01:26:50,843 --> 01:26:58,270
It simply doesn't behave like you might expect like two separate systems 

1147
01:26:58,575 --> 01:27:05,775
which are just independent, it has always been independent as they behave

1148
01:27:06,424 --> 01:27:10,878
Let's try, this is undoubtedly gonna get boring now

1149
01:27:11,209 --> 01:27:11,971
But let's do it anyway

1150
01:27:12,477 --> 01:27:15,846
This is plus minus, you go back to the original sandwich 

1151
01:27:16,770 --> 01:27:24,044
But instead of sigma z, let's do, let's do sigma x tau x

1152
01:27:25,929 --> 01:27:29,206
Alright so sigma x times tau x, it's easy to figure that out

1153
01:27:30,071 --> 01:27:37,393
Tau x flips the second entry, sigma x flips the first entry

1154
01:27:38,649 --> 01:27:42,163
Alright so this object here simply flips both entry

1155
01:27:42,825 --> 01:27:46,442
without changing any signs, it flips no entries at all

1156
01:27:47,045 --> 01:27:49,481
Well, it, what did I say, 

1157
01:27:54,374 --> 01:28:02,145
So what does it do? It looks like it interchanges these two, alright?

1158
01:28:02,885 --> 01:28:05,674
Interchanges these two, this one minus and this one plus 

1159
01:28:07,238 --> 01:28:08,819
Now what's the inner product?

1160
01:28:10,254 --> 01:28:13,552
<up down| with minus |up down>, minus one

1161
01:28:14,449 --> 01:28:18,385
<down up| , no, minus <down up|, with |down up>, minus one again

1162
01:28:18,936 --> 01:28:23,053
Again minus two over two, equals minus one 

1163
01:28:24,110 --> 01:28:25,877
Same exact thing as 

1164
01:28:34,883 --> 01:28:37,972
And what would you guess about sigma y tau y?

1165
01:28:38,708 --> 01:28:41,038
There was nothing special about the y axis, same thing 

1166
01:28:41,575 --> 01:28:44,529
Also has an expectation value equals to minus one

1167
01:28:45,400 --> 01:28:51,048
So alright, this again this is telling you very clearly this can't be a product state

1168
01:28:51,985 --> 01:28:56,064
For a product state, the expectation value of a product like this 

1169
01:28:56,264 --> 01:28:58,135
would just be the product of expectation values 

1170
01:28:59,571 --> 01:29:02,432
And that's not what is going on here 

1171
01:29:02,664 --> 01:29:05,403
So this is a highly entangled state, about as entangled as you can get 

1172
01:29:09,264 --> 01:29:11,350
>> .. for two observables?

1173
01:29:12,039 --> 01:29:14,428
>> Sorry, two operators, excuse me

1174
01:29:14,922 --> 01:29:18,205
>> So how I can figure that physics 

1175
01:29:18,530 --> 01:29:20,816
>> If follows back the operations?

1176
01:29:21,630 --> 01:29:24,535
In this case you, in this case remember

1177
01:29:24,985 --> 01:29:26,921
No you don't, not, in general no

1178
01:29:27,711 --> 01:29:32,364
In general it's just another operator, if it's Hermitian 

1179
01:29:32,619 --> 01:29:34,519
It's another observable, it can be measured

1180
01:29:34,733 --> 01:29:38,868
But in this case, since all sigma commutes with all taus

1181
01:29:39,564 --> 01:29:42,188
You can think about measuring them both simultaneously 

1182
01:29:43,272 --> 01:29:47,585
And in fact if you measure them both simultaneously you will simply discover

1183
01:29:48,275 --> 01:29:51,068
The obvious fact that they are oppositely oriented 

1184
01:29:52,593 --> 01:29:55,722
Let's do one more, let's do, since I haven't work it out

1185
01:29:55,971 --> 01:29:57,812
I've never work it out, and I will work it out right now 

1186
01:29:58,525 --> 01:30:02,531
Let's do tau, sigma x times tau z

1187
01:30:04,350 --> 01:30:08,899
Let's do sigma x, oops what do I write here sigma x tau x alright

1188
01:30:09,435 --> 01:30:16,109
Let's do sigma x times tau z and see what we get 

1189
01:30:16,664 --> 01:30:18,675
I don't know the answers so let's do it

1190
01:30:20,063 --> 01:30:25,026
I have a suspicion, I have a suspicion, but okay we will find out 

1191
01:30:26,387 --> 01:30:29,355
What happens when tau z acts on this guy here?

1192
01:30:31,662 --> 01:30:33,211
Tau z it just changes the sign right?

1193
01:30:35,228 --> 01:30:41,814
Tau z, if it's down you get minus one, if it's up you get plus one 

1194
01:30:46,492 --> 01:30:48,835
So you get, alright flip them around, yes you are right, good

1195
01:30:54,239 --> 01:31:00,547
Okay so tau z sees the down> here and makes it minus down>

1196
01:31:03,156 --> 01:31:05,224
And sees the up> here and leaves it alone

1197
01:31:06,911 --> 01:31:08,616
Okay now what does sigma x do? 

1198
01:31:09,281 --> 01:31:12,552
Sigma x flips the first entry 

1199
01:31:13,404 --> 01:31:20,401
And so it gives minus |down down>, minus

1200
01:31:20,995 --> 01:31:23,684
Now let's see it flips, minus |up up> right?

1201
01:31:28,416 --> 01:31:34,817
Now we have to take the inner product with this right over here

1202
01:31:35,585 --> 01:31:38,124
Well nothing over here matches anything over here 

1203
01:31:39,295 --> 01:31:41,504
So we see the answer in this case is zero 

1204
01:31:42,181 --> 01:31:43,187
We see the answer is zero 

1205
01:31:43,678 --> 01:31:52,348
The expectation value of sigma x times tau z or any unequal one

1206
01:31:52,574 --> 01:31:53,353
that's equal to zero 

1207
01:31:57,306 --> 01:31:59,140
What does it say? Well I don't know

1208
01:31:59,408 --> 01:32:02,734
The main reason I did it is because I was curious about what the answer was

1209
01:32:03,328 --> 01:32:05,209
But I also did it just as a illustration of 

1210
01:32:05,461 --> 01:32:08,036
the machinery of calculating expectation value 

1211
01:32:08,779 --> 01:32:11,449
But these ones tell you clear things 

1212
01:32:12,377 --> 01:32:15,542
What they tell you is that in a very definite sense

1213
01:32:17,127 --> 01:32:21,011
The any given component of sigma and tau

1214
01:32:21,358 --> 01:32:24,031
The x component or the y component or the z component 

1215
01:32:24,308 --> 01:32:25,519
are oppositely oriented

1216
01:32:29,535 --> 01:32:33,965
It's obvious here for the z component they are oppositely oriented huh

1217
01:32:34,737 --> 01:32:37,179
This is opposite to this, this is opposite to this 

1218
01:32:38,085 --> 01:32:41,912
So in any of the component, state vectors here

1219
01:32:42,397 --> 01:32:43,786
They are oppositely oriented


1220
01:32:44,422 --> 01:32:46,264
But we also found something else out

1221
01:32:46,491 --> 01:32:51,297
We found that they are oppositely oriented in the x basis or in the y basis

1222
01:32:51,959 --> 01:32:54,903
And one of the basis if you measure any one of the spins 

1223
01:32:56,690 --> 01:32:58,447
Any component of the up spin 

1224
01:32:58,966 --> 01:33:03,499
Basically it automatically tells you what the other one is

1225
01:33:03,769 --> 01:33:05,165
Well this is telling you, 

1226
01:33:05,363 --> 01:33:08,150
since it's been telling you the product is always equal to minus one 

1227
01:33:08,833 --> 01:33:12,825
In fact this is an eigenvector of the product, or any of these products 

1228
01:33:13,439 --> 01:33:16,363
It tells you if you measure any of the component for one of them

1229
01:33:17,515 --> 01:33:21,376
That you will get exactly the opposite if you measure the other one 

1230
01:33:22,253 --> 01:33:26,677
It's a correlation, it's a correlation and statistical correlation 

1231
01:33:27,390 --> 01:33:31,728
That if you measure one of the component of spin of sigma

1232
01:33:33,098 --> 01:33:37,978
and also the same component for the other one, they will be opposite

1233
01:33:39,094 --> 01:33:40,333
Of course it doesn't tell you much

1234
01:33:40,922 --> 01:33:44,403
If you measure the x component of this spin and the y component of that spin

1235
01:33:44,839 --> 01:33:45,794
That it doesn't tell you much, 

1236
01:33:46,133 --> 01:33:49,496
but it does tell you if you measure the same component of either of them

1237
01:33:49,762 --> 01:33:50,749
They would come out opposite 

1238
01:33:52,993 --> 01:33:54,998
>> Bell (inaudible)

1239
01:33:58,455 --> 01:34:02,164
>> dealing with these systems of two spin oriented in opposite direction 

1240
01:34:02,797 --> 01:34:05,729
>> isn't it will be a up down kind of state 

1241
01:34:06,189 --> 01:34:07,290
>> Is that right or is that so?

1242
01:34:07,725 --> 01:34:12,142
No the kind of state Bell dealt with was 

1243
01:34:12,387 --> 01:34:14,051
entangled state |up down> minus |down up>

1244
01:34:14,753 --> 01:34:15,741
It's exactly what he dealt with 

1245
01:34:20,270 --> 01:34:23,809
>>Is there any interaction with two of the state like this?

1246
01:34:24,560 --> 01:34:28,020
This, almost any interaction between the spin that we do a state like that 

1247
01:34:30,529 --> 01:34:35,988
>> Did you say that it's a statistical correlation?

1248
01:34:36,637 --> 01:34:38,481
Yes it's just the statistical correlation that 

1249
01:34:39,721 --> 01:34:50,349
Look, at this, to some extend, it's similar to a trivial observation 

1250
01:34:51,484 --> 01:34:56,142
The way I like to phrase it is I have a penny and a dime in my pocket 

1251
01:34:58,274 --> 01:35:00,784
I put my hand into my pocket I take them out I don't look at them

1252
01:35:01,044 --> 01:35:03,435
Without looking at them I give one to Alice and one to Bob

1253
01:35:04,618 --> 01:35:06,418
And then I take Alice and Bob to take off and 

1254
01:35:07,722 --> 01:35:12,632
They will go to moon or whatever you want to go

1255
01:35:13,708 --> 01:35:17,523
And it's gonna be the case that if Alice looks at her coin

1256
01:35:18,040 --> 01:35:20,488
She instantly knows what Bob's coin is alright?

1257
01:35:21,836 --> 01:35:23,513
She sees, if I got a dime he's got a penny

1258
01:35:24,472 --> 01:35:25,979
If I got a penny he's got a dime 

1259
01:35:28,248 --> 01:35:32,112
That in itself is not surprising, there is nothing

1260
01:35:33,170 --> 01:35:36,988
In the past they work together some correlation is established 

1261
01:35:37,307 --> 01:35:38,254
Now what was the correlation?

1262
01:35:38,706 --> 01:35:44,810
The correlation was a correlation that if Alice has a dime 

1263
01:35:45,585 --> 01:35:50,142
Then Bob has a penny and vice versa

1264
01:35:51,319 --> 01:35:53,020
No other possibilities were possible 

1265
01:35:55,861 --> 01:35:59,948
So there is a correlation between Alice's coin and Bob's coin

1266
01:36:00,167 --> 01:36:01,421
And there wasn't any anti-correlation 

1267
01:36:02,681 --> 01:36:06,164
If we call heads and tails plus one and minus one

1268
01:36:07,322 --> 01:36:10,260
Then if Alice is plus one Bob is minus one 

1269
01:36:10,706 --> 01:36:13,278
But the opposite is also possible, I don't know it advents 

1270
01:36:15,089 --> 01:36:18,327
And yet when Alice's far away and Bob is far awy

1271
01:36:18,624 --> 01:36:21,909
Alice look at the coin, and she instantly knows what Bob's is

1272
01:36:22,961 --> 01:36:26,407
I assure you that does not violate the special theory of relativity

1273
01:36:30,049 --> 01:36:33,119
And this kind of correlation is a, 

1274
01:36:35,331 --> 01:36:39,435
this is called the Einstein-Podolsky-Rosen correlation, or entanglement 

1275
01:36:41,804 --> 01:36:44,615
It's the quantum version of that same nickel and dime story 

1276
01:36:45,111 --> 01:36:46,183
Penny and dime story

1277
01:36:47,622 --> 01:36:49,280
>> You use nickel for real properties 

1278
01:36:49,544 --> 01:36:50,918
There are certainly a difference 

1279
01:36:51,390 --> 01:36:52,452
There are certainly a difference

1282
01:37:01,657 --> 01:37:06,141
That's just statistical correlation, it's tremendously common things in physics

1280
01:36:52,773 --> 01:36:56,631
But just the idea that by looking at something at one place

1281
01:36:56,898 --> 01:37:01,005
You instantly know something you didn't know with regard to other place

1283
01:37:06,407 --> 01:37:07,441
There's nothing special about it

1284
01:37:08,384 --> 01:37:13,962
The question is what is it that special and different about this entanglement?

1285
01:37:14,871 --> 01:37:17,317
And that's where John Bell found something interesting 

1286
01:37:17,818 --> 01:37:19,670
He didn't find non-locality, 

1287
01:37:20,045 --> 01:37:22,475
we are gonna talk next time about what non-locality means in physics

1288
01:37:22,746 --> 01:37:26,338
And what is it exactly that John Bell found?

1289
01:37:29,258 --> 01:37:32,732
But I'm gonna tell you right now it was not non-locality

1290
01:37:33,757 --> 01:37:37,382
It looks like non-locality and the reason is 

1291
01:37:37,746 --> 01:37:40,422
because you are trying to apply classical reasoning 

1292
01:37:40,888 --> 01:37:46,248
What John Bell said is you can't apply classical reasoning to quantum systems

1293
01:37:46,633 --> 01:37:47,791
That's what it came down to 

1294
01:37:48,130 --> 01:37:50,184
Quantum systems are quantum mechanical that 

1295
01:37:50,405 --> 01:37:52,977
if you try to apply classical reasoning

1296
01:37:53,950 --> 01:37:55,855
You are going to find inconsistency 

1297
01:37:57,451 --> 01:37:59,717
I can say in another way, I will tell you another way 

1298
01:38:00,154 --> 01:38:01,428
There is a way to say it

1299
01:38:07,540 --> 01:38:12,439
Imagine you have a computer and you try to simulate on that computer 

1300
01:38:13,379 --> 01:38:15,333
The quantum mechanics of a single spin 

1301
01:38:16,291 --> 01:38:17,747
Here's the way that you simulated 

1302
01:38:18,752 --> 01:38:24,707
You would be able to instruct the computer to orient it's apparatus 

1303
01:38:25,845 --> 01:38:27,067
in an arbitrary direction, 

1304
01:38:27,381 --> 01:38:29,590
in other words you would put in coordinates for a direction

1305
01:38:31,243 --> 01:38:33,274
And when you put in the coordinates for a direction 

1306
01:38:34,128 --> 01:38:39,248
That computer would do some calculation appeal to random number generator

1307
01:38:40,324 --> 01:38:42,564
And spit out a plus one or a minus one

1308
01:38:44,436 --> 01:38:48,540
There will be an algorithm that would say that 

1309
01:38:48,835 --> 01:38:51,143
if you do it over again immediately

1310
01:38:52,552 --> 01:38:54,435
It will spit out the same number again

1311
01:38:55,861 --> 01:38:58,310
That if you do it again, it will spit out the same number again 

1312
01:38:59,038 --> 01:39:03,483
But you might then say, alright now let me make it more interesting 

1313
01:39:04,291 --> 01:39:08,827
By adding in a mathematical magnetic field 

1314
01:39:09,329 --> 01:39:10,871
Which allows the spin to process 

1315
01:39:11,536 --> 01:39:15,685
What would you do? The computer would solve the Schrodinger equation 

1316
01:39:16,624 --> 01:39:19,479
Find out what the state was after a little bit of time 

1317
01:39:20,413 --> 01:39:22,896
Calculate the probability for up and down

1318
01:39:23,977 --> 01:39:25,780
Or whatever you decide to measure 

1319
01:39:26,444 --> 01:39:30,109
Calculate the probabilities appeal to the random number generator 

1320
01:39:30,712 --> 01:39:35,435
And spit out a number that was statistically exactly what you should get 

1321
01:39:35,746 --> 01:39:37,483
on the average for the quantum mechanics 

1322
01:39:38,400 --> 01:39:41,368
Somebody who came to this computer and started playing with it

1323
01:39:41,618 --> 01:39:45,104
And doing this experiment, rotating this apparatus in different directions

1324
01:39:46,198 --> 01:39:49,428
Turing on magnetic fields on other directions 

1325
01:39:49,732 --> 01:39:51,337
Waiting various amount of time 

1326
01:39:52,482 --> 01:39:56,535
And then telling the computer measure this or that component 

1327
01:39:57,418 --> 01:40:00,057
would reproduce exactly quantum mechanics

1328
01:40:00,300 --> 01:40:02,932
At least to the extent that random number generator are possible

1329
01:40:03,811 --> 01:40:04,795
They are not exactly possible 

1330
01:40:05,236 --> 01:40:07,130
but you know you can make pretty good random numbers

1331
01:40:07,683 --> 01:40:12,443
Alright so that would be a quantum, that wouldn't be a quantum computer

1332
01:40:12,836 --> 01:40:18,231
That would be a classical computer simulating quantum behavior 

1333
01:40:19,543 --> 01:40:23,895
Whereas somebody doing experiment on it can be fooled into thinking

1334
01:40:24,366 --> 01:40:27,545
that it's actually dealing with the single electron inside the computer

1335
01:40:28,025 --> 01:40:30,549
And doing a quantum experiment, you couldn't tell the difference 

1336
01:40:31,455 --> 01:40:36,047
Now the question is could you do the same thing for two spins?

1337
01:40:37,706 --> 01:40:40,899
In particular I want to do the same thing for two spins

1338
01:40:41,413 --> 01:40:44,658
And I want to do in such a way that one of the spins 

1339
01:40:45,712 --> 01:40:48,595
If all of it's data is located in one computer 

1340
01:40:50,107 --> 01:40:54,292
And all of the data for the other spin is located in the other computer

1341
01:40:54,534 --> 01:40:56,198
Now I can bring the computers together 

1342
01:40:57,168 --> 01:41:00,005
Have wires connecting them and so forth for a while

1343
01:41:00,943 --> 01:41:04,634
Do arbitrary things and then cut the wires and separate them 

1344
01:41:06,387 --> 01:41:12,308
Can I simulate the quantum mechanics of the two particle system 

1345
01:41:13,730 --> 01:41:17,641
In such a way that it would appear that one of the computer contains one particle

1346
01:41:18,187 --> 01:41:20,394
And the other computer contains the other particle

1347
01:41:20,877 --> 01:41:24,340
Now this is a classical computer, two classical computers 

1348
01:41:25,261 --> 01:41:27,438
And the answer is the only way you could do it 

1349
01:41:27,868 --> 01:41:31,996
would be to have wires connecting the computers at all times

1350
01:41:33,024 --> 01:41:35,832
You could not do it unless the computers 

1351
01:41:36,564 --> 01:41:41,431
are capable of interacting directly out of distance

1352
01:41:42,514 --> 01:41:46,609
But that does not mean that the system is non-local 

1353
01:41:47,677 --> 01:41:51,940
Non-local means the signal can propagate from one place to another 

1354
01:41:52,262 --> 01:41:57,206
What it's telling you is you simply can't simulate quantum mechanics 

1355
01:41:58,212 --> 01:42:00,280
in a local way by a classical computer 

1356
01:42:01,045 --> 01:42:03,129
The logic of quantum mechanics is different 

1357
01:42:03,821 --> 01:42:09,919
It's simply does not allow itself to be fake by a computer 

1358
01:42:10,862 --> 01:42:13,354
unless that computer is connected 

1359
01:42:14,127 --> 01:42:16,641
over large distances to everything in the world

1360
01:42:17,829 --> 01:42:20,109
Quantum mechanics managed to do the same thing 

1361
01:42:20,755 --> 01:42:22,565
without having wires connecting things

1362
01:42:23,839 --> 01:42:27,673
However you could ask the question does that constitute what physicists call

1363
01:42:27,910 --> 01:42:32,146
non-locality, or what Einstein would have called non-locality

1364
01:42:32,670 --> 01:42:35,844
It does not, it does not constitute in any way the ability 

1365
01:42:36,105 --> 01:42:37,678
to send a message from one place to another 

1366
01:42:38,980 --> 01:42:40,960
But we are gonna come to that, we will deal with that 

1367
01:42:43,235 --> 01:42:47,955
No it is fundamentally the inability to simulate quantum mechanics 

1368
01:42:48,222 --> 01:42:49,369
by classical computers, 

1369
01:42:49,693 --> 01:42:53,621
unless the classical computers are hook up in highly un-local ways

1370
01:42:55,081 --> 01:42:58,542
For more, please visit us at Stanford.edu

